iit physics lab manual (combined)

Experiment 1: Introductory Experiments

Density of Matter

The density of a material is an intrinsic quantity given by ρ = M/V , where M is the mass of thematerial and V is the volume of the material. In this exercise you will use vernier calipers and the balancesto measure the density of various sets of materials. Each material set includes four pieces of varioussizes. Start with one of the aluminum cylinders and measure (using the vernier caliper and balance) thedimensions and the mass of the cylinder. Note, that each of the measurements must be repeated at least3 times and by each of the students in your group. Repeat this whole procedure for all of the aluminumcylinders. Choose one other set of materials and repeat this experiment. The materials in the set includethe following: maple, polypropylene, nylon, acrylic, polyurethane, phenolic, pvc, teflon, and aluminum.

Obtain an average density for aluminum from your data (remember to properly take into accounterrors) and create a graph of mass vs. volume. Find the tabulated value for the density of aluminum andcompare to your results. Explain any difference. Repeat this for the other materials you measured. Writea brief description of the material you used. For example, describe its color, translucence, opacity, andtexture.

Hooke’s Law

This experiment is intentionally very basic in order to allow the student time to become acquaintedwith using a computer to write a laboratory report and to familiarize them with the Data Studio software.Pretend you are living in the 1600’s. Robert Hooke has just proposed his empirical relation for springsknown as Hooke’s Law,

F = −kx (1)

Very simply, this means the extension or compression of a spring, x, is linearly proportional to themagnitude of the force (F) that is exerted upon it. The negative sign indicates that the direction of therestoring force F is always opposite to the direction of the extension/compression x and the proportionalityconstant is k (the spring constant). You are a scientist at another university and are excitedly trying toreproduce his results. Quite frankly, you think the relation is too simple and you want to prove Hookewrong so your name can be indelibly inscribed in every general physics book of the 21st century.

In your laboratory, you have springs, different weights, rulers, and a force sensor. You know enoughabout gravity to conclude that the force exerted downward by a mass is directly proportional to the mass.To measure this force, we will use the “force sensor”. The force sensor can measure the pulling or pushingforce exerted on it. To set it up, connect the force sensor to the Science Workshop 750 interface and startup the Data Studio software on the computer. Choose the correct sensor from the sensor menu (ask yourTA about using this software in your experiment). Under the display settings in data studio you can choose“digits” to view to force value or “graph” to plot the force as a function of time. Click “start” to collectdata. Note: before collecting any data, press the “zero”/tare button to set the sensor to zero.

With these materials devise an experiment that will allow you to test Hooke’s Law and find the springconstant (k). Show your proposed procedure to the Lab Instructor and explain the data you plan toacquire. Include in your procedure the exact details of how you will analyze your data to test Hooke’sLaw. In your lab report, include a graph of Force versus ∆x. What does the slope of this graph represent?

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There are several questions that you should answer in your lab write-up:

1. Does Hooke’s Law depend on the spring being used?

2. Does Hooke’s Law hold for very large forces?

Tips To Write Laboratory Reports

The last part of this laboratory exercise will actually be done on your own. Since everyone is required totype their own laboratory reports, you will need to develop basic proficiency with: a word processor with anequation editor and a spreadsheet with graphing and at least linear regression capabilities. Openoffice.orghas free programs which meet these criteria and they can be used on any operating system platform. If youprefer to use other programs of these types, that is also acceptable. The Teaching Assistant will explainthe proper lab-report format. You can find a sample lab report here: http://agni.phys.iit.edu/~bcps/labs/resources/sample.pdf.

There are a few things that you must know to be able to efficiently write a polished laboratory report:

1. You should learn how to properly include (typeset) mathematical equations and Greek symbols.

2. You will be repeatedly required to fit your data with a best-fit straight line. You must know how todo this, including extracting the values of the slope, the y-intercept, and the ”goodness-of-fit” value(R2). R2 ranges from 0 to 1, where 0 is a bad fit and 1 is a perfect fit. The R2 value gives you anidea of how well the straight line represents your data points. R2 is defined as:

R2 =

∑(yfiti − y)2∑(yfiti − yi)2

(2)

where yi are the experimental data, yfiti are the fit values and y =∑yi/N is the mean of the

experimental data. Note that this is not the same as drawing a line graph, which simply connectsyour data points with straight lines.

3. Find out how repeated calculations can be done faster using a spreadsheet. For example, if T = 12mv

2,and you have 12 different measured values of v, learn how to make all of the calculations automatically.

4. Be careful with the number of significant figures you present in your report. This includes the datatables you present as part of your raw data and analysis. Figure out how to limit the number ofsignificant figures or decimal places of the numbers you show.

5. Try including a graph whenever possible. Graphs are a great way of explaining your data and showingrelationships you have obtained in the experiment. For example, try making a graph of F vs x foryour Hooke’s Law experiment, and then interpret the slope.

6. Always check your sig-figs and units in data tables and graphs and label axes correctly on yourgraphs.

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http://agni.phys.iit.edu/~bcps/labs/resources/sample.pdf

http://agni.phys.iit.edu/~bcps/labs/resources/sample.pdf

Experiment 2: Projectile Motion

In this lab we will study two dimensional projectile motion of an object in free fall - that is, an objectthat is launched into the air and then moves under the influence of gravity alone. Examples of projectilesinclude rockets, baseballs, fireworks, and the steel balls that will be used in this lab. To describe projectilemotion, such as the trajectory (path), we will use a coordinate system where the y-axis is verticallyupward and the x-axis is horizontal and in the direction of the initial launch (or initial velocity). Tosimplify projectile motion, we assume that the gravitational acceleration (g=9.8 m/s2) is constant, suchthat ax = 0 and ay = −g, and we will ignore any air resistance. The equations of motion in the x and ydirections for a projectile launched with a velocity v0 at an θ are given as

vx = vx0 (1)

x = x0 + vx0t (2)

vy = vy0 − gt (3)

y = y0 + vy0t−1

2gt2 (4)

where t is the time, and g is taken to be positive. The components of the initial velocity v0 arevx0 = v0cos(θ) and vy0 = v0sin(θ).

Experimental Objectives

In this lab you are given a mini-launcher, a time of flight sensor (TOF) pad, steel balls, carbon paper,a plumb-bob and rulers. The mini-launcher is equipped with a photogate sensor that is connected tothe time of flight sensor: When a steel ball passes through the photogate sensor, it starts a timer in thePASCO/Data Studio system that records the time it takes for the ball to hit the TOF sensor pad. Thetotal time of flight can be displayed using the “Digits” option under the display section in the Data Studiosoftware. The mini-launcher can be cocked to 3 different position settings and can therefore release theball with three different initial velocities. The protractor on the side of the mini-launcher can give youany desired launch angle. The carbon paper will be used to mark where the projectile lands, giving youan accurate measurement of the total horizontal distance traveled. With this equipment, complete thefollowing experiments (please read the warning at the bottom before doing the experiment):

• Using the set-up shown in Figure 1, determine the initial velocities of the ball released for the first twofiring settings, that is, the first two “clicks”. Measure the distance between the base of the platformto the TOF sensor accurately by using the plumb-bob to align the sensor with the trajectory. Inorder to minimize random errors, it is very important that all of your measurements be performedseveral times.

• Devise an experiment to measure the relationship between the range (horizontal distance of theprojectile motion) and the launch angle. Experimentally show which angle gives you the longestrange. What is the range and total time for the different velocity settings when θ = 0◦?

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θ

d

ymax

TOF

H

Figure 1: A projectile is launched at an angle θ off a platform with height H above the around. Theprojectile lands the on time of flight sensor (TOF), a distance d away from the base of the platform.

• Devise an experiment to measure the acceleration due to gravity using the TOF sensor, assuming youdo not know the initial velocities of the launcher (Hint: try a free-fall experiment). Do enough trialsto reasonably compare your results with the accepted value of g = 9.80 m/s2. What are possiblesources of error for this experiment?

WARNING: THE PROJECTILE LAUNCHER CAN SHOOT STEEL BALLS AT HIGH VELOCITIES.IF A PROJECTILE WERE TO HIT YOU IN THE FACE IT COULD CAUSE PERMANENT DAM-AGE! ALWAYS WEAR SAFETY GOGGLES WHEN OPERATING THE LAUNCHER! NEVER FIRETHE LAUNCHER WHEN SOMEONE IS DIRECTLY IN FRONT OF THE LAUNCHER, NO MATTERHOW FAR AWAY THEY SEEM TO BE! NEVER EVER POINT THE LAUNCHER AT YOUR FACE!VIOLATING ANY OF THESE SAFETY RULES WILL RESULT IN LOST POINTS AND A POSSIBLEEXPULSION FROM THE LABORATORY. HORSEPLAY IS NOT TOLERATED AT ALL! REMEM-BER, IT IS ONLY FUNNY UNTIL SOMEONE LOSES AN EYE! AND PLEASE DO NOT STEP ONTHE TIME OF FLIGHT SENSOR PAD.

A full lab report is not necessary for this lab. Answer the questions on the following page and turn itin with your signed datasheet.

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PHYS 123, Lab 2 Questions

Name: CWID:

Write your answers on a separate sheet and attach your signed datasheet when turning it in. You mustshow all of your work for full credit. Make it clear to me you understand what you’re doing. Any graphsor tables should be made via computer software and attached to this handout.

1. Answer the following questions using the data you acquired in this experiment:

(a) What are the two initial velocities for the first two firing settings (the first two “clicks”)? Makea table consisting of the initial velocities, its components vx and vy, the launch angles, the timeof flight, and the horizontal range.

(b) Consider the angle that gave you the longest range. What is the maximum height (ymax) reachedat this angle? What is the overall maximum height reached in your experimental data? Whichangle gave you the maximum height?

(c) Make a graph of launch angle vs. horizontal range for the second experiment. Label the axesappropriately with correct units.

(d) For the third experiment, how do your measured values of the gravitational acceleration compareto the accepted value of g = 9.8 m/s2? What are possible sources of error for this experiment?

(e) If the steel ball is shot vertically upward, how long would it take for it to hit the floor below?Calculate for both initial velocities.

2. Ideally, what kind of mathematical curve is the projectile motion trajectory? Describe two examplesof projectile motion which you have observed or experienced outside of this physics lab that followthis mathematical curve.

3. Are there two different launch angles that would give you the same range? How about the sameheight? Explain.

4. If the steel ball is shot horizontally off the table, how much time would it take the ball to hit theground for each of the velocity settings of the launcher? Explain your answer using the equations ofmotion and your experimental data. How does this relate to the ball being dropped vertically fromthe table top to the floor below?

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Experiment 3: Newton’s 2nd Law

In 1608, Galileo Galilei wanted to investigate the motion of falling objects. However, the objects felltoo fast making it extremely difficult to measure their velocities and accelerations. His solution was to rollthe objects down an inclined plane, thereby controlling the effective velocities, accelerations, times andforces. By the use of a frictionless air-track, we can make an even simpler system than Galileo’s and use itto test Newton’s second law of motion, F=ma. Our setup, similar to Galileo’s, will have an object slidingdown a frictionless inclined plane, as shown in Figure 1:

Fnet

W

θ

Figure 1: An object with weight W slides down a frictionless incline plane at an angle θ with respect tothe horizontal direction.

The object we will use is a “glider” whose weight, W, can be varied by slotting on more mass. Thereare two forces acting on the glider: the weight pointing vertically downward and a normal force N pointingperpendicular to the inclined plane. The weight of the glider can be resolved into horizontal and verticalcomponents with respect to the surface of the incline, as shown by the red dashed lines of Figure 1. Sinceonly motion along the incline is permitted, the component of the weight along the incline, Fnet, is theeffective force applied to the glider. According Newton’s second law

a =Fnet

m(1)

where m is the mass of the glider. This is the relation you will be testing with your experiments. Bylooking at the free body diagram of the sliding mass (see Figure 2), we can write the x and y componentsof the acceleration as

ax =Fx

m=mgsinθ

m(2)

ay =Fy

m=N −mgcosθ

m(3)

1

Fnet

W=mg

θ

N

x

y

Figure 2: Freebody diagram for the sliding glider.

Thus the acceleration of the glider along the incline can be calculated by knowing the angle θ. Theacceleration can also be determined by measuring the instantaneous velocity experimentally and makinguse of the kinematic relation between acceleration and instantaneous velocity:

a =v2 − v2o

2x(4)

where x is the distance travelled, vo is the initial velocity and v is the final velocity. The “instantaneous”velocity of a glider is measured (via Data Studio) by timing how long it takes the glider, with a flag ofknown length, to pass through a photogate: v=L/t ,where “L” is the length of the glider’s flag and “t”is the time to pass through the photogate. The computer system and Data Studio program will directlycalculate the velocity of the glider as it passes through the photogate, provided that you insert the correctflag length, L, before you start the program.


In this laboratory you have an air track, a glider, a photogate, wooden blocks to change the angle ofincline, rulers, and different weights that can be placed onto the glider. The photogate is connected to acomputer data acquisition system and velocity data can be collected using the Data Studio software (seemanual or ask your TA about using this software in your experiment).

• Devise an experimental procedure to test Newton’s second law (Eq. 1) assertion that the accelerationis proportional to the effective force applied to the glider. Take a thorough set of measurements whichwill permit you to calculate the acceleration as a function of the effective force applied. Make surethat you have consistent data and report error bars in your data analysis. Include appropriate graphswhich summarize your results.

• Devise an experimental procedure to prove to yourselves that it is impossible to use a simple inclinedplane to test the assertion that the acceleration is inversely proportional to the mass. Why is thisso?

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In order to minimize random errors, it is very important that each of your measurements to be per-formed several times. Show error bars on your graphs. Make sure that you discuss all possible sources ofsystematic error in your experiments. In particular, think about and discuss whether the photogate systemis measuring the true instantaneous velocity. Make sure to record all of the different masses, lengths, andangles you used and the velocities you obtained from the photogate.

Questions

Answer these questions in your lab report. Show all work.

1. Calculate the accelerations using Equations 2 and 4 for the two experiments. Compare your results.

2. Derive an expression for the normal force. Do a sample calculation of the normal force for one of yourdata sets. What happens to the normal force if the incline angle is completely vertical (θ = 90◦)?How about when horizontal?

3. Select one of the velocities you obtained from the second experiment. If the glider goes up the inclinewith that initial velocity, how far up the plane will it go? How much time would it take for the gliderto return to the starting point?

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Experiment 4: Newton’s 2nd Law - Incline Plane and Pulley

In this lab we will further investigate Newton’s 2nd law of motion by using an incline-pulley system.The incline-pulley system, shown in Figure 1, can be classified as a simple machine, that is, one of theclassic elementary devices that more complicated and advanced machines are built around. As shown inFigure 1, the acceleration of the mass along the inclined plane (M1) can be controlled by using a hangingcounterweight (M2) over the pully and/or varying the angle of the incline. The free body diagrams for thetwo masses are shown in Figure 2. We will use the airtrack to create a frictionless plane and also assumethat the pulley is frictionless with uniform tension in the string. With these assumptions, the accelerationof the two masses are the same (a1,x = a2,y). Applying Newton’s second law,

∑F = ma, to the freebody

diagram, we can write a system of equations describing the motion of the two masses:

m1a = T −m1gsinθ (1)

m2a = m2g − T (2)

Solving these equations for the acceleration:

a =(M2 −M1sin(θ))g

M1 +M2(3)

θ

M1

M2

Figure 1: A mass M1 slides along a frictionless incline of angle θ with a counterweight M2 passing over apulley.

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W1 = m1g

θ

Nxy T

W2 = m2g

T

y

x

a2,ya1,x

Figure 2: Freebody diagram for the two masses.


The objective of the lab is to experimentally test the theoretical acceleration (Eq. 3) and to study therelationship between mass and acceleration using a simple machine. In this lab, M1 will be a glider cartwhose velocity can be measured using a photogate, similar to the previous lab. The acceleration of the cartcan be determined from the velocity by using kinematic equations of motion. Thus, you can obtain theacceleration of the cart using the Data Studio software and then compare it with theoretical calculationsof Equation 3.

• Devise an experimental procedure to test the theoretical acceleration (Equation 3) of the incline-pulley system. Remember to record all of your variables thoroughly and each of your measurementsshould be performed several times to minimize any errors. Compare your measurements with thetheoretical calculations.

• Devise an experiment that could verify the inverse proportionality of the acceleration and the mass.Think of how you could achieve a consistent (not necessarily constant) applied force, independent ofthe mass of the glider. You do not necessarily have to know the details of the force, just make sureit is the same for all experiments.


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Name: CWID:

Write your answers on a separate sheet and attach your signed datasheet when turning it in. You mustshow all of your work for full credit. Make it clear to me you understand what you’re doing. Any graphsor tables should be made via computer software and attached to this worksheet.


(a) For the first experiment, create a data table for the different masses (M1,M2), the inclineangles, the velocities and accelerations obtained from the computer software, and the theoreticalaccelerations using Equation 3 of the lab manual.

(b) How do your measurements compare with the theoretical calculations? What are the sources oferror?

(c) Make a graph of experimental acceleration (aexp) versus (M2 −M1sin(θ))/(M1 +M2). Explainthe slope.

(d) Make a plot of your data that shows the inverse proportionality for the second experiment.Briefly comment on the slope.

(e) Can friction truly be ignored in this experiment? Explain using your data.

2. Why is it important that the string connecting the masses be parallel with the air track?

3. In 1589 Galileo dropped two different masses from the Leaning Tower of Pisa and observed theirtime of flights to be independent of mass. How does Galileo’s freefall experiment relate to the secondexperiment if there was no applied force? How does the addition of an applied force change things?

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Experiment 5: Atwood’s Machine

In 1784, George Atwood created a device to calculate force and tension and to verify the laws of motionof objects under constant acceleration. His device, now known as an Atwood’s Machine, consisted of twomasses, m1 and m2, connected by a tight string that passes over a pulley, as seen in Figure 1. When themasses are equal, the pulley system is in equilibrium, i.e. balanced. When the masses are not equal, bothmasses will experience an acceleration (indicated with red arrows in Figure 1). For simplification, we willassume the ideal pulley scenario, where the mass of the string is negligible and we ignore any frictionaleffects acting on the pulley. With these assumptions, the accelerations a1 and a2 are equal.

m1m2

a2

a1

Figure 1: An Atwood Machine.

When m1 > m2, the sign convention used is such that m1 accelerates in the downwards y directionand m2 moves upwards. Here we let the acceleration be positive. On the other hand, if m1 < m2, we willset the acceleration as negative. With this convention, we can derive a system of equations describing theacceleration of each mass by applying Newton’s second law, F = ma, to each mass individually. Lookingat the free body diagrams for the masses of the Atwood machine (see Figure 2), we can write the forces as

m1a = m1g − T (1)

m2a = T −m2g (2)

where T is the tension in the string and g is the acceleration due to gravity (g = 9.8 m/s2).

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m1m2

a

a TT

W1 W2

Figure 2: Free body diagrams for the masses of the Atwood Machine. The tension T is shown in blue andthe weight of each mass W is in green. Note that the tensions are the same and the direction of motion isindicated by red arrows.

Solving our system of equations for the acceleration:

a =(m1 −m2)g

m1 + m2(3)

The numerator (m1 − m2)g is the net force causing the system to accelerate, and the denominator(m1 + m2) is the total mass being accelerated. Thus,

a =Fnet

Mtotal(4)


The object of this lab is to study Newton’s laws of motion and to measure the acceleration due togravity using the Atwood machine. In this lab you are given various masses, string and a “smart pulley”that allows you to calculate and graph the position, velocity and the acceleration of the masses of theAtwood’s machine. To measure the acceleration using the pulley, you must select the “smart pulley”sensor in Data Studio and then view the velocity vs. time graph of each run/trial. You will notice asteady linear increase of the velocity as the mass falls, followed by sharp drop as the mass comes to a rest.Highlight the appropriate region when the mass is free falling and then select “linear fit” from the fit tab.This will give you the slope of the line which is a direct measurement of the acceleration.

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• Devise an experimental procedure to measure the acceleration of the masses on the Atwood’s machinefor 5 different mass combinations of m1 and m2, but keeping the sum of the masses m1+m2 constant.That is, if you decrease m1 by a set amount, increase the other mass (m2) so that the total mass isalways the same. You should compare the measured accelerations with those calculated from Eq. 3.

Verify that a plot of F vs a will give you a straight line with a slope equal to the total mass (see Eq.4). Don’t forget to include error bars on your graph.

• Devise an experiment to measure the gravitational acceleration constant g. Hint: use the equationsof motion ( i.e. y = 1/2at2 + v0t + y0), measuring the acceleration (a) and time (t) via Data Studio.Do at least 6 different runs where one mass is held constant while the other mass increases by 10 gincrements.

Make a plot of a vs (m1 −m2)/(m1 + m2) in your lab report. Explain what the slope of this graphrepresents.

Questions

Answer these questions in your lab report. Show all work.

1. Calculate the tensions in the string for the second experiment.

2. What is the relationship between the acceleration and the total mass when the force is held constant?

3. How do these measurements compare with calculations using equation 3? What sources of errorare most likely the cause of the discrepancies between your experimental data and your theoreticalcalculations? How well does the “ideal pulley” scenario hold? Explain.

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Experiment 6: Friction

In previous labs we studied Newton’s laws in an “ideal” setting, that is, one where friction and airresistance were ignored. However, from our everyday experience with motion, we know that friction mustbe taken into account for a realistic description of practical situations - it is something we cannot ignore.Frictional forces act between two surfaces and oppose their relative motion. They occur because of surfaceirregularities, such as defects, and molecular forces (or bonds) between the materials. In this lab we willstudy frictional forces between various objects on different types of surfaces.

There are two types of friction: kinetic and static. Kinetic friction is the friction between surfaces inrelative motion. When sliding an object across another surface, microscopic bumps and defects tend toimpede and resist the motion (even the smoothest surfaces are rough on the microscopic scale). This is thetype of force that brings a rolling ball to rest or a coasting car to a stop. Experimentally, it is observedthat the force of kinetic friction is proportional to the normal force acting between the surfaces: if youincrease the normal force, the surfaces are crushed more together, increasing the contact area, and thusincreasing the frictional force. Mathematically we can write the force of kinetic friction as

Fk = µkFN (1)

where FN is the normal force between the two surfaces in contact with one another and µk is the coefficientof kinetic friction. The coefficient of kinetic friction is a dimensionless quantity (no units) that dependson the properties of the two surfaces. µk ranges from 0.01 for very smooth surfaces to 1.5 for very roughsurfaces. So, for example, if we want to push an object with constant speed on a very smooth horizontalsurface (such as ice), we must apply around 1% of its weight, whereas if we wanted to push the object onrough surface (dry concrete) we might need to push the object with greater force than its own weight.

Static friction describes the frictional forces between the surfaces of two objects that are at rest withrespect to each other. The static friction between the two surfaces is described by the coefficient of staticfriction µs. Experimentally, is it found that the maximum value for the static frictional force is proportionalto the normal force between the two surfaces. Thus the static frictional force Fs is

Fs ≤ µsFN (2)

Since the objects are at rest with one another, more molecular bonds are able to form making theobject harder to move and so greater force is needed to start motion when compared to the kinetic frictioncase. Therefore µs is generally greater than µk. Graphically, this is shown in Figure 1: As you increasethe force, the static friction force increases linearly until the applied force F equals µsFN . After this pointthe object “breaks away” and the friction force falls to the kinetic friction value.

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nomotion sliding

fr

F

fr = μsFN

0

kineticstatic

Figure 1: Force of friction (fr) as a function of an external force F applied to an object that is initially atrest.


The purpose of this lab is to construct a relationship between frictional forces and the normal force onan object, to calculate the kinetic and static coefficients of friction for various objects and surfaces and toultimately gain a solid understanding of static vs kinetic friction.

In this lab you are given a pulley sensor that can measure acceleration, a force sensor, string, frictioncarts with different surfaces (cork, felt, and plastic), and different surfaces (sheet of aluminum, constructionpaper, and the table top) to drag the carts on.

1: Coefficient of Kinetic Friction

To study and calculate various coefficients of kinetic friction, we will use a pulley system as shown inFigure 2(a). The pulley (a “smart pulley”) is equipped with a sensor that allows you to measure and graphthe velocity of the masses as a function of time via the Data Studio software. With the velocity graph youcan obtain the acceleration of the mass system by finding the slope of the appropriate linear fit, similarto the Atwood lab (lab 5). Looking at the free body diagrams of our system (Figure 2(b)), we can writeNewton’s second law for each mass as

m1a = T − fr (3)

m2a = m2g − T (4)

Here we have assumed that the accelerations of the two masses are same by neglecting any frictional effectson the pulley making the tension in the string uniform. The kinetic frictional force fr is given by

fr = µkFN = µkm1g (5)

2

m2a

m1

(a)

m2

aT

W2

m1T

W1

fr

FN

a

(b)

Figure 2: (a) Pulley system used to calculate uk. (b) Free body diagrams for the pulley setup. Wi is theweight of the object and fr is the frictional force.

The system of equations (Eqs. 3-4) can be solved for the acceleration in terms of the masses (m1,m2),g, and µk

a =(m2 − µkm1)g

m1 +m2(6)

• Devise an experiment to calculate µk for various surfaces, making use of the smart pulley system, thefriction carts, and the different surfaces. Take measurements with all 3 carts (felt, plastic, and cork)on one of the surfaces. Use up to 3 different masses for each cart. Remember to do several trials foreach run to obtain consistent data.

2: Coefficient of Static Friction

To measure the static coefficient of friction µs we will use the force sensor. The force sensor recordsthe pulling (or pushing) in Newtons via the Data Studio software. Connect the force sensor to one of thefriction carts using string. With no force on the sensor press, the zero (tare) button before taking anymeasurements (this should only be performed once). Open the “graph” under the display section in DataStudio.

• With the force sensor setup and attached to the cart, start to slowly and carefully pull on the carton of the surfaces while monitoring the force value with the graph. You want to record the minimumforce needed for the friction cart to break away and start moving. Your graph should look similarto Figure 1 (note: depending on the force sensor setup, your graph might be upside down!). Haveevery member of the lab group try this. Repeat this procedure for all the friction carts (felt, plastic,cork) using at least 5 different masses for each cart.

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3: Coefficient of Kinetic Friction by Force Sensor

We can check our µk values obtained from the first experiment by making use of the force sensor.

• Devise and experiment to measure the coefficient of kinetic friction using the force sensor. Recordthe force, normal force and obtain µk graphically. How do your values compare with those fromexperiment 1? Repeat this experiment for the various carts with 3 different masses. Hint: This partmay be difficult at first, but draw a free body diagram of the cart and force sensor and then convinceyourself why you want the cart to move with a constant velocity (look at the velocity graph as aguide).


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Name: CWID:



(a) For the first experiment, create a data table for the different masses (M1,M2), the acceleration,and the calculated coefficient of friction µk. Remember to label the cart types (felt, cork, plastic)in your table and describe the surface.

(b) Do your measured values of µk make sense? Compare them with sample coefficients of friction(for various materials) found in your textbook.

(c) For the second experiment, what is the force that you are measuring? Create a plot of thismeasured force vs the normal force of the friction cart. Find its slope and explain what itrepresents.

(d) For the third experiment, make a data table consisting of the cart masses, any applied force,and the normal force. Using your data, create a graph that represents the coefficient of friction.

(e) How does your coefficient of friction from the third experiment compare with the one youobtained from the first experiment? What are the sources of error?

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Experiment 7: Conservation of Energy

One of the most important and useful concepts in mechanics is that of “Conservation of Energy”. Inthis experiment, you will make measurements to demonstrate the conservation of mechanical energy andits transformation between kinetic energy and potential energy.

The Total Mechanical Energy, E, of a system is defined as the sum of the Kinetic Energy, K, and thePotential Energy, U :

E = K + U. (1)

If the system is isolated, with only conservative forces acting on its parts, the total mechanical energy ofthe system is a constant. The mechanical energy can, however, be transformed between its kinetic andpotential forms. cannot be destroyed. Rather, the energy is transmitted from one form to another. Anychange in the kinetic energy will cause a corresponding change in the potential energy, and vice versa. Theconservation of energy then dictates that,

∆K + ∆U = 0 (2)

where ∆K is the change in the kinetic energy, and ∆U is the change in potential energy.Potential energy is a form of stored energy and is a consequence of the work done by a force. Examples

of forces which have an associated potential energy are the gravitational and the electromagnetic fieldsand, in mechanics, a spring. In a sense potential energy is a storage system for energy. For a body movingunder the influence of a force F , the change in potential energy is given by

∆U = −∫ f

i

−→F ·−→ds (3)

where i and f represent the initial and final positions of the body, respectively. Hence, from Equation 2we have what is commonly referred to as the work-kinetic energy theorem:

∆K =

∫ f

i

−→F ·−→ds (4)

Consider a body of mass m, being accelerated by a compressed spring. As you verified in the firstexperiment of the semester, the force exerted by a spring is given by Hooke’s Law, F = -kx. Thus, fromEquation 2, the change in potential energy as a spring is stretched or compressed is:

∆U = −∫ xf

xi

(−kx)dx =1

2k(x2f − x2i ) (5)

If we let the initial position xi = 0 (the spring’s equilibrium position and where its potential energy isdefined to be zero) and set xf = x, Equation 5 becomes

U =1

2kx2 (6)

where we have now defined a potential energy function, U , for the spring. The change in kinetic energy ofa body (Equation 4) under the acceleration of a force F = ma is given by:

∆K =

∫ xf

xi

(ma)dx =

∫ xf

xi

(mdv

dt)dx =

∫ vf

vi

mdv =1

2m(v2f − v2i ) (7)

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If the initial velocity of the mass is zero, then the kinetic energy at any given time is

K =1

2mv2 (8)

where v is the instantaneous velocity of the body. Note that this derivation is for any general force F , andnot just that specific to the spring. You can do the same derivation for the gravitational potential energyof an object at a height y by noting the force is F=mg and thus obtaining

U = mgy (9)

where the potential energy is chosen to be zero at height y=0.


In the laboratory you have an air track, glider, photogate, a spring launcher, a scale, wooden blocks,ruler, and a set of weights that can be placed on the glider. The photogate is connected to a computerdata acquisition system and velocity data can be collected using the “Datastudio” software, similar to Lab3 (see manual or ask your TA about using this software in this experiment). Using this equipment:

• Devise an experimental procedure to observe how the potential energy of the spring launcher isconverted to kinetic energy in the motion of the glider for different masses of the glider and differentcompressions of the spring. Determine if friction can truly be neglected in this system. Remember torecord all of your variables thoroughly and each of your measurements should be performed severaltimes to minimize any errors.

• Devise an experimental procedure to observe the transfer of potential energy from the spring launcherto gravitational potential energy of the glider. Verify energy conservation for this case. Don’t forgetto measure the spring constant. Summarize your results in your lab report with detailed data tablesand create appropriate graphs. Make sure that you have consistent data and report error bars inyour data analysis.

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Experiment 8: Mechanical Energy

In this lab we will study and investigate the concepts of potential energy, kinetic energy, conservativeforces and the conservation of energy in more detail. Potential energy is energy that is associated with andattributed to the configuration, position and arrangement of particles within an object that experiences aconservative force F (x). A conservative force is a force that does zero work on a closed path. That is, thework done by conservative force in going from point A to point B is independent of the path from A to B.Examples of conservative forces include gravity and the spring force. When a conservative force does workW on an object, the change in the potential energy ∆U is

∆U = −W (1)

If an object moves from x0 to xf , the change in potential energy is

∆U = −∫ xf

x0

F (x)dx (2)

For a spring we can use Hooke’s law, F = −kx, to obtain its potential energy from Equation 1:

U =1

2kx2 (3)

where k is the spring constant and x is the displacement of the spring from its rest state. Gravitationalpotential energy of an object of mass m at a height y is treated in the same manner by noting the force isF = mg and thus resulting in

U = mgy (4)

Here the potential energy is chosen to be zero at height y=0.

Experimental Methods and Objectives

Part 1: In this experiment you are given a frictionless horizontal airtrack, a glider, a hanging-masswith additional masses to add on, a spring, string and a ruler. You can take measurements of the velocityand force actin on the spring using the photogate and force sensor, respectively. We will start with a glidercart connected by string to the hanging mass over a pulley, as shown in Figure 1a.

Considering vertical positions and speeds at two different times, we can write down the conservationof the energy for the cart and hanging mass system as

1

2m1v

2i +

1

2m2v

2i + m2gyi =

1

2m1v

2f +

1

2m2v

2f + m2gyf (5)

where m1 is the mass of the cart, m2 is the mass of the hanging weight, and vi and vf are their respectiveinitial and final velocities. After releasing the hanging mass, it will fall from the initial height of yi to yf .We will assume that the string connecting the hanging mass to the gliding cart is inextensible and hasuniform tension, that is, if the cart moves a distance ∆x then the hanging weight will fall the same amount∆y (∆x = −∆y). The velocity of the cart can be measured by the photogate sensor (does the hangingmass have the same velocity?). This velocity can be verified by using Equation 5.

• Devise an experimental procedure to test the Conservation of Energy theorem using the above setup(Figure 1). Show that the work done by the falling mass is independent of the path taken.

1

m2

cart

h

(a)

m2

FS cart

(b)

Figure 1: (a) Arrangement for Part 1. The hanging mass is initially a height h off the floor. (b) Set up forPart 2. Here a spring is attached to the glider. The force on the spring can be measured with the forcesensor (FS).

Part 2: A spring will now be added to the setup as shown in Figure 1b. Assuming the string to beinextensible, as before, the spring stretches the same length as the falling mass.

• Devise an experimental procedure to calculate the total mechanical energy of this spring-cart systemand show whether or not the energy is conserved. What does the addition of a spring do to theoverall system? Don’t forget to measure the spring constant.


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Name: CWID:

Write your answers on a separate sheet and attach your signed datasheet when turning it in. You mustshow all of your work for full credit. Make it clear to me you understand what you’re doing. Any graphsor tables should be made via computer software and attached to this worksheet.


(a) For the first experiment, explain why work done by the falling mass is independent of the pathtaken.

(b) What did the addition of a spring do to the overall system? Is this spring force a conservativeforce?

(c) Write down the energy equation relations for the second experiment.

(d) Where does the maximum kinetic energy in parts 1 and 2 occur?

(e) Can we truly ignore friction in this lab? Explain using your data.

(f) Calculate the ratio of the kinetic and potential energies for parts 1 and 2. What do these ratiostell you about the conservation of energy?

(g) What effect would the release point have on the final velocity?

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Experiment 9: Momentum

Physics is often concerned with what are called “conserved” quantities. Mass and energy are twoexamples of quantities that must remain conserved for a closed system. Conservation of a quantity is aclue to a physicist that there is some underlying principle to be discovered. Perhaps the oldest and mostfamous conservation principle is the conservation of momentum. This is embodied in Newton’s First Law,written in 1687. It states that an object in motion will remain in motion unless acted upon by a net force.Conservation of momentum will be studied through one dimensional collisions.

One Dimensional Collisions

The concept of momentum is fundamental to an understanding of the motion and dynamics of anobject. The momentum of an object is defined to be

−→p = m−→v (1)

For multiple objects in a system, the total momentum is a vector sum of the individual momenta. As aconsequence of Newton’s second law

−−→Fext =

d−→pdt

(2)

For a closed system, the total momentum cannot change unless acted upon by an outside force. Thisconservation of momentum is a powerful tool for physicists to analyze the behaviors of systems of particles.The simplest application of this concept is in the one-dimensional collision between two particles. Thereare two special kinds of collisions which are particularly easy to analyze: the perfectly elastic and perfectlyinelastic collisions. While both of these processes conserve momentum, in the perfectly elastic collision thetotal kinetic energy, KE, is also conserved. Examples of perfectly inelastic collisions include objects whichcollide and stick together and objects which break apart due to internal forces.

In the analysis of a perfectly elastic one-dimensional collision, consider two objects with masses m1

and m2 and initial velocities v1 and v2. After the collision, the objects will have new velocities v′1 and v′2,where all velocities are assumed to be in the positive direction. Conservation of momentum demands thatthe total momentum must be the same before and after the collision. This can be stated as:

m1v1 + m2v2 = m1v′1 + m2v

′2 (3)

Since the kinetic energy is also conserved in this kind of collision, we have:

1

2m1v

21 +

1

2m2v

22 =

1

2m1v

′21 +

1

2m2v

′22 (4)

If we know the masses and the initial velocities, it is possible to solve for the final velocities of the twoobjects. After a number of algebraic manipulations, the solutions are:

v′1 =m1 −m2

m1 + m2v1 +

2m2

m1 + m2v2 (5)

v′2 =2m1

m1 + m2v1 +

m2 −m1

m1 + m2v2. (6)

Again, all velocities are presumed to be along the positive direction. If a velocity is negative, it is thendirected along the negative direction.

1

The perfectly inelastic (“sticky”) collision is somewhat easier to analyze as only Equation 3 can beused. The energy equation used in the analysis depends on which case is being studied. If the collisionstarts with a single object which breaks into two, then we have v1 = v2. If there are initially two objectswhich end up stuck together, then we have v′1 = v′2. The analysis is then straightforward.

Most collisions are neither perfectly elastic nor perfectly inelastic but partially elastic. This means thata certain fraction of the kinetic energy is lost to the system but the objects do not stick together. In thiscase, it is valuable to define a quantity called the coefficient of restitution

e =v′1 − v′2v2 − v1

(7)

For a perfectly elastic collision, e = 1 and for a perfectly inelastic collision (starting with two bodiesand ending with one), e = 0. For most real collisions 0 < e < 1.


In the laboratory you have air track, two gliders, two photogates, a scale, and additional masses thatcan be placed on the gliders. The glider carts have velcro on the ends to create a “sticky” collision orrubber bands for an elastic collision. The photogates are connected to a computer data acquisition systemand velocity data can be collected using the Data Studio software (ask your TA about using this softwarein your experiment and setting up the two photogates). Remember, velocity is a vector, so you must assignit a direction - the Data Studio software cannot do this. Using this equipment:

• Devise an experimental procedure to observe and verify linear momentum and energy conservationlaws. Repeat your experiment for different masses and different velocities of the glider (includingzero initial velocity for one of the gliders). Do five different collisions and repeat each one to getconsistent data.

• Devise an experiment to study “perfectly” inelastic collisions. Verify whether or not momentum andenergy are conserved in this type of collision.


2


Name: CWID:



(a) For the two experiments, create data tables for the different cart masses (M1,M2), the initialcart velocities, the final velocities, the initial and final momentums and kinetic energies. Give abrief description of the collisions.

(b) Explain if momentum and kinetic energy are conserved in the collisions. What are the sourcesof error?

(c) Determine if the collisions are perfectly elastic or inelastic by using the concept of the coefficientof restitution.

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Experiment 10: Circular Motion

Previously we discussed two dimensional projectile motion, but another example of motion in twodimensions is that of circular motion. The simplest case is when the radius and velocity of the moving bodyare held constant. However, even though the speed is constant, the direction of the velocity is continuallychanging and the body is, therefore, always undergoing acceleration. Imagine a body, originally at somepoint on a circle designated by radius ro and angle θ, with a velocity, −→v , tangential to the circle (see Figure1). Its position may be written as:

−→r = rocosθi+ rosinθj (1)

θ

r

v

y

x

Figure 1: A particle moving at constant speed v in a circular path of radius r.

Since the particle is under uniform circular motion, the angle θ changes with time as:

θ = ωt (2)

where ω is the (constant) angular velocity. Thus, combining Equations 1 and 2, we obtain the timedependence of the body’s position

−→r = rocos(ωt)i+ rosin(ωt)j (3)

The velocity and the acceleration of the body are then obtained by applying Equation 3 with v = d~r/dtand a = d~v/dt:

−→v = −ωrosin(ωt)i+ ωrocos(ωt)j (4)

−→a = −ω2rocos(ωt)i− ω2rosin(ωt)j (5)

By looking at Equations 4 and 5, it is clear that the position, velocity and acceleration vectors all haveconstant magnitudes as a function of time (ro, ωro, ω

2ro, respectively). The direction of the acceleration isalways perpendicular to the direction of the velocity, which is always perpendicular to the direction of theposition vector. The acceleration always points towards the center of the circle.

1

Since the moving body is experiencing an acceleration towards the center of the circle, then there mustbe a net force in that direction to keep it under acceleration. By applying Newton’s second law (Note:The equations deal only with the magnitudes, so no vector notation is needed):

F = ma = mω2ro = mv2

ro(6)

This is the equation we will study in this lab.


The rotating platform used in this experiment is shown in Figure 2. There is a spring on the centralrotation axis of the system which provides the force required to keep the brass bob in uniform circularmotion. You can adjust the spring length and the position of the bob. The position of the bob duringrotation can be monitored with the orange marker/indicator as it aligns with the reference slot. Theplatform is connected to the computer interface through a rotational motion sensor. You will be able torecord the angular velocity of the rotation using the computer interface. Your TA will explain how tooperate the platform and collect data with the Data Studio software. Besides the platform, you will needa set of different hanging weights.

spring

bob

rotating platform

removable balancing mass

Figure 2: The rotating platform.

• Devise an experimental procedure to verify that the accelerating force is proportional to the squareof the angular velocity of rotation. Hint: Do not spin the rotating platform with the hanging massattached. Make a plot of F vs. ω2 and explain the slope in your lab report.

• Devise an experimental procedure to verify that the force that keeps the bob in uniform circularmotion is proportional to the radius of the rotation.

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Experiment 11: Torque

Up until now we have dealt with mostly translational motion in one dimension. Now we will focus onthe dynamics of rotational motion about a fixed axis. When a force ~F is applied to body about its axis arotation or pivot, a twisting or rotating force arises called torque. The magnitude of the torque, τ is givenas the cross product between the applied force and the “lever arm”, which is the length from the axis ofrotation to the point where force is applied. Thus if ~F is exerted on a point described by the positionvector ~r, the magnitude of the torque is

τ = ~r × ~F = rFsin(θ) (1)

where θ is the angle between ~r and ~F . For simplification, it is convenient to have and express ~r and ~Fperpendicular to one another. In terms of Newton’s second law, the net torque is given by

τ = Iα (2)

α is the angular acceleration about the axis due to the force, and I is moment of inertia. In the same wayas the mass for translational dynamics, the moment of inertia is intimately related to how hard it is tomake a system move with the application of a particular force (torque). The larger the moment of inertia,the harder it is to rotationally accelerate a system, the more energy is stored in a rotational motion andthe larger the angular momentum.

For a single point particle of mass m at a distance r from the axis of rotation, the moment of inertiais found to be I = mr2 from a simple analysis of kinetic energy. For a system of N particles, the momentof inertia is given by:

I =

N∑i

mir2i (3)

where ri, once again, is the shortest distance (perpendicular distance) to the axis of rotation. For acontinuous distribution of mass, this definition my be generalized to an integral over the entire massdistribution

I =

∫r2i dm (4)

The moment of inertia of common objects such as cylinders, spheres, etc. is usually given in tables (seeyour Textbook) with the axis of rotation passing through the center of mass. To calculate the moment ofinertia of the body along another parallel axis, the parallel axis theorem may be applied: I = ICM +md2,where d is the shortest distance (perpendicular distance) from the center of mass to the actual axis ofrotation.

1


In this laboratory you have a rotating platform with an attachment in the shape of a rail, large masseswhich can be attached to the rotating rail, various masses, string, a pulley and a computer with theScientific Workshop interface connected to the rotational measurement sensor that allows you to measurethe rotational acceleration via Data Studio. The string can be wound around 3 different spool settings(bobbins), each with a different radii. Using this equipment:

• Devise an experiment to measure the moment of inertia of the rail by varying the length of leverarm, keeping the applied force constant. Convince yourself what the “lever arm” is in this set-up.

• Devise an experimental procedure to verify Newton’s second law for rotational motion. First you willneed to verify that torque τ is proportional to angular acceleration α for fixed moment of inertia.Then you will need to verify that for a constant torque, the moment of inertia is inversely proportionalto angular acceleration. Make a graph for both of these cases and explain the slopes.

Questions

Answer these questions in your lab write-up. Show all your work.

1. Calculate the inertia of the rotational platform and of the masses which can be attached to the rotatingrail at various distances from the axis of rotation and compare it the experimental measurements.

2. Can friction truly be ignored in this experiment? Explain using your data.

2

Experiment 12: Angular Momentum

In this lab we will study the conservation of angular momentum by studying the moment of inertiasof various systems. The angular momentum, ~L, of a particle with a mass m and a linear velocity ~v withrespect to a fixed point, typically the origin, is given as

~L = ~r × ~p = m(~r × ~v) (1)

The ~r term is once again the position vector locating the particle relative to a fixed point. Thus, we canwrite the magnitude of the angular momentum as L = rmv sinθ, where θ is the angle between ~r and ~p.

The angular momentum component of a rigid body rotating about a fixed axis is given as

L = Iω (2)

where I is the moment of inertia and ω is the angular speed. Similar to translational motion, we can writeNewton’s second law in angular form, that is in terms of torque and angular momentum as

~τ =d~L

dt(3)

Consequently, if the net external torque acting on the system is zero, the angular momentum remainsconstant (~L = constant):

~Linitial = ~Lfinal (4)

This is the fundamental law of conservation of angular momentum that you will be studying in thislab. We can express this in term of the moment of inertia and angular speed as

Iiωi = Ifωf (5)

where the final moment of inertia can be obtained using the parallel axis theorem.


In this lab experiment you have a rotating disk attachment, a short cylindrical tube which fits into agroove of the rotating disk attachment, a mass hanger, various masses, string, a pulley and a computer withthe Scientific Workshop interface connected to the rotational measurement sensor. Using this equipment:

• Devise an experimental procedure to measure the moment of inertia of the disk attachment and theshort cylindrical tube. How do these measured values agree with the expected theoretical momentsof inertia as calculated using the parallel axis theorem? You can find the equations in your textbook.

• Use the measured moments of inertia to study the conservation of angular momentum in a rotationalcollision between the disk attachment and the cylindrical tube. You can set the initial angular speedby hand, and then, gently and carefully, drop the tube onto the disk.

1

Conceptual Test

This part of the experiment requires no calculations, yet it may be the most difficult part to discussin your write-up. Your skill in describing what you observe and giving a reasonable explanation of yourobservations using the physics involved will be tested here. CAUTION: SAFETY GLASSES MUST BEWORN FOR THIS PART OF THE EXPERIMENT.

• Set up the rotating platform to include the tilted metal ramp. Note: you must have the counter massattached to the other end to ensure that the rotating platform is balanced.

• Display the ω versus time graph on the computer screen.

• Place a single ball at the top end of the metal ramp.

• Give the ramp a small push so that the metal ball stays at the end of the ramp. THE RAMP DOESNOT HAVE TO BE SPUN VERY FAST TO ACHIEVE THIS! BE VERY CAREFUL!

• Start recording the ω versus time graph. As the angular velocity of the ramp decreases, carefullyobserve what happens to the ball and how this corresponds to what you observe on the ω versus tgraph.

• After the ball rolls down the ramp, press the STOP button to stop recording.

• Be sure to print out this graph so that you may refer to it as you are writing your report. (Hint: tohelp you with your description and explanation, it may be beneficial to properly annotate this graphso that you may refer to parts of the graph more easily.)

• Repeat the procedure described in the above paragraph, but instead of using one ball, place threeballs at the top of the ramp. The apparatus may need to spun a little faster, but not much faster. Itis important to be very cautious.

• In a clear and concise manner, describe your observations of the single ball experiment. Explain howthe conservation of angular momentum can account for the behavior of the graph obtained. In thesame manner, describe and explain your observation of the three-ball experiment.


2


Name: CWID:



(a) Using the parallel axis theorem, write down the equations for the moment of inertia of a shortdisk attachment and the short cylindrical tube. Calculate the moments of inertia using themeasured values of mass and radius.

(b) How well do the experimental values agree with the expected theoretical moments of inertiawhen calculated using the parallel axis theorem?

(c) How well is the angular momentum conversed in the second experiment? Is this collision elasticor inelastic? Explain using energy relations.

2. In a clear and concise manner, describe your observations of the single ball experiment. Explain howthe conservation of angular momentum can account for the behavior of the graph obtained. In thesame manner, describe and explain your observations of the three-ball experiment.

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iit physics lab manual (combined)

Documents

force sensor

force value

set of materials

sensor menu

correct sensor

data studio software

graph of mass

density of aluminum