How to Design an Experiment and Most Common AP Physics 1 Labs

The following steps outline a general procedure for designing an experiment.

a) Make a MATHEMATICAL MODEL of the experimental situation that relates what you want to determine

(Z), to other variables. The model is obtained by applying physics to the experimental situation to derive an

expected equation or model that contains the unknown variable, Z. That model or equation guides your

experimental design; in other words it tells you what needs to be measured to find Z. [Note that in order to

model the experimental situation, simplifying assumptions usually need to be made (such as neglecting air

resistance or friction) and these assumptions are a source of experimental error).]

Example1: you want to design an

experiment to determine the kinetic

coefficient of friction,k, between a

block and the table. An equation

that contains k is given by

fk = kFN.

This is a model.

Example2: You need to determine the acceleration due to gravity, g, given

a car on a ramp, a meterstick and stopwatch. To design the experiment, you

need to make a model of the car on the ramp using physics. By applying

Newton’s 2nd Law to the car traveling down the ramp, one gets the equation:

acar = gsin

This is a model of the experimental situation. It involves an assumption –

that the force of friction between the car and ramp is negligible. (Note:

Another model could be obtained using conservation of energy: gh = ½ vB2)

b) Use the model to SELECT TWO VARIABLES TO MEASURE:

Example1: Given the model

fk = kFN.

you could experimentally determine k by

measuring the 2 variables:

fk, the force of friction and

FN, the normal force on the block

Example2: Given the model of the experimental situation,

acar = gsin

you could experimentally determine g by measuring the 2

variables:

acar, the acceleration of the car and

, the angle of the incline.

c) EXPERIMENTAL DESIGN: To design the experiment, decide which measured variable is independent

and which is dependent. If the variable cannot be measured directly with accuracy, measure something else

that can be used to determine that variable. For each variable, describe what and how it’s measured. Control

and document other variables. If a measurement has a lot of uncertainty (variability), repeat and average a

few identical trials. Measure at least 6 independent data points (6 different pairs of indep, dep variables).

Example1: Indep var: FN

Dep var: fk

Procedure (what and how):

FN: FN = mg if pull block along horizontal surface.

fk: cant measure directly, but if pull block with

spring scale at constant speed then Fpull = fk.

Constants and controls – same block and table used

throughout, same surface area of block in contact

with table throughout.

1. Measure weight of block (=FN). Using a spring

scale, drag block at a constant speed along the

table so that the reading on the spring scale is

equal to fk.

2. Add mass on top of the block and repeat step 1 to

measure fk for at least 6 different values of FN.

Example2: Indep.Var:

Dep.Var: acar

Procedure (what and how):

: Cant measure with a meterstick, but can measure h and

d along incline and use those to determine (sin = h/d).

acar: cant measure directly with meterstick or stopwatch,

but can measure distance car travels along incline, x,

(starting from rest) and the time it takes, t (with a

stopwatch), and use those to determine acar (x= ½at2).

Constants and controls – same car and ramp used

throughout (car and ramp have minimal friction)

1. Car is released from rest at top of incline. Acceleration

of car, acar is determined for at least 6 different ramp

angles, (as described above).

d) ANALYSIS - Plot and fit the dataset: Use your model to guide the analysis of the data. Make a graph of

the data (with dependent variable on y axis and independent variable on x-axis). Using your model which

gives you the expected relationship between the variables you measured, fit the data to an appropriate

function. Compare the bestfit equation (experimental results) to the model in order to determine the

unknown quantity or to support what you expected.

Example1:

Model or expected: fk = kFN.

Indep var: FN

Dep var: fk

Plot fk vs FN (y vs x)

Fit with a line (because expected relationship is linear)

Compare experimental bestfit to expected model:

slope of bestfit equation (FN/fk) represents uk

y-intercept of bestfit represents error.

Example2:

Model or expected: acar = gsin

Indep var:

Dep var: acar

Plot acar vs sin (y vs x)

Fit with a line (because the expected relationship is linear)

Compare experimental bestfit to expected model:

slope of bestfit equation (a/sin) represents g

y-intercept of bestfit represents systematic error due to

the unaccounted for friction between the car and track

(expect that when =0, a = 0. If a=0 at some nonzero ,

then there is an unaccounted for force in the model and

that was friction)

e) ERROR ANALYSIS: Find %error of experimentally determined value by comparing it with an accepted

value. Cite and discuss sources of systematic errors that would cause the %error. Systematic errors come

from assumptions that were made in applying the model to the experimental situation and from systematic

experimental errors.

Example1: Compare the experimentally

determined k to an accepted value if available to

determine %error. Most likely systematic errors

that caused the %error:

- Notable assumption made in this experiment was

that Fpull = fk. However, if the object is not

moving at constant speed, then Fpull > fk and that

leads to overestimate of fk and k

- Possible experimental systematic errors – Fpull not

parallel to surface and so not just balancing fk

Example2:

Compare the experimentally determined g to 9.81m/s2 to

determine %error. Most likely systematic errors that caused

the %error:

- Notable assumption made in this experiment was that there

is no friction between the car and the track. In reality, the

friction would cause a=0 at a small angle. Unaccounted for

friction would lead to an underestimate in g because the

slope that we assumed to be g based on the model would be

<g if there were friction (a and were measured, there

were no model-based assumptions in those values).

Common Experiments in AP Physics 1 (most you did, some were demonstrated)

See if you can use steps a)-e) above to design experiments for each of the labs below. On the following pages,

a design is outlined for each lab that follows steps a)-e). At the end is a question with an AP-style experiment

for you to design.

1. Determine the velocity and acceleration of a uniformly accelerating object

2. Determine the launch speed of a projectile shot from a launcher

3. Experimentally determine acceleration due to gravity, g

a) Free fall expt

b) Using an Atwood machine

c) using a Car on incline

d) Using Simple harmonic motion of a pendulum

4. Verify Newton’s 2nd Law using a modified Atwood setup

5. Experimentally determine coefficients of friction

6. Experimentally determine the speed of an object moving in UCM

7. Determine the acceleration of an elevator

8. Conservation of Momentum – NEED TO PUT THIS IN (also what happens to E)

9. Experimentally determine a spring constant (and/or determine whether a rubber band behaves like a

Hookean spring)

10. Experimentally determine the speed of a bullet using a ballistic pendulum

11. Determine the mass of a meterstick (or an unknown mass) using static equilibrium (given a known mass and

a ruler)

12. Experimentally determine the moment of inertia of an object

13. Determine the mass of a penny using the Atwood machine

14. Determine the speed of sound using a column (O/C or O/O), a tuning fork and a meterstick.

15. Determine the resistance of a circuit element (and show whether it is ohmic or nonohmic)

Experiment 1: Determine the velocity and acceleration of a uniformly accelerating object

Determining Velocity

a) MODEL: we expect vav=x/t

b) Select variables: Guided by the model, vav could be determined by measuring x (position) and t (time)

from a starting position and time.

c) Experimental Design: Collect at least 6 x-t points by

measuring position with a meterstick and time with a stopwatch

OR by measuring x and t with a motion sensor

OR by measuring v directly with a photogate

OR by measuring x and t with a video that can be analyzed frame by frame

d) Analysis: Plot x vs t (t is always on horizontal axis whether it is the independent variable or not)

Compare experimental graph to expected model: slope between two points, x/t is vav between those

points.

If v is constant, then the data is fit to a line and the bestfit slope is the experimentally determined

velocity.

If v is not constant, the x-t graph will not have a constant slope, will be curved. vav can then be

determined between each pair of successive points to plot a v-t curve.

e) Error Analysis: Compare to a known speed to find %error.

Determining acceleration of a uniformly accelerating object

There are a couple common ways to determine acceleration

Measure velocity

a) MODEL: we expect that a=aav=v/t

(or v=v0+at)

b) Select Variables: Guided by the model,

a could be determined by measuring v

and t.

c) Experimental Design: Collect at least 6

v-t points: Measure v and t with a

motion sensor

OR measure v and t by recording a

video that can be analyzed frame by

frame

d) Analysis: Plot v vs t (t is always on

horizontal axis whether it is the

independent variable or not).

Fit to a line (from model expect a linear

relation betw v and t).

Compare model and bestfit:

Slope of the line, v/t represents

experimentally determined a.

y-intercept represents initial velocity, v0.

Measure position

a) Model: For uniform acceleration, we expect that

x=x0+v0t+½at2.

b) Select variables: Guided by model, a could be determined

by measuring x and t.

c) Experimental Design: Collect at least 6 x-t points:

Measure position with a meterstick and time with a

stopwatch

OR by measure x and t with a motion sensor

OR by measure x and t with a video that can be analyzed

frame by frame

d) Analysis: Plot x vs t (t is always on horizontal axis

whether it is the independent variable or not).

Fit plot to a quadratic function (because expected x-t is

parabolic).

Compare the experimentally determined best fit equation,

x=A+Bt+Ct2, to the expected model (x=x0+v0t+½at2) to

determine what the bestfit coefficients, A, B, and C,

represent. From the comparison, x0, v0 and a can be

determined.

(Alternatively, if x0 and v0 are 0, could plot a linearized

graph of x vs t2. Slope of the x-t2 graph can be used to

determine a (slope of x-t2 graph is ½a))

e) Error Analysis: Compare experimentally determined a to a known acceleration to find %error. Systematic

errors that cause %error in come from assumptions made and from systematic experimental errors:

- One notable assumption made in the model-guided analysis is that acceleration is uniform.

Experiment 2: Determine the launch speed of a projectile shot from a launcher (using just a meterstick)

a) Model: Easiest to measure launch speed from a horizontally launched projectile because the launch

velocity only has one component (vx). Expected x=vxt so could measure x, (the range) and t (flight

time). However, if t is small, it’s difficult to measure t without lots of uncertainty. Instead could measure

y because it is also related to t: y= ½gt2. Combining the two, x=vx(√(2y/g))

Assumptions in the model: used kinematics assuming that the projectile is only under influence of gravity

(air resistance and was neglected)

b) Select Variables: Guided by model, launch speed, vx, could be determined from the horizontally launched

projectile by measuring x (range) and y (height).

c) Experimental design: Horizontally launch projectile and measure x (range) and y (height) with a

meterstick. Repeat a few times taking the average x (because it would have significant variability or

uncertainty).

d) Analysis: Use the model to guide the analysis – in other words, plug the data (xav and y) into expected

equation to determine an average launch speed, vx.

e) Error Analysis: Compare to an accepted speed (found by directly measuring the speed with a photogate

attached to the launcher) to find %error. Systematic errors that cause %error in launch speed come from

assumptions made in the model and from systematic experimental errors:

- One notable assumption made in the analysis is that the projectile is only under the influence of gravity

ay = g and ax = 0). In reality, there is air resistance (ax<0 and ay<g) and not accounting for it causes the

determined launch speed to be systematically overestimated (larger than accepted).

Experiment 3: Experimentally determine acceleration due to gravity, g

There are lots of ways to measure g

1. Determine g in a free fall experiment (see Experiment 1: determining acceleration of uniformly

accelerating object)

2. Determine g using an Atwood machine

a) Model of the Atwood Machine: Apply Newtons laws to the Atwood machine to find expected Atwood

acceleration Fnet syst/Msys = aCM =(m2-m1)g/(m1+m2). Assumptions in the model – the pulley is

massless (no rotational inertia)

b) Select variables: Guided by the model, g could be determined by measuring a and the masses, m1 and m2.

c) Experimental design:

m1 and m2 : measure with a balance

a: cant measure directly but because the forces and acceleration are constant, can measure distance a

block falls, y, (starting from rest) and the time it takes, t (with a stopwatch), and use those to determine

a (y= ½at2 assuming no air resistance). (See measuring acceleration in Experiment 1 for other ways

to measure a)

Repeat several times to get aav (or measure a for at least 6 different m1,m2 sets).

d) Analysis: Use the model to guide the analysis – in other words, plug the data (aav, m1, m2) into the

expected equation to determine an average value of g. (or if a was measured for several m1,m2 values,

plot a vs (m2-m1)/(m1+m2) and fit to a line. Slope of line is g)

e) Error Analysis: Compare experimentally determine g to 9.81m/s2 to find %error. Systematic errors that

cause %error in g come from assumptions made and from systematic experimental errors

- A notable assumption made in the model is that the pulley is massless (no rotational inertia). In

reality, it has mass and rotational inertia and not accounting for it causes the system mass to be

systematically too small. Using the model, g was found as the slope of the graph:

g=a(m1+m2)/(m2-m1). a and m’s were measured not calculated with the model so the model-

based assumption does not affect those values. The system mass is really bigger than m1+m2 and

so the assumption of a massless pulley leads to a slope that is an underestimate of g (smaller than

accepted value).

- Minor error is in the assumption of no air resistance in calculating a. That would lead to a small,

minor overestimate in a and g because the air resistance was minimal.

3. Determine g using a car accelerating down an incline (See Example 2 in the first section)

4. Determine g using simple harmonic motion of a pendulum

a) Model of the pendulum swinging back and forth: Assuming the pendulum acts like a simple harmonic

oscillator, expected period: T = 2√(L/g). Assumptions in the model – no energy is lost as the

pendulum swings (ideal massless string, no air resistance) and the angle of swing is small.

b) Select variables: Guided by the model, g could be determined by measuring T, period, and L, length of

pendulum.

c) Experimental design:

L (Indep.Var): measure with a meterstick

T (Dep.Var): measure with a stopwatch (because T is short and subject to lots of uncertainty, minimize

error by measuring 5T to determine T)

Collect at least 6 T-L pairs of data. Measure T for at least 6 different L

Controls and constants: mass of pendulum bob, starting amp kept constant.

d) Analysis: Plot T vs L (y vs x) and fit to a power function (because it is expected that T depends on √L).

Compare the experimentally determined best fit equation, T=AL0.5, to the model (T = 2√(L/g)) to

determine g.

Alternatively, could plot a linearized graph of T vs √L or T2 vs L. Slope of the linearized graph can be

used to determine g.

e) Error analysis: Compare experimentally determined g to 9.81m/s2 to find %error. Systematic errors that

cause %error in g come from assumptions made and from systematic experimental errors:

- One notable assumption made in the analysis is that the pendulum exhibits SHM.

Experiment 4: Verify Newton’s 2nd Law (a ~ Fnet and a ~ 1/m)

a) Model: Newtons 2nd Law: a = Fnet/M

Experimental setup: modified Atwood machine - system of car (mcar)+onboard mass (m) on smooth track

attached to a hanging mass, mh; total mass of system = M. Applying Newtons 2nd Law to the experimental

situation: Fnet = mhg (if friction between car and track is neglected)

b) Select variables: It is expected that a is directly proportional to Fnet (when M is constant) and that a is

inversely proportional to mass (when Fnet is constant). Measure acceleration of an object or system in

response to Fnet and mass.

c) Experimental design:

Part 1: Measure a vs Fnet (keeping M constant)

Fnet (Indep.Var): Fnet is provided by weight of

hanging mass (=mhg). Measure mh with balance.

a (Dep,Var): measure v-t with motion sensor; slope of

v-t is a (also see Experiment 1 for other ways to

determine a)

For system of car+onboard mass on smooth track

attached to a hanging mass, mh, measure mh, mcar

and m (sum is M). Total M must remain constant

Collect at least 6 a-Fnet pairs of data by measuring a

for at least 6 different values of Fnet (different

amounts of hanging mass). In all 6 measurements,

keep total system mass, M, constant by moving

mass from car to hanging mass so that mass of

accelerating system is constant.

Part2: Measure a vs M (keeping Fnet constant)

M (Indep.Var): Measure total accelerating mass

(mcar+m+mh) with balance.

a (Dep,Var): measure v-t with motion sensor;

slope of v-t is a (also see Experiment 1 for

other ways to determine a)

For system of car with onboard mass on smooth

track attached to a hanging mass, mh, measure

mh, mcar and m (sum is M). mh (mhg=Fnet) must

remain constant

Collect at least 6 a-M pairs of data by measuring

a for at least 6 different values of M (different

amounts of onboard mass). In all 6

measurements, keep Fnet constant by keeping the

hanging mass constant.

d) Analysis:

Part 1: Measure a vs Fnet (keeping M constant)

Plot a vs Fnet (y vs x)

Fit to a line (because it is expected that a is directly

proportional to Fnet).

Compare the experimental bestfit to expected model (a =

Fnet/M):

Slope of bestfit (a/Fnet ) represents 1/M, the inverse of

the system mass

y-intercept of bestfit represents error. Expect that a=0 at

Fnet=0. But that is assuming that the hanging mass

were indeed equal to Fnet (we did not make any

assumptions about a, that was measured). However

because the friction between the car and track was

neglected, the graph does not show a 0 y-intercept (see

graph below). Rather a = 0 when there is a small +Fnet

which shows that the hanging mass is balanced by

another neglected force – friction between car and track.

Part2: Measure a vs M (keeping Fnet

constant)

Plot a vs M

Fit to power function (because it is expected

that a is inversely proportional to M when

Fnet is constant).

Compare the experimentally determined

best fit equation (a=kM-1), to the model (a =

Fnet/M), Newtons 2nd. Comparison shows

that the bestfit coefficient represents the

experimentally determined Fnet exerted by

the hanging mass. Alternatively, could plot a

linearized graph of a vs 1/M; slope of the

linearized graph can be used to determine the

constant Fnet.

See graphs below

e) Error analysis: Compare the experimentally determined system mass to the accepted and directly

measured mass to determine %error. Systematic errors that cause %error in mass come from

assumptions made and from experimental systematic errors:

- Notable assumption made in this experiment was that there is no friction between car and track and

that the hanging mass was the net force. In reality, there is friction and not accounting for it causes

Fnet to be systematically too large which causes the experimentally determined total mass, M to be

overestimated (acceleration was directly measured, no assumptions made)

- Possible experimental systematic errors – track was not horizontal which would lead to a

measurement of a that was systematically too large or too small.

Experiment 5: Experimentally determine coefficients of friction

1. Coefficient of kinetic friction (See Example 1 in the first section) (there are many ways to determine k

based on applying models to various experimental situations. For example, see if you could design an expt

to determine the coefficient of kinetic friction between a block and an incline when block is sliding down

using either Newtons laws or energy to model the situation)

a

FnetNeglected friction

a

Msys

a

1/Msys

OR

2. Coefficient of static friction: Below are a couple common ways to s

Pulling Method

a) Model: It is expected that fs ≤ SFN

b) Select variables: Guided by the model,

could determine S, by measuring max fS

and FN.

c) Experimental Design

FN (Indep var): FN = mg if pull block along

horizontal surface

max fs (Dep var): cant measure directly, but

if pull block with spring scale while it

remains at rest, then Fpull = fk

Measure weight of block (=FN). Using a

spring scale or force meter, Apply a

gradually increasing horizontal force to an

object AT REST until it just starts to move.

Measure the max Fpull (= fS) to remain at

rest.

Measure fS max for same object with at least 6

different added masses (6 different FN).

d) Analysis: same as Example1 in 1st

section.

Angle of Repose Method – determine s (max value) of

block at rest on an incline (using just a meterstick)

a) Model: By applying Newtons Law to the experimental

situation (all forces balanced, a=0), one gets the model

s = tan ( is incline angle). (The angle of repose, ,

is defined as the incline angle at which an object just

starts to slide down an inclined plane; in other words, it

is the incline angle where Fgx = fs max.)

b) Select variables: Guided by the model, s could be

determined by measuring , the angle at which the block

just starts to slide (angle of repose).

c) Experimental design: Place block at rest on an incline

and slowly increase the incline angle until the object just

begins to slide. Measure that angle of repose, , with a

protractor OR determine by measuring the height of

the block and its distance along the incline to the ground

(sin=h/d). Repeat several times and average

d) Analysis: Use the model to guide the analysis – in other

words, plug the data (av) into expected equation to

experimentally determine s.

e) Error analysis: Compare the experimentally determined coefficient to an accepted value if available to

determine %error

Experiment 6: Experimentally determine the speed of an object moving in UCM

For a conical pendulum like the flying pigs, there are two ways to determine the speed of the pig:

Method 1

a) Model: By applying Newtons 2nd Law to the conical

pendulum, one can get an expression for the speed:

v = √(grtan) where is the angle the string makes with the

vertical. Assumptions made in the model: pig flies in UCM

which is a pretty good approximation

b) Select variables: Guided by the model, to determine v, we

can measure the r and of the flying pig.

c) Experimental design:

r: it’s very difficult to accurately measure r because the middle

of the circle needed to measure r is not visible or tangible.

Instead, measure d, the diameter of the circle (r=d/2).

: it’s very difficult to directly measure because the vertical

line needed to measure is not tangible. Instead, measure L,

length of pendulum and d, the diameter of the circular flight

(sin=r/L).

Method 2

a) Model: For UCM, sav=v =d/t = (2r)/T

b) Select variables: Guided by the model, v

could be determined by measuring r and

T of the flying pig.

c) Experimental design:

r: it’s very difficult to accurately

measure r because the middle of the

circle needed to measure r is not visible

or tangible. Instead, measure d, the

diameter of the circle (r=d/2).

T: measure T with a stopwatch.

Repeat measurements with high

variability several times and take

averages

d) Analysis: Use the model to guide

Repeat measurements with high variability several times and

take averages.

d) Analysis: Use the model to guide the analysis –plug the data

(rav and av) into expected equation to experimentally

determine v.

analysis – plug the data (rav and av) into

expected equation to experimentally

determine v.

e) Error analysis: Compare the two experimentally determined v’s to determine %difference Systematic

errors that cause %difference come from assumptions made and from experimental systematic errors

Experiment 7: Determine the max acceleration of an elevator

a) Model: Apply Newtons 2nd Law to an object of mass m in an elevator; it is expected that

FN = mg ± ma where a is the acceleration of the elevator (it is + if a is positive, - if a is negative).

b) Select Variables: Guided by the model, to determine a, can measure FN and m.

c) Experimental design:

m: measure using a balance

FN: Measure using a bathroom scale, force plate, or measure m with a balance (FN=mg)

Measure m of an object while at rest. Take same object and measure max/min FN as elevator accelerates.

d) Analysis: Use the model to guide analysis – plug the data (m and FN) into expected equation to

experimentally determine a.

e) Errors Analysis: Compare experimentally determined elevator acceleration to an accepted value to find

%error.

Experiment 8: Experimentally determine the spring constant of a spring (or a rubber band).

There are two ways to determine the spring constant experimentally

Static Method

a) Model: For an ideal (Hookean) spring, it is expected

that |FS| = k|x|.

b) Select variables: The spring constant, k, could be

determined by measuring FS and x.

c) Experimental design:

x (Indep.Var): stretch spring horizontally with a

spring scale or vertically with a hanging weight.

Measure x from equilibrium with a meterstick.

Fs (Dep.Var)

i. Horizontal spring – using a force meter or spring

scale, apply a pulling force to stretch the spring to

x and hold at rest so that FS = Fpull.

ii. Vertical spring – using hanging mass, apply vertical

force to stretch the spring x so that FS = mg

Collect at least 6 FS-x pairs of data

d) Analysis: Plot FS vs x and fit to a line (because it is

expected that FS is directly proportional to x).

Compare the experimental bestfit to model (|FS| =

k|x|)

Dynamic Method (mass oscillating on a spring)

a) Model: An object of mass m on an ideal spring

will oscillate in SHM when pulled from

equilibrium and released. It is expected that

the period of oscillation is TS = 2√(m/k).

Assumptions made with this model – there is

no energy lost as the mass oscillates back and

forth and the spring obeys Hooke’s Law

b) Select variables: Guided by the model, k

could be determined by measuring TS and m.

c) Experimental design: For at least six different

masses m (indep variable, measured with a

balance) measure S (dependent variable,

measured with a stopwatch).

d) Analysis: Plot T vs m and fit to a power

function (because it is expected that T depends

on √m). Compare the experimentally

determined best fit equation, T=Am0.5, to the

expected equation (TS = 2√(m/k)), one can

determine what the coefficient of the best fit

Slope of the line, FS/x represents k, the spring

constant.

NOTE: the FS-x plot is NOT linear for elastic

materials like rubber bands. A rubber band is not

“Hookean”; therefore, the elastic force it exerts is not

a restoring force.

equation represents and use that to determine k.

Alternatively, could plot a linearized graph of

T vs √m or T2 vs m. Slope of the linearized

graph can be used to determine k.

e) Error analysis: Compare experimentally determined k to accepted value if available to find %error.

Systematic errors that cause %error come from assumptions made and from experimental systematic errors

Experiment 9: Experimentally determine the speed of a bullet/ball using a ballistic pendulum

a) Model: In order to derive a model or expected equation for the speed of a launched ball, v0, the appropriate

physics needs to be applied to the experimental situation. Conservation of momentum (of the

ball/pendulum system) can be applied during the collision and conservation of energy (of the

ball/pendulum/Earth system) can be applied during the swing of the pendulum. Energy (of b/p/E)is NOT

conserved during the collision (it’s perfectly inelastic) and momentum (of b/p) is NOT conserved during

the swing (as the bullet/pendulum swing up, they are losing momentum to the earth because the pendulum

is fixed in place). By separating the process into parts (collision and swing) and using Cop and CoE

appropriately, you can derive an expected equation for the speed of the ball before collision in terms of the

max height the pendulum rises to: v0 = ((mball + Mpend)/mball)(2gh) where h is the max height pendulum

swings to after collision (you should be able to derive this).

b) Select Variables: Guided by the model, v0 could be determined by measuring the masses of the ball and

pendulum and the max vertical height of the pendulum after collision h.

c) Experimental design: .

mball and mpend : measure with a balance.

h: It’s very difficult to accurately measure h directly because it is not a tangible height and you would have

to do a very crude estimation. Instead, measure L, length of pendulum with a meterstick (Note: for the

ballistic pendulum with significant mass in the shaft, L is the distance to the CM and not to the bob) and

measure max, the max angle at max height (with an attached rotary sensor). From L and max, h can be

determined with trig (h=L-Lcosmax). Repeat max measurement several times to determine hav.

One way to measure h directly - take a video and analyze frame by frame. Repeat h measurement

d) Analysis: Use the model to guide analysis – plug the data (mball, mpend and hav) into expected equation to

experimentally determine v0 an average bullet/ball speed.

e) Errors Analysis: Compare to an accepted speed (found by directly measuring the speed with a photogate

attached to the launcher) to find %error.

Experiment 10: Determine the mass of a meterstick (given a known mass and a meterstick for

measurement)

a) Model: One can balance the meterstick (mass Mms) and a known mass, m, on a pivot (edge of table) so that

it is in static equilibrium. Applying Newtons 2nd Law to the rotational equilibrium, it is expected that the

sum of the torques must be zero: (Mmsg)rms+(mg)r = 0 where r is measured from pivot (r=0) and could be

positive or negative depending on the direction of the torque.

b) Select Variables: Using the model and given m, Mms could be determined by measuring the distances rms

(distance of meterstick CM to the pivot) and r (distance of m to pivot),

c) Experimental design: A meterstick and a known mass m placed on it at some position r, are placed on a

pivot (or the edge of a table) so that the system of meterstick and known mass is in static equilibrium.

Measure r values with a meterstick. Be careful to include whether r is + or – depending on the direction of

torque around the pivot. Setup the meterstick and mass system in several different equilibrium positions and

measure the r values.

d) Analysis: Use the model to guide the analysis . Shown are 2 ways to analyze

1. Plug the data (m, r and rms) into expected equation to experimentally determine Mms for each set of data.

Use average value of Mms.

2. Model: mr =Mmsrms. Plot r vs rms and fit to a line (because it is expected from the model that these

variables are linearly related) Compare experimental bestfit to model (r = (Mms/m)rms) in order to

determine what the slope and intercept of bestfit represent and to determine Mms. Comparison shows

that slope represents Mms/m. Intercept should be 0 (if it is very close to 0, it represents random

experimental error; if it is a significant value, it represents some systematic error)

e) Errors Analysis: Compare experimentally determined mass to the known value to find %error.

Extension experiment: Given a meterstick, a known mass and an UNKNOWN mass, determine the mass of

the meterstick (as above) and the UNKNOWN mass. See if you can design the experiment using steps a)-e) and

clearly writeup your experimental design.

Experiment 11: Experimentally determine the moment of inertia of a rotating object or system of

rotating objects.

There are two ways to determine the rotational inertia of an object

Experimental, Dynamic Method

a) Model: We can apply Newtons 2nd to a rotating system of objects:

net = Isys

b) Select variables: Guided by the model, the rotational inertia, Isys,

could be determined by measuring net and of a rotating object or

system of objects.

c) Experimental design: Apply various net (indep var) to an object

free to rotate by hanging various masses, m (measured with

balance), on a string wrapped around the axle at a distance rp from

rotation axis. Determine the resulting of the object (dependent

variable). (draw a diagram of the setup with all variables labelled).

Collect at least 6 pairs of net - data points.

(Dep.Var): cant measure directly. Instead, use an attached rotary

sensor to measure -t as object rotates (the slope of -t graph is

)

net (Indep.Var): difficult to measure directly because it consists of

the torques produced by the string tension and the torques due to

the friction of the rotary axle, pulley and other parts of the setup.

If friction of the axle and various pulleys that bend the tension

are neglected (an assumption that introduces error), then

net = T = rpT and this can be determined by measuring rp (radius

of axle string is wrapped around) and from the acceleration of

the falling mass: T = mg-ma = mg-m(rp). Then net can be

determined by measuring m, the falling mass (with a balance), rp

Theoretical Method

a) Model: The rotational inertia of a

system of objects rotating around

an axis can be calculated if the

rotational inertias of the

components of the system are

known theoretically. For example,

the rotational inertia of a system

containing a disk (ID= ½MDRD2)

and a ring (IR = ½MR(Ri2+ Ro

2) is

the sum of the individual I’s:

Isys = ID + IR

b) Select variables: Guided by the

model, Isys could be determined by

measuring the masses of each

object in the system (MD and MR)

and the relevant distances from the

rotation axis (R values)

c) Experimental design: measure the

masses with a balance and the R’s

with a meterstick.

d) Use the model to guide analysis –

plug the data (Ms and Rs) into

expected equation to determine the

net

Neglected torque due to friction

(radius of pulley string is wrapped around and (as described

above)

d) Analysis: Plot net vs (y vs x) and fit to a line (because it is

expected that net is directly proportional to ).

Compare the bestfit equation to the model (net = Isys).

Slope of the line, net/a is Isys, the rotational inertia of the rotating

system.

Y-intercept: The graph should show a direct relationship. A

significant non-zero intercept is caused by a systematic error. The

value of net was based on an assumption that net=T. However

because the axle and pulley frictions were neglected, the graph

does not show a 0 y-intercept (see graph below). Rather = 0

when there is a small +net which shows that the torque produced

by the tension is balanced by another neglected torque– friction of

the axle- that’s what the y-intercept represents.

theoretical Isys.

e) Error Analysis: There are not

many errors in this method, just

small random measurement errors.

Systematic errors would also be

very small if the disc and ring had

small scratches and nicks that

changed their true I a bit.

e) Error analysis: Compare experimentally determined I to the theoretically determined I (close to accepted)

to find %error. Systematic errors that cause %error come from assumptions made and from experimental

systematic errors

- Notable assumption made in this experiment was that there is no axel

friction torque In reality, there is friction and not accounting for it

causes net to be systematically too large which causes I to be larger

than the accepted value ( was directly measured, no assumptions

made)

- Another assumption was that the experimentally determined I was just

the I of the system of objects; it was assumed that other rotating parts

in the setup like the pulleys, added no I to the rotating system.

Therefore the experimentally determined I includes those unaccounted for I’s and is an overestimate.

Experiment 13: Determine the speed of sound using a tuning fork, a meterstick and a variable length

Closed-Open tube

a) Model: The experimental setup (shown at right) typically uses a tuning

fork that produces sound at one given frequency (instead of tuning

fork, can use a tone generator of constant frequency). A tube is closed

at one end by placing it in water; the length of the tube can be changed

by putting more/less of the tube in the water. When the tuning fork is

placed over the tube closed at one end, the sound will get loudest at the

length that produces resonance or a standing wave. Apply physics to

the experimental situation to make a model: The shortest tube length

that produces resonance is the fundamental and its length is L = ¼1.

The expected speed of sound in the tube is v = f = 4Lf = v.

b) Select Variables: Guided by the model, knowing the frequency of the tuning fork, v could be determined by

measuring L when there is resonance at the fundamental.

c) Experimental design: Use a tuning fork or tone generator of known frequency and increase length of tube

from 0 by raising it out of water until you hear resonance at 1st harmonic. Measure resonant length with a

V

I

nonohmic

ohmic

meterstick. Repeat several times and take average L.

More accurate way to do this experiment is to measure the lengths at 2 successive harmonics:

expected L = /2 = v/2f

d) Analysis: Use the model to guide analysis – plug the data (f and Lavr) into expected equation to determine

an average value of v, the speed of sound in air.

e) Error Analysis: Compare experimentally determined speed of sound to the known value (343m/s) to find

%error. One of the systematic errors in this experiment that was not accounted for in the model is that the

resonant wave extends out of the tube a bit and the amount it extends over depends on the diameter of the

tube. Neglecting this effect results in an underestimate of L and therefore a v that is less than the accepted

value. (by finding the resonance at 2 successive harmonics and L as described above, this error is

subtracted out)

Experiment 14: Determine the resistance of a circuit element (and show whether it is ohmic or

nonohmic)

a) Model: Each circuit element obeys Ohms Law V = IR

b) Select variables: Guided by the model, R could be determined by measuring V across and I through a

circuit element,

c) Experimental design: Connect a battery or voltage source across a circuit element (resistor or light bulb).

V: Measure, the voltage across the bulb or resistor with a voltmeter connected in parallel to the element.

I: Measure, the current through the bulb or resistor with an ammeter connected in series to the element.

Change the voltage across the circuit (add batteries or change voltage) to collect at least 6 V-I pairs of data

d) Analysis: Plot V vs I.

Compare the experimental graph to the model (V = IR)

Slope of the curve represents R, the resistance of the circuit element

If the V-I plot is linear and the slope is constant, then the circuit element

is “ohmic”; in other words, the resistance of the device is constant for a

range of currents and voltages (Resistors are ohmic for a range of

currents and voltages).

If the V-I graph is not linear, then the slope (R) is not constant and the

circuit element is “nonohmic”; in nonohmic devices such as lightbulbs,

the resistance usually increases with current through the device.

e) Error analysis: Compare experimentally determined R to an accepted value to find %error.

EXPERIMENT FOR YOU TO DESIGN: (the AP test has one experimental design free response question)

A student makes the hypothesis that the %energy lost when a ball bounces off the ground increases with the

height it is dropped from.

Using steps a)-e) design an experiment to test the student’s hypothesis.

1. Make a model of the experiment with a clearly labeled diagram

2. Describe the procedure

- what variables will be measured and how will each be measured

- which variable is indep, which is dep

- Give a clear and concise step by step procedure.

3. Explain how the data will be analyzed:

- What will be plotted?

- How will the graph be analyzed to address the hypothesis?

- What will the graph look like if the data supports the students hypothesis?

4. What are the main sources of systematic error in your experiment that would cause %error?