L07 Rotational Motion and the Moment of Inertia 1

Pre-Lab Exercise Full Name: Lab Section: Hand this in at the beginning of the lab period. The grade for these exercises will be included in your lab grade this week.

In class we derived the moment of inertia about the center of mass of a homogeneous rod of mass M, length L and width W to be Icm rod = (1/12) M L2. Now, derive the expression for the moment of inertia about the center of mass of a homogeneous rectangular plate of mass M, length L and width W and show that Icm plate = (1/12) M [ L 2 + W 2 ]

As shown below, consider the plate as being made up of an infinite number of differential rods, each of mass dM, length L, and width dy, and each having a moment of inertia about their own center of mass given by dIcm = (1/12) dM L2.

By using the Parallel Axis Theorem, each differential rod has a moment of inertia about the axis P given by dIp = dIcm + dM h2 where h is equal to y.

Since we are dealing with a homogeneous plate, you can relate the mass dM to its area dA = L dy as the total mass M is related to the total Area (L W) and then “sum all dIp. Show all work here.

L07 Rotational Motion and the Moment of Inertia 2

Rotational Motion and the Moment of Inertia Full Name: Lab Partners’ Names: Lab Section:

Introduction: This experiment deals with some of the characteristics of rotational motion. You will be using the relations between linear variables (x, v, a, F) and angular variables (θ, ω, α ,τ) as well as the angular equivalent of Newton’s Second Law (τ = I α) throughout this experiment. In doing so, you will determine the moment of inertia of a plate and an aluminum disc both experimentally and by calculation. Materials Pasco Rotary Motion Sensor with pulley Aluminum Disc 50 g hanger . Rectangular plate Meter stick Balance Vernier Calipers Set of Masses Procedure: 1. Calculate the Moment of Inertia of the Plate and Disc at their center of mass points. Record all your information below and in the table on page 6.

1.1 Find and record the mass and radius of the aluminum disc. Calculate its moment of inertia.

Aluminum Disk:

Mass (kg)

Radius (cm)

I (kg*m^2)

1.2 Measure the mass, length and width of the aluminum plate. Calculate its moment of inertia. Record the ID number that is stamped on the plate.

Rectangular Plate

ID Number :

Mass (kg)

Length (cm)

Width (cm)

I (kg*m^2)

1.3 Find the radius of the large and small pulley by first finding the diameters

using a vernier caliper.

Small Pulley Large Pulley Radius (cm)

L07 Rotational Motion and the Moment of Inertia 3 2. Setting Up the Equipment:

2.1 The aluminum disc is free to rotate about an axis as shown below. It passes through a rotary motion sensor and is attached to a small shaft and a pulley. The sensor can measure the shaft’s rotation as a function of time. A light cord is wrapped around the pulley and is attached to a hanging hook weight. The hook weight which is held at rest and then is let go, moves downward with a constant acceleration.

2.2 Connect the Science Workshop interface to the computer, turn on the interface and then turn on the computer. This may have already been done for you.

2.3 Open the Data Studio software package by double clicking on the icon from the desktop menu. 2.4 In the Experiment Setup screen double- click the Rotary Motion Sensor picture. This will add it to the interface box diagram, showing the proper connections. Connect the leads as indicated. 2.5 Click on the Rotary Motion Sensor icon in the wiring diagram. Under the General tab, set the sample rate to 50 Hz (fast). Under the Measurement tab, select Angular Velocity (rad/s). Close the setup window.

2.6 Under the Displays Menu on the lower left, select an angular velocity vs time graph.

2.7 The rotary motion system (pulley and detector) should be assembled for you when you enter the lab. Thread the cord through the hole in the side of the pulley. At first, the cord should be wrapped on the larger pulley. Connect the other end to the mass hanger.

L07 Rotational Motion and the Moment of Inertia 4

3. Calculating the Moment of Inertia for the Systems:

3.1 Draw an extended free body diagram (EFBD) below for the disc-shaft-pulley and an FBD for the hook weight mass point MH.

3.2 From the free body diagrams above, write Newton’s 2nd Law for the pulley system (Eq. 1) and for the hook weight (Eq. 2).

3.3 Since the linear acceleration a of the hook weight is related to the angular

acceleration α of the pulley system by a= α /R, rewrite Eq. 2 as Eq. 3 with this substitution.

3.4 Combine your Eq. 1 and Eq. 3 by eliminating T, and solve for the moment of inertia, I of the system. Your equation for I should only be a function of the hanging mass MH, the radius of the pulley R, the angular acceleration α, & g.

L07 Rotational Motion and the Moment of Inertia 5 4. Measuring angular accelerations:

4.1 With just the pulley attached to the rotary motion detector, wind the string on the large pulley. Place the hook weight on the string hanging off the pulley making sure that it is stationary and start collecting data for a few seconds before letting the mass fall. Stop collecting data just after the mass hits the bottom. The first portion of your graph indicates the initial descent of the 50 gram hook weight.

4.2 Select a portion of the sloping straight line on the angular velocity vs. time graph and perform a linear fit on it using the fit/linear command in the menu immediately above the graph. Note the slope. It represents the angular acceleration of the rotary detector. Label it as αdet and record it in the table below as Run 1.

4.3 Repeat steps 4.1 & 4.2 with the rectangular plate fastened to the pulley so that the plate’s center of mass lies on the axis of rotation. The slope represents the angular acceleration of the plate and the rotary detector. Label it as αplate + det and record it as Run 2.

4.4 Repeat steps 4.1 & 4.2 with the aluminum disc fastened to the pulley system so that its center of mass lies on the axis of rotation. . The slope represents the angular acceleration of the disc and rotary detector. Label it as αdisc + det and record it as Run 3.

4.5 Repeat 4.1 thru 4.4 with the cord wound on the small pulley this time. Record the angular accelerations as Runs 4, 5, and 6 respectively.

4.6 Print Run 1,2 & 3 graphs with all 3 slopes displayed and include them in your post lab report.

5. Finding the Moment of Inertia of the Plate and Disc Experimentally: 5.1 The moment of inertia of just the plate, Iplate, can be calculated using your

equation in 3.4 by finding an expression for Iplate + det minus Idet. Write this expression here. It will be a function of mh, g, r, αplate and αdet. Do the same and write an expression for just the disc.

5.2 Calculate and record Iplate Idisc and Idet. On a separate page stapled to this writeup, show all your detailed calculations for a) Iplate using data from Runs 1 & 2, and 4 & 5, b) for Idisc using data from Runs 1 & 3 and 4 & 6, and c) for Idet using your Run 1 data.

5.3 Find the percent difference between your experimental moments of inertia of

the plate and of the disc, and your calculated values as the “accepted” value.

L07 Rotational Motion and the Moment of Inertia 6

Plate ID Number: Disc Mass (g) =

Plate Mass (g) = Disc Radius (cm) =

Plate Length (cm) = Icalc disc (kgm2) =

Plate Width (cm) = Small Pulley Radius = cm

Icalc plate (kgm2) = Large Pulley Radius= cm

5.4 What physical reasons might there be for the difference in 5.3?