P~ysICS 121 E~xPERIMENT No. 6



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PHYSICS 123 EXPERIMENT NO. 6

CONSERVATION OF ANGULAR MOMENTUM

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Introduction:

In this experiment we study torque, moment of inertia, and angular momentum conservation with a rotating platform.

Equipment:

One rotating platform with photogate, one pulley with clamp, one iron disk with handle, one interface box, one computer with timing program, small masses, vernier caliper.

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Method:

Observing the angular momentum of an object, the rotating platform, under an external torque will enable us to measure the moment of inertia of the object using the equation

(external = I(

Where (external is the net external torque, ( is the angular acceleration, and I is the moment of inertia. After the moment of inertia is determined, conservation of angular momentum will be investigated. We drop a mass onto the rotating platform and measure the angular velocity , ( , before and after the "drop". We test angular momentum conservation using the relation

Ii(i= If(f

where i and f standard for initial and final.

Part I Measurement of the Moment of Inertia of the Rotating Wheel and Attached cylinder:

large wheel

axis of rotation

pulley pulley

cylinder with string

wound around it

mass m

The apparatus used for this lab is sketched above. A small cylinder with string wound around it is attached to a large wheel. A mass m attached to the string provides an external torque, τexternal, which gives the system, cylinder-wheel, an angular acceleration α which you measure by measuring the rate of increase of the angular velocity. The photo gate mounted on the wheel registers whenever one of the 4 black radial strips on the plastic rim of the wheel blocks the light beam of the photo gate. From the 900 angle between the strips the angular velocity is computed. The system is not friction free, the spinning wheel will stop after a period of time. A frictional torque, τfric, opposes the external torque thus making a smaller net torque, τnet.

First you measure αfric by measuring the rate of decrease of the angular velocity of the initially spinning wheel. You use αfric in a subsequent measurement of the moment of inertia I of the system, cylinder-wheel. I can be calculated from the following formula:

[pic] (3)

where m is mass attached to the string, r is the radius of the cylinder, g is the gravitational accelaration and α is the angular acceleration of the wheel due to the net torque acting on the wheel, external torque minus frictional torque.

Connect the photogate to the interface box and set up the computer in MOTION TIMER mode (see procedure 3. to 5. in lab “Acceleration" ). Measure the angular velocity, ω, of the freely turning table for 20 or 30 seconds. Hit “ENTER” on the keyboard to see a data table of time values. Hit “ENTER”, select “Graph Data”, and select “Velocity vs. Time”, and then select “Other (specify the length)”.Your computer screen will show “Enter length (in meters)”, this is actually the distance between timings, ignore the “in meters” on the computer screen and enter the angular distance (in radians) between the two pieces of tape on the rotating disk. Once you have put in the angular distance, hit “ENTER” to bypass the “SELECT GRAPH STYLE” screen. Then select “Automatic Scaling, Axis Starts at 0” for both horizontal and vertical scaling. Your graph should show a smooth straight line with a negative slope. To get out of the graph screen, hit “ENTER” and select “Display Table of Values”. On the screen, you should see a table of values of time t and angular velocity ω (the units for angular velocity is in radians/s, just ignore the units for linear velocity on your computer screen).

Graph ( vs. time in your lab book, fit a best line through your data points and calculate the angular acceleration (deceleration actually), (fric , due to the frictional torque.

Question 1: Is the frictional torque dependent on the velocity ?

II. MEASUREMENT OF MOMENT OF INERTIA, I: Wrap a piece of string around the cylinder and measure the length of the string to determine the cylinder’s circumference. Use the circumference to find the radius of the cylinder. Attach a mass to the free end of the string, and wind the string neatly around the cylinder. Loop the string over the pulley and release the mass. Record angular velocities as above and calculate the angular acceleration, α . Calculate the moment of inertia, I ,from the equation

[pic]

Derive this equation.

III. CONSERVATION OF ANGULAR MOMENTUM:

Remove the string from the rotating table. Get the computer and interface box ready for data taking as you have done previously and then spin the rotating table. While the table is spinning, drop the iron disk onto it from a small height (less than 1cm) above the table. You want the rim of the disk to match the rim of the wheel as closely as possible. Obtain the graph of angular velocity ω vs. time from the computer by hitting “ENTER”, select “Graph Data”, and select “Velocity vs. Time”, and then select “Other (specify the length)”. Make sure that your angular distance, which you put in earlier as 2п/4, is still correct in the computer. Hit “ENTER” to bypass the “SELECT GRAPH STYLE” screen. Then select “Automatic Scaling, Axis Starts at 0” for both horizontal and vertical scaling. You should get a graph that looks roughly like this:

Given the mass ,M, of the iron disk and measuring its radius R, calculate its moment of inertia using

I = 1/2MR2

Compare the initial angular momentum of the table/disk system, Ii(i , with the final angular momentum after the drop, If(f . Determine whether your experiment (within its error limits) confirms angular momentum conservation.

Questions 2:

What assumptions are your calculations of the moment of inertia based on ? If you drop the disk a couple of cm off center, what effect will this have on your result ? Explain Qualitatively.

What you need:

1. Radius of rotating table

2. Angular distance between tape pieces

3. Graph of ω vs t

4. Calculation of slope of graph to find αfric

5. Calculation of α and I

6. Derivation of Inertia equation

7. Radius and mass of disk, calculation of its moment of inertia

8. Measurement of ωi and ωf

9. Check of conservation of angular momentum

10. Answers to the questions

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Disk with handle

Vernier Caliper

Photo gate

Mass with string

Interface Box

Computer Terminal

Cylinder for wrapping string around

Rotating table

Pulley

t

[pic]i: take the average of 4 points here, get the error from (max-min)/2; do the same for [pic]f

ωf

the negative slope is caused by [pic]

ignore these points

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