Physics 250 Laboratory: Rotational Motion - Penn State University

Physics 250 Laboratory: 

Rotational Motion 

Score: _____ 

Section #:______ 

Name:_____________________________ 

Name:_____________________________ 

Name: _____________________________ 

Lab-Specific Goals: 

• Examine the angular displacement, angular velocity, and angular acceleration of a rotating 

disc. 

• Relate linear and rotational motion. 

• Determine the coefficient of static friction between the turntable surface and a penny. 

• Predict the magnitude of the angular velocity that causes a penny at a given radius to slip. 

Equipment List: 

Rotating Platform with attached Rotary Motion Sensors 

Photogates 

Stickers located at two different radii 

Pulley 

String 

Hanging mass and hanger 

Penny 

Ruler 

Index card piece and putty (for flag on disc) 

Introduction and Pre-Lab Questions: 

Rotational motion, where an object rotates about some 

internal axis, is a common form of motion: the earth 

rotating, an ice skater doing a turn, a car’s wheels spinning, 

etc. Rotational motion is closely connected to circular 

motion since any point on the rotating object undergoes 

circular motion around the axis. One of the goals of this lab 

activity is to explore and understand this connection. 

v t 

Δθ 

s 

r 

ω 

© 2009 Penn State University 

1

The radius of a particle undergoing circular motion is always a constant. The angular position, 

however, will change with time depending on the motion of the particle. The angular quantities 

(θ, ω, α) for the object as a whole are related through geometry (specifically the radius) to the 

quantities for the point of interest (s, v t , a t ): 

Arc distance traveled 

Tangential velocity 

Tangential acceleration 

S 

v t 

a t 

s 

Δ θ = Angular displacement 

r 

v 

ω ≡ Δθ = t 

Δt 

r 

Angular velocity 

a 

α ≡ Δω = t 

Δt 

r 

Angular acceleration 

The relationships between the angular position θ, angular velocity ω, and angular acceleration α 

are exactly the same as the relationships previously determined for one-dimensional motion. 

Write below the angular equivalents of the X, V, and V 2 equations for constant (angular) 

acceleration: 

X : 

V : 

V 2 : 

A point on a rotating object experiences two kinds of acceleration: 

1) Centripetal acceleration (inwards 

towards the center of the circle) 

a 

c 

2 

vt 

= = ω 2 r 

r 

2) Tangential acceleration 

(tangent to the circle) 

a t = rα 

The total acceleration is the vector sum of a c and a t : 

a 2 = a c 2 + a t 

2 


2

Q1) At a given instant, a disc is rotating with an angular velocity of 10 rad/s and accelerating at 2 

rad/s 2 . Consider a particle on the disc with mass 0.05 kg and a distance 0.2 meters from the axis. 

a) What is the centripetal acceleration of this point? 

b) What is the tangential acceleration of this point? 

c) What is the total acceleration of this point? 

d) What is the net force acting on this point? 

e) Which of these types of acceleration will change over time for this object? Why? 

Q2) Two pennies start on a spinning disc – one near the center and one near the edge of the disc. 

Which penny will be more likely to fall off the disc and why? 

Q3) A disc is initially at rest, but is undergoing a constant counter-clockwise angular 

acceleration. Make qualitative sketches for the angular position θ, angular velocity ω and angular 

acceleration α as a function of time. 

θ 

t 

ω 

t 

α 

t 


3

Introductory Activity: Using the Rotating Disc 

1. Measure the radius of the circle created by the outer blue sticker, R o (from center of disc to 

center of blue circle). 

R o = ___________________ meters 

2. Measure the radius of the circle created by the inner yellow sticker, R i . 

R i = ___________________ meters 

3. Carefully rewind the string around the base of the turntable and place the string over the 

pulley with the hanging mass attached. If your turntable apparatus has multiple radii around 

which the string can be wrapped, but sure to use the same one every time. Make sure the 

string remains on the pulley. 

4. You will need to use the same hanging mass the entire time. For Activity 2, the hanging mass 

must be large enough to cause a penny to slip at both the blue and yellow positions, but small 

enough to keep it from slipping too quickly. Find a hanging mass that fits these criteria: 

Hanging mass: m = ___________________ kg (typically about 150 grams) 


4

Activity 1: Kinematics of Circular Motion 

Develop a procedure to collect the data required to graph the angular velocity (ω) versus time for 

this system. Some hints: 

(1) you can use pulse mode with one photogate to determine the time it takes to make the 

first, second, third, etc. rotations and from that determine θ(t). From these measurements 

you can find α and then ω(t). 

(2) you can use gate mode with one photogate to find the tangential speed v t (and thus the 

angular velocity ω) at the end of each rotation and from this determine α and then ω(t). 

(3) You may need to do multiple runs to get one full set of data since the photogate timer will 

only store one or two times for a single run. 

Sketch and explain your procedure below. (Start with the system initially at rest.) 


5

Label the headings in Table 1 to match the data and calculations that you will be doing. Then 

collect your data and fill in the table. Show your calculations for your first measurement below: 

Table 1: Data for Lab Activity 1 

Now graph your data on the next page. What data point do you already know without collecting 

any data? (And how do you know this point must be there?) 


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Graph 1: Angular velocity ω as a function of time 


7

Draw a best fit line (making sure to go through the origin) to your angular velocity as a 

function of time graph. 

What is the angular acceleration α in your system? Explain below how you find α and what 

value you get. (Record your hanging mass and value for α on the Class Data Sheet.) 

What is the tangential acceleration of a point at the very edge of the disc? (Show how you 

determine this from α.) 

What is the angular displacement Δθ for the disc from t=0 to the maximum time shown on your 

graph? (Show how you determine this from your graph.) 


8

Activity 2: The Dynamics of Circular Motion 

You are going to predict the angular velocity at which the penny will slip from one radius using 

your measurements of when the penny slips at a different radius. Because the penny will be 

spinning it will have an inwards centripetal acceleration of 

a c = v t 2 /r = ω 2 r. 

Since the disc is angularly accelerating, it also has a tangential acceleration 

a t = rα. 

Remember that tangential acceleration is a measure of the rate change of the tangential (and 

therefore, angular) speed of the object, i.e., the magnitude of the velocity vector. Centripetal 

acceleration is a measure of the rate at which the direction of the object changes at every 

moment. 

The magnitude of acceleration is then given by the vector sum of a c and a t . Since they are 

perpendicular to each other, we can use the Pythagorean theorem: 

a 2 = a c 2 + a t 2 . 

α 

ω 

α 

ω 

v a a c 

a t 

f s 

Figure 1 (Left): a view from above of the turntable, showing at one instant the penny’s instantaneous acceleration 

and velocity. We break the acceleration into centripetal and tangential components. (Right): the only horizontal 

force acting on the penny to provide it with the acceleration necessary to move with the turntable is that of static 

friction. Because it is the only horizontal force acting (and the vertical forces balance) we can say that the static 

frictional force must be the net force on the penny. 

The penny is held in place by static friction. Recall that static friction is a variable force, able to 

provide resistance up to a particular maximum value: 

f s,max = µ s n 


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Part I: Determine the Coefficient of Static Friction on the Turntable 

1. Place a penny at a distance, R o , from the center of the turntable. [Note: Do not place the 

penny directly on top of the blue sticker, just at the same radial distance from the center.] 

2. Release the turntable from rest and determine the angular velocity of the turntable when 

the penny slips. (Remember, you already know the angular velocity as a function of time 

from Activity 1!) [You can use the photogate timer as a stopwatch, by pushing to 

start/stop button.] 

You will want to do several runs to get a good average time (and thus average ω): 

______________ _______________ ________________ _______________ 

______________ _______________ ________________ _______________ 

3. For the moment just prior to when the penny slipped, calculate the linear (tangential) 

velocity, the tangential acceleration, and the centripetal acceleration of the penny. (Note: 

The tangential acceleration will likely be much smaller in magnitude than the centripetal 

acceleration.) Record your results and explain your calculations in the table below. 

4. Determine the coefficient of static friction between the turntable surface and the penny. 

Record your results and explain your calculations in the table below. Clearly and 

completely explain your method of calculating µ s . Record your value for µ on the Class 

Data Sheet. (Note: you do not need to find the mass of the penny to solve for µ.) 

Quantity Result Explanation of how Result was obtained… 

R o = Radius (meters) 

ω = Angular Velocity (rad/s) 

This radius was measured using a ruler in the 

introductory activity. 

α = Angular Acceleration (rad/s 2 ) Measured in Activity 1. 

v t = Tangential Velocity (m/s) 

a t = Tangential Accel. (m/s 2 ) 

a c = Centripetal Accel. (m/s 2 ) 

a = magnitude of acceleration 

(m/s 2 ) 

µ s = Coefficient of Static Friction 


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Physics 250 Laboratory: Rotational Motion - Penn State University

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