How can we measure this angular acceleration? What are the directions of an object’s velocity and acceleration vectors when it is subject to non-uniform circular motion?

10 – 1

Experiment 10

Newton’s Second Law for Rotation

Questions .

How do we give something a constant angular acceleration? How can we measure this angular acceleration? What are the directions of an object’s velocity and acceleration vectors when it is subject to non-uniform circular motion?

Concepts .

When an object moves in a circle and its angular speed (omega, 𝜔) changes, the object experiences an angular acceleration (alpha, 𝛼). In straight-line motion, a net force makes something accelerate. For rotation, a net torque gives an object an angular acceleration. In this experiment, you will apply a constant torque to a circular disk that can rotate on a low friction bearing. The disk will ‘spin up’ to a higher and higher angular speed. You will calculate the disk’s constant, angular acceleration and relate it to the applied torque.

Figure 1. Figure 2.

Figure 3.

v 1

v 2

Δ !

v / Δt Δ !

v 1

v 2

acen atan

a = Δ !

v / Δt( )

10 – 2

Figure 4.

In Figure 1, a particle moves clockwise in

non-uniform circular motion. Its angular and

linear speeds increase. The subscripts 1 and 2

refer to early and later times.

Figure 2 shows the change in the particle’s

velocity vector. The acceleration vector is

just a scaled version of the change in the

velocity vector. The scaling factor is the time

interval.

Figure 3 shows how the acceleration

vector has tangential and centripetal

components. The tangential component is

given by , and as usual .

The radius of the circle and the angular

acceleration are constant. So, the tangential

component of the acceleration remains

constant in time. However, due to the speed-

squared term, the centripetal component

increases with time.

Figure 4 shows both the velocity (blue)

and acceleration (red) vectors of the particle

at four successive times. The velocity vector

is always tangential and grows linearly in

length. The acceleration vector also grows in

length. It starts out tangential but becomes

more and more centripetal.

For a constant angular acceleration, the angular speed,

w, obeys a rotational kinematic

equation similar to straight-line motion: . If the disk starts from rest, = 0.

The linear and rotational versions of Newton’s Second Law are extremely similar:

and . Analogous to the mass m in the linear version, the rotational mass

of the object about a given axis of rotation is called the object’s rotational inertia or the moment

of inertia, I. Here the word moment is a statistical term that measures how spread out the mass is

distributed from the axis of rotation. A dumbbell shaped object is more difficult to rotate than a

solid sphere of the same mass. This is because more of the dumbbell’s mass is farther from the

middle of the dumbbell, which is typically where the torque is applied. Differently shaped

objects have different rotational inertias. Also, the same object has different moments of inertia

depending on the choice of the rotation axis. We will use a solid disk of mass M and radius R

with an axis perpendicular to the plane of the disk and through the disk’s center. In this case, the

moment of inertia is I = ½MR2 .

a tan = r

α acen = v 2

ω(t) =

ω 0 +

αt ω 0

Fnet = m !

a !

τ net = I !

10 – 3

A low-detail diagram of the apparatus is shown in Figure 5. A hanging mass falls and

accelerates the disk by unwinding a string wound around one of three small (stepped) pulleys mounted to the disk. A photogate is used to measure the angular speed of the vertical pulley.

From this you can use Logger Pro to calculate the linear speed of the string and its acceleration.

Let us take as the linear speed of the string; the sub-script V stands for the vertically mounted pulley and S for the stepped pulley. Then the angular speed of each pulley is written as follows.

Each r is the radius of each pulley.

and (4)

The tension in the string is the force that exerts the torque. However, this tension is not simply the weight of the hanging mass. We will ignore the mass of the vertical pulley and all friction. Then Newton’s Second Law gives for the falling weight:

Ignoring the mass of the vertical pulley allows us to write . (5)

Applying the definition of torque to the disk and using the rotational version of N2 gives:

Substituting for the tension gives: .

Substituting from Eqn. (5) and solving for the linear acceleration gives

(6)

Method .

Figure 5 depicts the second part of this week’s apparatus: a heavy disk that can rotate in a horizontal plane on a low friction bearing. A two-claw clamp and a support rod for a pulley have been omitted for simplicity. The acceleration of the string is measured by a photogate mounted around the pulley and connected to the Lab Pro interface. A similar arrangement was used in the Newton’s Second Law experiment.

ωV = v

ω S = v

T − mg = −ma

a =

αrS

τ ≡ rS

T = I

τ = rS mg − m

αrS( ) = I

a = mgrS

I + mrS

10 – 4

Figure 5. The Pasco Introductory Rotation Apparatus.

Procedure .

1) Use a Vernier caliper to measure the diameter of whichever stepped pulley you want to use. Divide the diameter by two to find r S .

2) Measure the mass M of the plastic disk. The disk’s mass is greater than 610 grams,

which is the maximum mass the triple beam balance can measure. However, hanging a

tare weight on a small metal stud found at the end of the scale extends the balance’s

range. These are called tare weights. One tare weight adds 500 grams to the reading. In reality, this mass is 147.5 grams but its torque on the apparatus yields a 500 gram

equivalent mass. Using the same proportions, a 200-gram hooked mass can be used as a tare weight. This will add 678 grams to the scale reading.

3) Measure the radius, R, of the plastic disk. We will ignore the slight difference in the

moment of inertia formula that would result from including the shape of the stepped

pulley. Calculate the moment of inertia of the disk using I = ½MR2 .

4) Set-up the apparatus as in Fig. 5. Use white, kite string or fishing line as the string. Do

not tie the string to the screw at the top of the stepped pulley; just wrap it around the

screw a few times. This way you can easily remove the string. Then thread it through

the holes drilled in the pulley and then wrap the string around the chosen pulley. For

example, if you use the largest diameter pulley, the string should go through three holes.

5) Setup the software for using the photogate and the 10-spoke pulley by executing these menu commands: Experiment > Set Up Sensors > Show All Interfaces >

> Set Distance or Length… > Ultra Pulley 10

Spoke In Groove.

6) Also increase the sampling rate to 50 samples / sec. Execute these commands:

Experiment > Data Collection.

7) Adjust the length of the string so the weight hanger does not strike the floor when all of the string has unwound. Place a foam pad beneath the hanging weight to avoid damage if

stepped pulley

rotating disk

hanging

mass

vertically

mounted

pulley

10 – 5 the string breaks. Use slotted masses and the Beck mass hangers. The plastic 5-gram mass hangers are fragile and can break too easily.

8) Lower a mass before recording data. Examine the apparatus to make sure the disk

accelerates, and that friction is minimal. Use Logger Pro to measure the linear

acceleration of the 10-spoke pulley for five different falling masses. Do not use more

than 150 grams of hanging mass.

9) Use Logger Pro to acquire a velocity vs. time graph for when the hanging weight falls.

Use the slope of a tangent line feature to obtain five accelerations for one run.

Record your data in a spreadsheet. Later, calculate the average and sample standard

deviation for the five accelerations.

10) Change the amount of hanging mass and perform additional trials.

11) For each trial, calculate the linear acceleration from Eqn. (6). Compare it to the measured value. Enter these theoretical accelerations into the spreadsheet.

12) Calculate the differences between all experimental and theoretical accelerations.

Compare each difference to the standard deviation in each experimental value. If the

standard deviation is as large or larger than the difference, then the results agree to within experimental error. There may be no need to propagate errors through Eqn. 6.

13) Comment on sources and types of errors in your discussion.

How can we measure this angular acceleration? What are the directions of an object’s velocity and acceleration vectors when it is subject to non-uniform circular motion?