# Physics

NORTHERN ILLINOIS UNIVERSITY

PHYSICS DEPARTMENT

Physics 253 – Basic Mechanics Fall 2016

Lab #11

Lab Writeup Due: Mon/Tue/Wed/Thu, Nov. 28/29/30, Dec. 1, 2016

Read Giancoli: Chapter 10 (Lecture Notes #13)

Rolling

Apparatus

In this experiment a sphere, disk, and cylinder are rolled down an inclined plane

with a raised guide to keep it on the track. Two photogates are positioned over the track

to measure the velocity of each object at the position of each photogate. Each photogate

only records the elapsed time between the when the object enters and leaves the

photogate. The experimenter must determine the width d of the object as seen by the photogate detector to determine the velocity through each photogate.

Theory

Velocity is the time rate of change of position of an object. If the width, d , of an object and the time, t , it takes to pass a point are both known, the average velocity is

ave d

v t

(1)

Angular velocity is the time rate of change of the angle of a rotating object,

measured in radians per second (rad/sec). For an object that rolls without slipping the

angular velocity is related to its linear velocity as

v d

R R t

(2)

The resistance of an object to a force (the inertia of an object) is caused by the

object’s mass. The resistance of an object to a torque (a force that causes a rotation) is

caused by the object’s moment of inertia (which is related to the object’s mass and how

far the mass is from the axis of rotation: 𝐼~𝑚𝑅2). This is why when a figure skater on ice brings in her arms she rotates faster—her moment of inertial is smaller because more

of her mass is closer to her body (her effective radius decreases). Decreasing the moment

of inertia makes it easier for her to rotate (her rotational inertial decreases).

Objects in motion possess kinetic energy K . If the object is rolling it has kinetic energy due to the forward motion of its center of mass, CMK , and its rotation, rotK .

Translational kinetic energy is based on the mass and velocity, 21

2CM CM K mv .

Rotational kinetic energy about the center of mass is based on the moment of inertia and

angular velocity, 21 2 CMrotK I .

2 21 1

2 2 CM CMK mv I (3)

Using Eq. (2) we can convert velocity to angular velocity:

2 2 2 2 2 2 1 1 1 1

2 2 2 2 CM CMK mR I I mR I (4)

Notice that we have derived the parallel-axis theorem: 𝐼 = 𝐼𝐶𝑀 +𝑀ℎ 2 where, in the

situation for this lab, the object rolls about an axis at the point where it touches the ramp,

thus ℎ = 𝑅.

As a disk rolls down a slope the gravitational potential energy, gU mgh , is

converted into kinetic energy and thermal energy thermalE caused by sliding rather than

rolling. If the initial and final angular velocities are i and f , then the relationship for

the conservation of energy is:

𝐾𝑖 + 𝑈𝑖 = 𝐾𝑓 +𝑈𝑓 + ∆𝐸𝑡ℎ𝑒𝑟𝑚𝑎𝑙

. 1

2 𝐼𝜔𝑖

2 +𝑚𝑔ℎ𝑖 = 1

2 𝐼𝜔𝑓

2 +𝑚𝑔ℎ𝑓 + ∆𝐸𝑡ℎ𝑒𝑟𝑚𝑎𝑙 (5)

This equation [and using Eq. (4)] can be rearranged to solve for the moment of inertia

about the center of mass, 𝐼𝐶𝑀. Sliding (or slipping) is relatively small (hopefully), so neglecting the thermal energy it generates gives:

. 2 2 2

2 ( )f i CM

i f

mg h h I mR

(6)

Data Collection

(1) Measure the mass and diameter of each object. Record this in your logbook.

(2) Measure the height of the track at each photogate. This gives you fh and ih .

Carefully measure the distance between the photogates (make certain they are

roughly 80 cm apart). This gives you L . Put a short strip of paper tape 10 cm up the track from the 1st photogate. This will be the starting point for your measurements

Also measure the height of this starting point: this give you starth .

(3) Put a short strip of paper tape on the track between the upper photogate. Holding the sphere in place, mark on the tape the center of the sphere when it just begins to

interrupt the photogate beam as indicated by the red light. Next, allow the sphere to

roll down the track through the gate until it no longer interrupts the photogate beam

(the red light turns off) and mark on the tape the center of the sphere at this point.

Record the distance between the marks as sphere id . Do the same for the disk and the

cylinder to get disk id and

cylinder id . This gives you the width d of each object as seen

by the photogate detector.

(4) Repeat Step (3) for the lower photogate and record the distances sphere fd ,

disk fd , and

cylinder fd in your logbook.

(5) Start the Logger Pro program, and release the sphere from rest at the starting point (10 cm from the 1st photogate). Record the times measured by the computer (both in

an Excel spreadsheet and in your lab notebook). Make certain to label which pair of

times are for the upper and lower photogate.

(6) Repeat the measurement from the same starting point a total of six times. Find the

average final angular velocity f . Record this calculation in your lab notebook.

(7) Repeat Step (6) for the disk and the cylinder.

Analysis

Everything up to the race is to be completed before you leave lab. Show the TA your results from (1) to (8) before leaving lab. Writup due is a Word document.

(1) Find CMI for the sphere, disk, and cylinder using Eq. (6).

(2) Using the table on the last page of this handout, calculate the theoretical values of

CMI for the sphere, disk, and cylinder.

(3) Compute the percent error of CMI for each object:

100% Experimental Theoretical

Error Theoretical

(4) Comment on the accuracy of your measurements and why there may be a discrepancy.

(5) Which one of the objects is the fastest? (That is, which one makes it to the bottom of the ramp first?)

(6) Test your prediction by racing the sphere, disk, and cylinder (2 at a time) to see which one is fastest. Use the wooden ramp located at the front of the lab class. Did

all go according to your prediction?

(7) Give a simple explanation as to why the fastest object won the race.

(8) Using Eq. (6), find CMI for the sphere, disk, and cylinder using your starting point

10 cm in front of the 1st photogate (where 0i ). Do you get the same values as

before?

(9) Compute the percent error of CMI for each object for these new values. Comment on whether there is an improvement or not.

(10) Draw a free body torque diagram, and write down Newton’s 2nd Law for translations and rotations. The rest of the questions below are only for the sphere.

(11) Using Newton’s 2nd Law for rotations, solve for the angular acceleration of the sphere.

(12) Insert this angular acceleration into the kinematical equation of motion to

determine the angular velocity (Eq. M2): 0 t .

(13) Computer what f should be at the bottom photogate when (a) 0 is the value at

the starting point (which is zero), (b) when 0 is the value at the 1 st photogate.

(14) How do these values agree with your measured value of f for the sphere? Give

the percent error.

(15) Calculate the coefficient of friction (and its uncertainty) using the measured value

of f (for I use the value in the table below for a sphere) assuming slipping occurs from start to finish.

(16) Calculate the coefficient of static friction assuming no slipping occurs. Note: I did this in class.

(17) Is slipping (or sliding) occurring? [Use your results of Parts (13) to (16)]