Ph170A Spring 2006 Prof. Pui Lam

Homework #7: Rotational Kinematics, Rotational Dynamics, Conservation of mechanical energy

in rolling without slipping.

______________________

1. (30 points) Rotational Kinematics and Dynamics

a. The rotational kinematics/dynamics of a rigid object about a fixed axis are analogous to the 1-

D linear kinematics/dynamics. Fill in the table below the equivalent rotational variable and/or

equation which are analogous to their linear counterparts.

For constant a:

€

v

(

t

)

at

v

o

Conservation of mechanical energy (rolling

without slipping):

Conservation of mechanical energy (only

linear motion):

€

1

2

mv

i

2

U

i

1

2

mv

f

2

U

f

Work-Kinetic energy Theorem (assume

constant force):

€

F

d

1

2

mv

f

2

−

1

2

mv

i

2

€

K

1

2

mv

2

p=mv

F=ma

For constant a:

€

x

(

t

)

1

2

at

2

v

o

t

x

o

a=dv/dt

v=dx/dt

x(t)

Rotational of a rigid object about a fixed axis

1-D linear motion

b. What is the angular acceleration and angular speed of the second-hand of a good clock in

€

rad

s

2

and

rad

s

respectively? How does the angle depend on time? Suppose the length of the

second-hand is 0.1m, what is the tangential acceleration, centripetal acceleration, and tangential

speed of the tip of the second-hand?

c. Suppose you have a bad clock which speeds up “uniformly”, let the instantaneous angular

speed of the second-hand be given by

€

ω

(

t

)

2

π

60

0.0002

t

. Answer similar questions as in part

b, except now we are interested in the instantaneous values.

1

2. (30 points) A “spinning firework”

Suppose a “spinning firework” consists of a rod of length 10 cm and mass 0.1kg and two

“rockets” attached at both ends at 120

o

, see figure below. Each rocket has a mass of

0.3 kg. There is a pivot at the middle of the rod which allows the firework to spin freely. The

rockets were lit. The left rocket provided a thrust of 5N and the right rocket provided a thrust of

6 N. The rocket stayed on for 2 seconds

Assume no gravity, no friction and assume negligible loss of mass (the burned fuel).

We want to find out how fast the system is spinning at all time (i.e.

€

ω

(

t

)

).

The principle is:

€

Torque generates angular acceleration:

τ

(

t

)

I

α

(

t

)

. So we have to calculate

the torque about the pivot point

€

τ

(

t

)

, then the rotational inertia about the pivot point

€

I

, then

deduce the angular acceleration

€

α

(

t

)

, then use kinematics to deduce

€

ω

(

t

)

. Here are the steps:

a. Calculate the net torque

vector

about the pivot point

for 0< t < 2 s ? (Express the vector in

terms of unit vectors)

b. Calculate the rotational inertia of this system about the axis of rotation (which passes through

the pivot point)?

c. Calculate the angular acceleration

vector

for 0< t < 2 s? What is the angular acceleration

vector

after t=2s ?.

d. Deduce the expression for the angular velocity (

€

ω

(

t

)

) for all time. In particular, what is the

angular velocity

vecto

r at t=5 s?

e. How many revolutions the firework has spun around from t=0 to t=5 s?

f. If the pivot point were located at the left rocket, what is the torque, the rotational inertia, and

the angular acceleration about this axis of rotation? (Do the torque and rotational inertia depend

on the location of the pivot point? A special case where the total torque does not depend on the

pivot point is when the two forces are equal and opposite, i.e. net force equal zero; try it!)

120

o

120

o

x

y

z

pivot

.

exhaust

exhaust

left rocket

right rocket

____________________________________________________________________

3. (30 points)“Atwood Machine’ revisit - combination of linear and rotational motion (no

slipping)

Two masses m1=5 kg and m2=5.1kg are connected by a rope which passes over a pulley of

radius r=5 cm. Because of the unbalance weights, the masses accelerate.

m1

m2

T1

T2

(a) First assume the rope and the pulley are massless and the hinge of the pulley is frictionless,

find the acceleration of the system and find the tension T1 and T2. Are T1 and T2 equal?

In reality, the pulley has mass. Now assume the rope is massless but the pulley has a mass of

2kg and a radius of 10 cm (the pulley can be approximated as a uniform circular disk). Again

find the acceleration of the system and find the tension T1 and T2 assuming the rope does slip.

2

If you have trouble, here are the steps that guide you though it:

(b) What is the rotational inertia of the pulley about its axis of rotation?

(c) What is the kinematic relationship between the angular acceleration of the pulley and the

acceleration of the system.

(d) Set up F=ma for m1 and m2 and set up

€

τ

I

α

for the pulley. (How many equations and how

many unknowns? What is the connection between linear acceleration of the system and angular

acceleration of the pulley under no slipping condition)

(e) Solve the equations to find the acceleration of the system and the tension T1 and T2. Are T1

and T2 equal?

(Note: If we take into account that the rope has mass, then the problem is more complicated, the

acceleration would not be constant).

(f) In real applications, an Atwood machine is used to measure the acceleration due to gravity

Suppose this Atwood machine is brought to the surface of planet X. The observed acceleration

of the system is 0.1 m/s

2

. Find the acceleration due to gravity of this planet.

__________________________________________________________________________

4. (30 points) Rolling without slipping and conservation of mechanical energy

A solid, uniform circular disk (radius=R) is initially at rest on top of an incline at a height h and

angle

€

θ

. It rolls down the incline without slipping (

meaning there is friction, but this is static

friction because there is no slipping)

.

θ

h

a. Find the center of mass acceleration of the disk. (Hint: apply Newton’s second law to the

center of mass and apply the torque equation for rotation about the

center of mass. Need to solve two equations and two unknowns). Find the magnitude of the

frictional force.

b. Find the center of mass velocity when the disk reaches the bottom of the incline using result of

part a.

c. Now use conservation of mechanical energy to find the the center of mass velocity when the

disk reaches the bottom of the incline. (You should get the same answer as in part b and hence

verify that friction does NOT take away mechanical energy when rolling without slipping. That

is why rolling is so efficient!).

d. A more detail analysis of the conservation of energy - Calculate the work done by gravity

and by the frictional force and verify that the work-kinetic energy theorem is obeyed by the

center of mass motion (the center of mass KE is reduced by frictional work). Now calculate

the “rotational work” and verify the work-kinetic energy theorem for the rotational motion. You

will see that frictional force remove energy from the center of mass motion and put it into

rotational motion without any loss

if there is no slipping.

e. Repeat part c for a solid sphere.

________________________

5. Extra credit (20 points):

Use integration to find the rotational inertia of a uniform solid

sphere, verify that

€

I

3

5

MR

2

3

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