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Physics C Rotational Motion Name:_______________
AP Review Packet
Linear and angular analogs Torque
Linear Rotation Torque is the rotational analog of force.
x position A “twist” (whereas force is a push or pull).
x displacement Torque is a vector)
v velocity = r F
a tangential acceleration = r F sin
T R: moment arm length
F: force
Vectors in rotational motion
Use the right hand rule to determine direction of the : angle between moment arm and point of
vector! application of force.
= I (think F = ma)
Don’t forget centripetal acceleration! : torque
2
a = a = v /r I: rotational inertia
R c
: angular acceleration
Kinematic equations for angular and linear motion.
Kinematic Equations 1 Work in rotating systems
W = • (think W = F•d)
v = v + at rot
o W : work done in rotation
= + t rot
o : torque
Kinematic Equations 2
x = x +v t + 1/ at2 : angular displacement
o o 2
= + t + 1/ t2
o o 2 Power in rotating systems
Kinematic Equations 3 Prot = • (think P = F•v)
2 2
v = v +2a(x x )
o o Prot : power expended
2 2
= +2( )
o o : torque
: angular velocity
Rotational Inertia
Rotational analog of mass Static Equilibrium
For point masses = 0 F = 0
I = mr2
I: rotational inertia (kg m2) Angular momentum
m: mass (kg) For a particle
r: radius of rotation (m) L = r p
For solid objects For a system of particles
2 L = Li
I = r dm For a rigid body
L = I (think P = mv)
Parallel Axis Theorem
I = I + M h2 Conservation of Angular Momentum
cm
I: rotational inertia about center of mass Angular momentum of a system will not change
M: mass unless an external torque is applied to the system.
h: distance between axis in question and axis L = L
B A
through center of mass I = I (one body)
B A
l = l (system of particles)
b a
Kinetic Energy
K = ½ M v 2 Angular momentum and torque
trans 2 cm = dL/dt (think F = dP/dt)
K = ½ I
rot 2 2 : torque
K = ½ M v +½ I
combined cm L: angular momentum
t: time
Rolling without slipping uses both kinds Torque increases angular momentum when parallel.
2 2
K = ½ M vcm +½ I Torque decreases angular momentum when antiparallel.
v = r Torque changes the direction of the angular momentum
K = ½ M v 2 + ½ I v 2/R2
cm cm cm vector in all other situations.
or
2 2 2
K = ½ M R +½ I Precession
cm
The rotating motion made by a spinning top or gyroscope.
Precession is caused by the interaction of torque and
angular momentum vectors.
= dL / dt
= r F
Physics C Rotational Motion Name:_______________
AP Review Packet
Base your answers to questions 4 and 5 on the
MULTIPLE CHOICE following situation.
PRACTICE PROBLEMS An object weighing 10 N swings at the end of a
rope that is 0.72 m long as a simple pendulum. At
1. ________ A wheel spinning at 3 m/s uniformly the bottom of the of the swing, the tension in the
accelerates to 6 m/s in 4 s. Its radius is 20 cm. string is 12 N.
How far around the wheel will a speck of dust
travel during that interval? 4. ________What is the magnitude of the
A) 6 m D) 18 m centripetal acceleration at the bottom of the
B) 9 m E) 30 m swing?
C) 12 m A) 2 m/s2 D) 12 m/s2
2
B) 4 m/s E) 22 m/s
2. ________ If an object of radius 3 m that C) 10 m/s2
experiences a constant angular acceleration
starting from rest, rotates 10 rads in 2 s, what 5. ________What is the speed of the object at the
is its angular acceleration? bottom of the swing?
2 2 A) 0.6 m/s D) 2.4 m/s
A) 2.5 rad/s D) 10 rad/s B) 1.2 m/s E) 7.2 m/s
2 2
B) 5 rad/s E) 15 rad/s C) 2.0 m/s
2
C) 7.5 rad/s
Base your answers to questions 6 and 7 on the
picture below, which represents a rigid uniform
rod with a mass of 6kg and a length of 1.0 m is
pivoted on the right end. It is held in equilibrium
by an upward force of 40 N.
3. ________ A bicycle moves at constant speed
over a hill along a smoothly curved surface as
shown above. Which of the following best
describes the directions of the velocity and the
acceleration at the instant it is at the highest
position?
A) The velocity is towards the right of the page 6. ________How far from the left end of the rod
and the acceleration is towards the top of the should the force be placed to maintain
page. equilibrium?
B) The velocity is towards the right of the page A) 10 cm D) 40 cm
and the acceleration is towards the bottom of B) 20 cm E) 50 cm
the page. C) 25 cm
C) The velocity is towards the right of the page
and the acceleration is towards the bottom 7. ________What force is applied to the rod by
right of the page. the pivot?
D) The velocity is towards the right of the page A) 10 N D) 60 N
and the acceleration is towards the top right B) 20 N E) 100 N
of the page. C) 40 N
E) The velocity is towards the top right of the
page and the acceleration is towards the
bottom right of the page.
Physics C Rotational Motion Name:_______________
AP Review Packet
8. ________A uniform wooden board of mass 10 12. ________What is the moment of inertia of a
M is held up by a nail hammered into a wall. A spinning object of radius 0.5 m and mass 6 kg
block of mass M rests L/2 away from the pivot. moving at 5 m/s, if it has a kinetic energy of 100 J?
Another block of a certain mass is hung a 2 2
A) 1 kgm D) 8 kgm
distance L/3. The system is in static 2 2
B) 2 kgm E) 20 kgm
equilibrium. 2
C) 4 kgm
13. ________Which of the following objects has the
least kinetic energy at the bottom of the incline if
they all have the same mass and radius?
A) cylinder D) all have the same
B) sphere E) not enough information
C) hoop
14. ________Which of the following objects has the
What is the measure of the mass labeled "?" ? greatest rotational kinetic energy at the bottom of
A) M D) 3M the incline if they all have the same mass & radius?
2 2 A) cylinder D) all have the same
B) M E) 2M B) sphere E) not enough information
3 C) hoop
C) M
2 15. ________A solid cylinder of radius .2 m and mass
2 kg is at rest at a height 1.2 m at the top of an
9. ________The angular velocity of a rotating inclined plane making an angle 60° with the
disk with a radius of 2 m decreases from 6 rads horizontal. Assuming no slipping, what is the
per second to 3 rads per second in 2 seconds. speed of the cylinder at the bottom of the incline?
What is the linear acceleration of a point on A) Zero D) 6 m/s
the edge of the disk during this time interval? B) 2 m/s E) 10 m/s
2 C) 4 m/s
A) Zero D) 3/2 m/s
2 2
B) –3 m/s E) 3 m/s
2 16. ________What is the ratio of the moment of
C) –3/2 m/s inertia of a cylinder of mass m and radius r to
the moment of inertia of a hoop of the same
mass and same radius?
10. ________A solid sphere of radius 0.2 m and A) 1:1 D) 1:4
mass 2 kg is at rest at a height 7 m at the top of B) 1:2 E) 4:1
an inclined plane making an angle 60° with the C) 2:1
horizontal. Assuming no slipping, what is the
speed of the cylinder at the bottom of the 17. ________A 4 kg object moves in a circle of
incline? radius 8 m at a constant speed of 2 m/s. What
A) Zero D) 6 m/s is the angular momentum of the object with
B) 2 m/s E) 10 m/s respect to an axis perpendicular to the circle
C) 4 m/s and through its center?
2
11. ________A spinning object with moment of A) 2 N•s D) 24 m /s
2
inertia I increases in angular speed from = 0 B) 6 N•m/kg E) 64 kg•m /s
to in t seconds. What is the average power C) 12 kg•m/s
a
delivered to the object during this interval t?
2
A) I /2t
a 18. ________A solid cylinder with diameter 20 cm
B) I 2/t
a has an angular velocity of 10 m/s and angular
C) I 2/2t
a 2
2 2 momentum of 2 kgm /s. What is its mass?
D) I /t
a A) 0.1 kg D) 5 kg
2 2
E) I /2t
a B) 1 kg E) 10 kg
C) 2 kg
Physics C Rotational Motion Name:_______________
AP Review Packet
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