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AP Physics C I.E
Circular Motion and Rotation
Centripetal force and centripetal
acceleration
Centripetal force is the net force
(sum of the forces) that keep an
object moving in a circular path.
It is always directed towards the
center of the circle.
Ex. A 10.0 kg mass is attached to a string that has
breaking strength of 200.0 N. If the mass is whirled in
a horizontal circle of radius 80.0 cm, what maximum
speed can it have? Assume the string is horizontal.
Ex. A roller coaster car enters the circular loop portion
of the ride. If the diameter of the loop is 50 m and the
total mass of the car (plus passengers) is 1200 kg,
find the magnitude of the force exerted by the track on
the car at a) the top of the track and b) the bottom of
the track. Assume the speed of the car is 25 m/s at
each location.
Ex. A Izzy-Dizzy-Throw-Up ride (The Gravitron) has a
radius of 2.1 m and a coefficient of friction between the
rider’s clothing and wall of 0.40. What minimum
velocity must the Gravitron have so the rider doesn’t
fall when the floor drops?
Ex. A stock car of mass 1600 kg travels at a constant
speed of 20 m/s around a flat circular track with a
radius of 190 m. What is the minimum coefficient of
friction between the tires and track required for the car
to make the turn without slipping?
FT
θ
Ex. The ball above makes a horizontal circular
path, while the tension in the string is at an angle θ
to the horizontal as shown above. Write an
expression for the centripetal force on the ball.
Minimum speed at the top of the
circle for objects making a
vertical circular path
• The only force at the top of the path is
weight
• For a roller coaster, the normal force is
zero
• For an object at the end of sting, tension
is zero
Translational motion and rotational
motion
Rotation of a body about a
• Rigid body – all parts are locked together
and do not change shape (a CD but not
the sun; an iron bar but not a rubber hose)
• Fixed axis – axis that does not move (a
CD but not a bowling ball)
Rotational Kinematics
Ex. A rotating rigid body makes one complete
revolution in 2.0 s. What is its average angular
velocity?
Ex. The angular velocity of a rotating disk increases
from 2.0 rad/s to 5 rad/s in 0.5 s. What is the average
angular acceleration of the disk?
Ex. A disk of radius 20.0 cm rotates at a constant
angular velocity of 6. 0 rad/s. What is the linear
speed of a point on the rim of the disk?
Ex. The angular velocity of a rotating disk of radius
50 cm increases from 2.0 rad/s to 5.0 rad/s in 0.50 s.
What is the tangential acceleration of a point on the
rim of the disk during this time interval?
Summary for angular motion
Translational
Rotational
Relationship
Dis.
s
θ
s = rθ
Vel.
v
ω
v = rω
Acc.
a
α
a = rα
The kinematics “Big Four” and their
corresponding equations for
rotational motion
Linear
Angular
Ex. An object with an initial speed of 1.0 rad/s rotates
with a constant angular acceleration. Three seconds
later, its angular velocity is 5.0 rad/s. Calculate the
angular displacement during this time interval.
Ex. Starting with zero initial angular velocity, a sphere
begins to spin with constant angular acceleration
about an axis through its center, achieving an angular
velocity of 10 rad/s when its angular displacement is
20 rad. What is the sphere’s angular acceleration?
Our goal – write Newton’s
Second Law for a rotating object
First, consider torque – that
which creates rotation
F
Calculating torque – note that it is a cross
product.
Most of the torque problems on
the AP C exam involve rotating
an object that is spherical or
cylindrical. Therefore, the force
that produces the torque is 90º to
the lever arm.
An easy example: A student pulls down with a force
of 40 N on a rope that winds around a pulley with
radius of 5 cm. What is the torque on the pulley?
Ex. What is the net torque on the cylinder below which
rotates about its center?
F1 = 100 N
F2 = 80 N
8 cm
12 cm
Now, let’s look at rotational
inertia
Rotational inertia shows how
the mass of a rotating object is
distributed about the axis of
rotation
For a point mass
Ex. Three beads, each of mass m, are arranged along
a rod of negligible mass and length L. Find the
rotational inertia when the axis of rotation is through
a) the center bead and b) one of the beads on the end.
The parallel-axis theorem
For a continuous object
Ex. Find the rotational inertia for a uniform rod of
length L and mass M rotating about its central axis.
Ex. Use the parallel-axis theorem to find the
rotational inertia of the road about one of its ends.
Newton’s Second Law
Translational
Rotational
Ex. A block of mass m is hung from a pulley of radius
R and mass M and allowed to fall. What is the
acceleration of the block?
Moments of Inertia for Common
Shapes
Kinetic Energy and Rotation
Rolling motion (without slipping)
Ex. A cylinder of mass M and radius R rolls without
slipping down an inclined plane that makes an angle θ
with the horizontal. Determine the acceleration of the
cylinder’s center of mass, and the minimum coefficient
of friction that will allow the cylinder to roll without
slipping down the incline.
Ex. A cylinder of mass M and radius R rolls without
slipping down an inclined plane of height h and
length L. The plane makes an angle θ with the
horizontal. If the cylinder is released from rest at the
top of the plane, what is the linear speed of its center
of mass when it reaches the bottom of the incline?
Angular momentum
Ex. A solid uniform sphere of mass M = 8.0 kg and radius
R = 50 cm is rotating about an axis through its center at
an angular speed of 10 rad/s. What is the angular
momentum of the sphere?
Ex. A child of mass m = 30 kg stands at the edge of a
small merry-go-round that is rotating at a rate of 1.0
rad/s. The merry-go-round is a disk of radius 2.5 m
and mass M = 100 kg. If the child walks toward the
center of the disk and stops 0.50 m from the center,
what is the angular velocity of the merry-go-round?
Examples of centripetal force
Ex. Igor, a cosmonaut on the International Space
Station, is in a circular orbit around Earth at an altitude
of 520 km with a constant speed of 7.6 km/s. If he has a
mass of 79 kg a) what is his centripetal acceleration
and b) what force does the Earth’s gravity exert on him
at this location?
Ex. In 1901, as a circus stunt, Dare Devil Diavola rode
his bicycle in a loop-the-loop. If the loop was a circle
with a radius of 2.7 m, what is the minimum speed
Diavola could have had at the top of the loop and still
remain in contact with the loop?
Q? When you ride a Ferris wheel at a constant
velocity, what are the directions of the centripetal
acceleration and normal force at the highest and
lowest points of the ride?
Ex. A Gravitron has a radius of 2.1 m. The coefficient of
static friction between the rider’s clothing and the wall is
0.40. What minimum speed must the ride have so a
passenger doesn’t fall when the floor drops?
Ex. A stock car with a mass of 1600 kg travels at a
constant speed of 20 m/s around a flat circular track with
a radius of 190 m. What minimum coefficient of friction
between the tires and track is required for the car to
make the turn without slipping?
Ex. A grindstone rotates at a constant angular
acceleration of 0.35 rad/s2. At t = 0 s the angular
velocity is −4.6 rad/s and the reference line is at θo = 0
a) At what time after t = 0 is the reference line at 5.0
revolutions? b) Describe the rotation between 0 s and
the time found in c) At what time does the grindstone
momentarily stop?
Ex. You are operating the Gravitron (apparently you
flunked out of college and are now a “Carney”) and spot
a rider who is about to hurl. You decrease the angular
speed from 3.40 rad/s to 2.00 rad/s in 20.0 rev at a
constant angular acceleration. a) What is the angular
acceleration? b) How much time does this decrease in
angular speed take? c) Does the rider hurl?
Linear and angular periods
Ex. A cockroach rides the rim of a moving merry-goround. If the angular speed is constant, does the
cockroach have a) radial acceleration? b) tangential
acceleration? If the angular speed is decreasing does
the cockroach have c) radial acceleration d) tangential
acceleration?
Ex. A centrifuge is used to prepare astronauts for high
accelerations. If the radius of the centrifuge is 15 m, at
what constant angular speed must the centrifuge rotate
for the astronaut to have a tangential acceleration of
11g? What is the tangential acceleration (in terms of g) if
the centrifuge accelerates from rest to the angular speed
found above in 120 s?
Kinetic Energy of Rotation
Ex. The spheres are free to rotate about the axis shown.
Rank each sphere according to its rotational inertia.
Ex. a) For the figure shown, find the rotation of inertia
about the center of mass. b) What is the rotational inertia
about an axis through the left end of the rod and parallel
to the first axis?
Ex. Large machine components that experience
prolonged high speed rotations are tested for the
possibility of failure using a spin test in a cylinder. A
solid steel rotor (disk) with mass of 272 kg and a radius
of 38.0 cm was accelerate to an angular speed of
14 000 rev/min when it exploded. How much energy was
released by the explosion?
Center of Mass
Center of Mass for two particles
Center of Mass for more than two
particles
Newton’s Second Law for Rotation
Ex. A uniform disk with a mass of 2.5 kg and radius of
20.0 cm is fixed on an axle. Friction between the axis
and disk is negligible. A block with mass of 1.2 kg is
attached to a massless cord that is wrapped around the
rim of the disk. Find the a) acceleration of the falling
block b) the angular acceleration of the disk and c) the
tension in the cord.
Ex. A uniform thin rod of length L and mass M is
attached to a frictionless pivot at one end. The rod is
held in a horizontal position and released. Find a) the
angular acceleration immediately after it is released and
b) the force exerted on the rod by the pivot at this time.
Rolling and Angular Motion
An interesting fact about a rolling
tire
Kinetic energy of a rolling object
Ex. A uniform solid cylindrical disk with a mass of 1.4
kg and radius of 8.5 cm rolls smoothly across a
horizontal table with a speed of 15 cm/s. What is the
total kinetic energy of the disk?
Ex. A uniform ball of mass 6.00 kg and radius R rolls
without slipping along a ramp that makes an angle of
30.0º with the horizontal. The balls is released from rest
on the ramp at a vertical height of 1.20 m. a) What is the
speed of the ball at the bottom of the ramp? b) What is
the magnitude and direction of the frictional force on the
ball?
Torque
Ex. A cat walks along a uniform plank that is 4.00 m long
and has a mass of 7.00 kg. The plank is supported by
two sawhorses, one 0.440 m from the left end and the
other 1.50 m from the right end. When the cat reaches
the right end, the plank just begins to tip. What is the
mass of the cat?
Ex. A hiker has broken his arm and rigs a temporary
sling stretching from his shoulder to his hand. The cord
holds the forearm and makes an angle of 40.0º with the
horizontal where it attaches to the hand. Assuming the
forearm and hand are uniform with a total mass of 1.30
kg and length of 0.300 m, find the tension in the cord.
Angular Momentum
Only the tangential component of
an object’s linear momentum is
used to calculate angular
momentum
Ex. A solid uniform sphere of mass M = 8.0 kg and
radius R = 50 cm is revolving around an axis through its
center at an angular speed of 10.0 rad/s. What is the
angular momentum of the spinning sphere?
Conservation of Angular
Momentum
Ex. A child of mass m = 30 kg stands at the edge of a
small merry-go-round that rotates at 1.0 rad/s. The
merry-go-round is a disk of radius R = 2.5 m and mass
M = 100 kg. If the child walks toward the center of the
merry-go-round and stops 0.5 m from the center, what is
the angular velocity of the merry-go-round?
Ex. A 34.0 kg child runs with a speed of 2.80 m/s
tangential to the rim of a stationary merry-go-round. The
merry-go-round has an inertia of 512 kg∙m2 and radius
of 2.31 m. When the child jumps on the merry-go-round
it begins to rotate. What is the angular speed of the
system?