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 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 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?

FTFT 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 TranslationalRotational 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? F 1 = 100 N F 2 = 80 N 12 cm 8 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 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/s 2. At t = 0 s the angular velocity is −4.6 rad/s and the reference line is at θ o = 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-go- round. 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 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 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 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∙m 2 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?