Example 1
A 0.14 kg baseball with an initial speed of 28 m/s rebounds with a speed of 34 m/s after being struck with a bat. a) If the duration of contact between ball and bat was 2.1 ms, what was the average force on the ball? b) What was the average force on the bat?

Example 2
Leslie is rolling along on her 4.0 kg skateboard with a constant speed of 3.0 m/s when she jumps off the back and continues forward with a velocity of 2.0 m/s relative to the ground. This causes the skateboard to go flying forward with a speed of 15.5 m/s relative to the ground. What is Leslie's mass?

Example 3
A 12.0 g bullet is fired horizontally into a 101 g wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having a spring constant 152 N/m. The bullet becomes embedded in the block. If the initial speed of the bullet is 273 m/s, by how much does the bullet-block system compress?

Example 4
Two cars approaching each other along streets that meet at a right angle collide at the intersection. After the crash, they stick together. If one car has a mass of 1450 kg and an initial speed of 11.5 m/s and the other has a mass of 1750 kg and an initial speed of 15.5 m/s, what will be their speed and direction immediately after impact?

Example 5
Two air track gliders of equal mass are initially moving in the same direction along a straight line. The rearmost glider has an initial speed of 3.0 m/s and the forward glider has a speed of 2.0 m/s. If the collision is elastic, what are the speeds and directions of the gliders after the collision?

Example 6
A roller in a printing press turns through an angle q(t) given by q(t) =at2 - bt3, where a = 3.20 rad/s2 and b = 0.500 rad/s3. a) Calculate the angular velocity of the roller as a function of time. b) Calculate the angular acceleration of the roller as a function of time. c) What is the maximum positive angular velocity, and at what value of t does it occur?

Example 7
The angular speed of the rotor in a centrifuge increases from 420 rad/s to 1420 rad/s in a time of 5.00 s. a) Obtain the angle through which the rotor turns. b) What is the magnitude of the angular acceleration?

Example #8
Discus throwers often warm up by standing with both feet flat on the ground and throwing the discus with a twisting motion of their bodies. Starting from rest, the thrower accelerates the discus to a final angular velocity of 15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves on a circular arc of radius 0.810m. Find the magnitude of the total acceleration of the discus just before it is released.

Example #9
The Atwood’s machine below consists of masses m1 = 1.35 kg and m2 = 1.15 kg and a solid pulley with a diameter of 6.70 cm and a mass
mp = 0.546 kg. When released from rest, what is the speed of the heavier mass after it has fallen a distance of h = 1.45 m?
Treat the pulley as a uniform disk.
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Example 10
A hollow cylinder has mass m, an outside radius R2, and an inside radius R1. Show that its moment of inertia about its symmetry axis is given by I = 1/2 m ( R22 + R12 ).

Example #11
The force below creates a torque of magnitude 45 Nm on the bar. Using the data in the drawing, determine the magnitude of the force F.

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Example 12
Compute the torque about the origin for the force F = -mgj acting on a particle at r = xi + yj, and show that this torque is independent of the y coordinate.

Example 13
The mechanism shown is used to raise a crate of supplies from a ship's hold. The crate has total mass 50.0 kg. A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius 0.25 m and a moment of inertia I = 2.9 kgm2 about the axle. The crate is suspended from the free end of the rope. One end of the axle pivots on frictionless bearings; a crank handle is attached to the other end. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius 0.12 m, the cylinder turns, and the crate is raised. What magnitude of the force F applied tangentially to the rotating crank is required to raise the crate with an acceleration of 0.80 m/s2?

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Example 14
A block with mass m = 5.00 kg slides down a surface inclined 36.9o to the horizontal. The coefficient of kinetic friction is 0.25. A string attached to the block is wrapped around a flywheel on a fixed axis at O. The flywheel has mass 25.0 kg and moment of inertia 0.500 kgm2 with respect to the axis of rotation. The string pulls without slipping at a perpendicular distance of 0.200 m from that axis. a) What is the acceleration of the block down the plane? b) What is the tension in the string?

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Example 15
A solid disk is rolling without slipping on a level surface at a constant speed of 2.50 m/s. If the disk rolls up a 30.0o ramp, how far along the ramp will it move before it stops?

Example 16
a) Calculate the magnitude of the angular momentum of earth considered as a particle orbiting the sun. The mass of the earth is 5.97 x 1024 kg. Treat it as moving in a circular orbit of radius 1.50 x 1011 m at a speed of 2.98 x 104 m/s. b) Calculate the magnitude of the angular momentum of the earth due to its rotation around an axis through the north and south poles. Treat the earth as a uniform sphere of radius 6.38 x 106 m that makes one revolution in 24.0 hours.

Example #17
A playground merry-go-round has a disk-shaped platform that rotates with negligible friction about a vertical axis. The disk has a mass of 201 kg and a radius of 1.8 m. A 36 kg child rides at the center of the merry-go-round while a playmate sets it turning at 0.25 rev/s. If the child then walks along a radius to the outer edge of the disk, how fast will the disk be turning?

Example 18
The pistons in an internal combustion engine undergo a motion that is approximately simple harmonic. If the amplitude of motion is 3.5 cm, and the engine runs at 1700 rev/min, find a) the maximum acceleration of the pistons and b) their maximum speed?

Example 19
An object oscillates with an amplitude of 5.8cm on a horizontal spring of force constant 1.8 kN/m. Its maximum speed is 2.20 m/s. Find a) the mass of the object b) the frequency of the motion, and c) the period of the motion.

Example 20
A 1.0 g bullet is fired into a 0.50 kg block attached to the end of a 0.60 m nonuniform rod of mass 0.50 kg. The block-rod-bullet system then rotates in the plane of the figure, about a fixed axis at A. The rotational inertia of the rod alone about that axis at is .060 kgm2. Treat the block as a particle. (a) What then is the moment of inertia of the block-rod-bullet system about point A? (b) If the angular speed of the system about A just after impact is 4.5 rad/s, what is the bullet's speed just before impact?

Example 21 (#10.69, page 339)
A large 16.0 kg roll of paper with radius R = 18.0 cm rests against the wall and is held in place by a bracket attached to a rod through the center of the roll. The rod turns without friction in the bracket, and the moment of inertia of the paper and rod about the axis is 0.260 km2. The other end of the bracket is attached by a frictionless hinge to the wall such that the bracket makes an angle of 30.0o with the wall. The weight of the bracket is negligible. The coefficient of kinetic friction between the paper and the wall is 0.25. A constant vertical force F = 60.0 N is applied to the paper, and the paper unrolls.
a) What is the magnitude of the force that the rod exerts on the paper as it unrolls?
b) What is the magnitude of the angular acceleration of the roll?

Example 22
A 10 g particle undergoes SHM with an amplitude of 2.0 mm, a maximum acceleration of magnitude 8.0 x 103 m/s2, and an unknown phase constant. What are (a) the period of the motion, (b) the maximum speed of the particle, and (c) the total mechanical energy of the oscillator? What is the magnitude of the force on the particle when the particle is at (d) its maximum displacement and (e) half its maximum displacement?