Spring 2009 FOURTH MIDTERM -- REVIEW PROBLEMS

Transcription

1 Physics 10 Spring 009 FOURTH MIDTERM -- REVIEW PROBLEMS George Williams A data sheet is provided. The extent of coverage of Ch. 9 will be discussed in class. Problems involving integration to calculate COM may be on this test. Not for exam 4 but keep for exam 5: 61-63, 66-67, 76-77, A car, weighing 000 pounds, is traveling at 30 miles per hour. Calculate its momentum, in British units. Calculate the kinetic energy, in joules, of the car in. A 10.0 kg man moves with velocity 1.5 m/s. Find its momentum. 3 Convert kg m/s of momentum to British units. A golf ball of mass 100 g, initially at rest, is given a velocity of 35 m/s. Calculate the impulse delivered by the golf club.. A big truck weighing 0,000 pounds is traveling at 60 miles per hour. Calculate its momentum (in British units). Calculate the kinetic energy, in joules, of the same truck. A 3.0 kg mass moves with velocity 7.5 m/s. Find its momentum. 4 Convert.0 10 kg m/s of momentum to British units. A golf ball of mass 100 g, initially at rest, is given a velocity of 45 m/s. Calculate the impulse delivered by the golf club. 3. Find the momentum of a 10 kg object moving at a speed of 35 m/s. Find the momentum of a 56 pound object moving at a speed of 35 ft/s. Find the x coordinate of the center of mass of a system consisting of a 10 kg object at x = 0 and a 0.55 kg object at x = +8 m. A baseball of mass 0. kg, is thrown by a pitcher at a speed of 45 m/s. It is struck by a bat and returns directly toward the pitcher at 45 m/s. Find the impulse applied to the ball. 4. Find the center of mass of the system shown. Calculate the momentum (in British units) of a truck that weighs 755 lbs moving at 55.0 mi/hr. Calculate the kinetic energy, in joules of the truck in. A baseball, of mass 0.50 kg, is thrown at 15 m/s. The batter strikes it, and the new velocity is 150 m/s back along its initial direction. Calculate the impulse delivered to the ball. An object of mass 3.75 kg is traveling at 1.50 m/s in the positive x-direction. A second object of mass 1.5 kg, travels at 4.50 m/s in the negative x-direction. They collide in a completely inelastic collision. Calculate the kinetic energy lost. 5. A ball of mass 1.0 kg moves to the left with a velocity of 9.70 m/s. It strikes a wall and bounces directly back with a velocity of 8.75 m/s. Calculate the magnitude of the impulse given to the ball by the wall. Calculate the amount of energy converted to heat in the impact above. Convert 375 ft pounds into the proper SI unit. A football player of mass 100 kg is moving with a speed of 8.75 m/s. Calculate the magnitude of his momentum. Find the x-coordinate of the center of mass of a system consisting of 1.5 kg at x = m,.75 kg at x = m and 3.5 kg at x = +.5 m.

2 6. Two particles, one having twice the mass of the other, are held together with a compressed spring between them. The energy stored in the spring is 60 J. How much kinetic energy does each particle have after they are released? Assume that all the stored energy is transferred to the particles and that neither particle is attached to the spring after they are released. Find the ratio of the velocities of the two particles. 7. Two.0 kg masses, A and B collide. The velocities before the collision are v = 15^i + 30^j m/s and v = -10^i + 5.0^j m/s. After the collision v = -5.0^i + 0^j m/s. B What is the final velocity of B? How much kinetic energy was gained or lost in the collision? A 8. A bullet of mass 4.5 g is fired horizontally into a 1.8 g wooden block at rest on a horizontal surface. The coefficient of kinetic friction between block and surface is 0.0. The bullet comes to rest in the block which moves 1.8 m. Find the speed of the bullet. 9. A block of wood of mass M is suspended on a string of length L. A bullet of mass m, strikes the block and sticks in it. The bullet is fired horizontally. As a result, the block describes a complete circle about the point of attachment of the string. At the top of the loop, the tension in the string is (M + m)g. Find the original velocity of the bullet. A 10. A bullet of mass m strikes a block of wood of mass M on an inclined plane and is embedded in it. The coefficient of friction between the block and the plane is The block slides up the plane a distance d. Find the original velocity of the bullet. 11. An artillery shell is traveling horizontally at a velocity of 150 m/s. The shell explodes and breaks into three pieces. A has a mass of 75 kg and a horizontal velocity of 50 m/s. B has a mass of 75 kg and a downward velocity of 130 m/s. C has a mass of 10 kg. Find the magnitude and direction of the velocity of C. (For direction find the angle as defined in the drawing.) Find the energy added to the system. 1. Cylinder A of mass 0.50 kg, is dropped.50 m onto cylinder B of mass 3.5 kg, which is resting (v = 0) on a spring whose 3 spring constant is 4.5 kn/m ( N/m). (Assume a Hooke's law spring.) When A strikes B, they stick together. Determine the maximum displacement of B from its original position. Find the energy lost during impact.

3 13. A block whose mass is.00 kg is at rest on the inclined plane shown. The coefficients of static and kinetic friction are 0.60 and 0.550, respectively. A bullet, traveling parallel to the plane in the direction of the arrow, strikes the block and passes through it, emerging with 1/3 of its original velocity. The mass of the bullet is 4.00 g and its velocity originally is 350 m/s. Does the block reach the bottom of the plane? Show why or why not. If it does, calculate its velocity at the bottom. If it does not, calculate its final displacement from its initial position. 14. A block of mass m slides from rest down a frictionless circular track as shown. It strikes a second block of mass 3m exactly at the bottom, and they stick together. Find the largest value of, the angular position after the collision. When is at its largest value as in, find the normal force the track exerts on the block. 15. Car A is traveling north as shown at 45.0 mi/hr. Car B is traveling in the direction given at 35.0 mi/hr. The mass of A is 50 kg, and the mass of B is 350 kg. They collide in a completely inelastic collision. Find the velocity (magnitude and direction) just after the collision. If the coefficient of kinetic friction is 0.55, how far will the wreckage slide? 16. Mass 1 rests on a frictionless inclined plane supported by a Hooke's law spring of constant k = 111. N/m. Mass slides down the plane and collides with 1 and sticks to it. If the maximum compression of the spring is 35.0 cm from its length just before impact, calculate the distance moves from rest before colliding with 1. M = 1.5 kg, M = 0.75 kg Car A is traveling east at 45.0 mi/hr. Car B is traveling north at 55.0 mi/hr. The masses of the cars are M A = 00 kg and M B = 3750 kg. They collide and the wreckage sticks together. Calculate the direction the wreckage moves (angle in the diagram). If the coefficient of friction is taken to be 0.65, how far does the wreckage slide (in feet)?

4 18. A bullet of mass 8 g is fired into a block of mass 1.35 kg. The velocity of the bullet is 75 m/s and it stops inside the block. The block M is part of a pendulum of pendulum length L =.00 m as shown. Calculate the maximum value of the angle that the pendulum achieves. Calculate the tension in the string when =. max/ 19. Find the y coordinate of the center of mass of the object shown. The object is a sheet metal of density and thickness t. Find the mass of this object. 0. Calculate the x coordinate of the center of mass of object shown. The object is the lined region. It is a sheet of thickness t and density. 1. The cross-hatched figure shown has a thickness t, a uniform density mass and it is bounded by the horizontal x axis, the line x = x and the curve y = x. o Calculate the x coordinate of the center of mass of the figure shown. (Express your answer as a number times x o.) Calculate the y coordinate of the center of mass of the same object. (Express your answer in terms of x.) o. A steady stream of water strikes a scale and bounces off as shown. The magnitude of the velocity of the water stream is unchanged. The stream of water flows at 7.50 kg/s with a speed of.3 m/s. What does the scale read (in N)? A bucket that can hold exactly 6.00 kg of water is placed on the scale. The same hose is directed into this bucket in the same direction. What does the scale read at the instant the bucket is full?

5 3. A mass m 1 is attached to the ceiling by a Hooke's law spring with spring constant k. It is initially at rest. A second mass, m, is given an initial velocity v o. It strikes m 1 and sticks to it after m has moved a distance h vertically. Calculate the maximum vertical distance traveled by m after the collision. (Numerical answer.) 4. Convert.30 horsepower into watts. Convert 65 foot-pounds into joules. (You will have to work out the conversion factor from the definitions and the ones given.) A car is moving at 50.0 mi/hr. If it has a mass of 1750 kg, calculate its linear momentum in SI units (metric). A baseball is thrown at 45.0 m/s. The batter hits it directly back towards the pitcher at 65.0 m/s. If the mass is kg, calculate the magnitude of the impulse given to the ball. A car of mass 100 kg is traveling at 60.0 mi/hr. The brakes are put on and it skids to a stop in 418 ft. Find the power being dissipated as heat (in watts) after is has skidded 400 ft, assuming constant deceleration. 5. The object shown has a uniform density, and a constant thickness t in the z direction. Calculate the x-coordinate of the center of mass in terms of A, B and numbers, as needed. 6. When mass A is placed on a long Hooke's law spring and allowed to come to rest, the spring is squeezed 1.0 cm from its equilibrium length. A mass B is given an initial velocity v o = downwards at a point 3.75 m above A. When B strikes A they stick together. Use energy methods where appropriate. How much kinetic energy is lost in the collision of A and B (numerical answer)? Calculate the maximum displacement of the spring from the point labeled y = 0 after the collision (numerical answer). m A = 4.1, m B = 1.75 kg, v o = 6.0 m/s

6 7. Calculate the conversion identity between joules and ft-pounds. We find that the potential energy of an object in the Earth's gravity is given by U = -(A/R) where A is a constant and R the distance to the center of the Earth. Calculate the force in the R direction associated with this potential energy. Calculate the energy (in joules) represented by a power of 1735 kilowatts acting for 35 seconds. A car skids with wheels locked for 165 ft before stopping. The mass of the car is 500 kg. The coefficients of friction are s = and k = Calculate the kinetic energy of the car (in joules) when it just starts to skid. A car of mass 500 kg is traveling at 75.0 mi/hr. The brakes are put on and it skids with the wheels locked to a stop in 35 ft. Find the power being dissipated as heat (in watts) after it has skidded 16.5 ft Given a potential energy function U = Ax + Bx + Cx where A, B and C are constants. Calculate the force described by this function as a function of x. 3 A spring follows the force law F = -kx. If k = 1.5 N/m, find the energy stored in the spring when it is compressed by 0.1 m. A block slides with constant velocity on an inclined plane at an angle of 7. Find the coefficient of kinetic friction. Convert 153 J to ft-lbs. A 1.5 kg rock is dropped from 15.7 m above the surface of a lake. While in the lake the rock falls at a constant velocity of 5.00 m/s. Find the energy lost to heat when the rock reaches 10.0 m below the surface of the lake. 9. On the loop-the-loop shown a block of mass m slides without friction. The block starts with a speed v o at a height of 6R from the bottom of the loop. (R is the radius of the loop.) v is given by v = 3 R g. o Find the velocity of the block at point A. Find the normal force on the block at point B. o 30. A block of mass m is launched in the frictionless circular loop-the-loop shown. Given that the spring constant is k, the radius R and the mass m, find the distance the spring must be compressed before launch if the normal force on the block at top of the loop is mg. 31. A massless Hooke's Law spring has a unstretched length of.5 m. When a 10.0 kg mass is placed on it, and slowly lowered until the mass is at rest, the spring is squeezed to.00 m length. The same 10.0 kg mass is dropped from a height of 6.5 m above the spring. What is the maximum value of the compression of the spring? (Numerical answer.) What is the velocity of the block after the spring has been compressed 0.5 m. (Numerical answer.)

7 3. A car weighing 000 pounds is moving at 30.0 mi/hr. Calculate its linear momentum in SI units. Calculate the x-coordinate of the center of mass of the system shown. A golf ball at rest is struck and leaves with a velocity of 110 m/s. If the mass of the ball is kg, calculate the impulse applied to the ball. A car of mass 1000 kg is traveling west at 15.0 m/s. A second car of mass 1500 kg is traveling east at 0.0 m/s. If they collide head-on and stick together, calculate the kinetic energy lost in the collision. Calculate the kinetic energy, in joules, of the car in. 33. A mass of m =.55 kg is suspended at rest by a massless spring as shown. When the mass is attached to the unstretched spring it stretches the spring by 1.45 cm. A bullet is fired from below so that it strikes the mass and remains in it. If the mass of the bullet is 3.0 grams and the velocity of the bullet when it enters the block is 85 m/s, calculate the maximum height of the block from y = 0 in the drawing. 34. Calculate the y-coordinate of the center of mass of the object shown. The object is a uniform sheet of thickness t and density. 35. Calculate the linear momentum of a 17.0 kg object moving at 15. m/s. A golf ball of mass kg is given a velocity of 85.0 m/s, starting at rest. Calculate the impulse applied to the ball. Calculate the kinetic energy, in Joules, of a car of mass 1500 kg moving at 60 mi/hr. Convert 875 ft pounds into the proper SI unit. Find the center of mass of the system shown. m =.75 kg, m = 1.5 kg, m = 0.75 kg. 1 3

8 36. In a ballistic pendulum experiment a bullet is shot into a mass M and remains inside of it. If the pendulum swings to a maximum angle of = 17.0, calculate the initial velocity of the bullet. M =.50 kg, m = 0.1 kg, L = 1.10 m. bullet 37. A car has a kinetic energy of 700,000 J. If its mass is 1500 kg, calculate its linear momentum. Find the x coordinate of the center of mass of the system shown. A tennis ball of m = kg is at rest when struck and given a velocity of 55.0 m/s. Calculate the impulse applied to the ball. A baseball of mass = 0.50 kg is moving at 60.0 m/s. It is struck by a bat and now moves at 75.0 m/s in exactly the opposite direction. If the time of collision with the bat is 0.00 s, calculate the average force applied to the ball. Calculate the linear momentum of a car whose weight is 500 pounds if it is traveling at 100 ft/s. 38. The object shown is a sheet of material of density 3 bounded by the line y = 0, x = x o, and the curve y = ax. The thickness perpendicular to the paper varies according to the relationship t = Cx. C, a and x o are constants. Calculate the x coordinate of the center of mass of this object. Express your answer in terms of numbers, x o, c and a as needed. 39. On the frictionless, horizontal air track we have used in class, are two carts. Cart 1 has an initial velocity of v o, and cart is initially at rest. They collide in a completely elastic collision. Take 3M 1 = M (M 1 = b M ). For full credit on this problem yout6o MUST derive any equation used starting with first principles, i.e., conservation laws, as appropriate. Calculate the final velocities of M 1 and M after the collision in terms of numbers and v o. How much energy is lost in this collision?

9 40. The spring shown has an unstretched length of 1.5 m. When mass m is placed on it, and all oscillations have died out, its length is 1.10 m. Block m 1 is dropped on m from a height of 6.5 m. It lands on top of m and stays there. m = 1.10 kg; m = 3.7 kg 1 Calculate the length of the spring at maximum compression. Determine the energy lost to heat. Find the length of the spring with both masses on it after all oscillations have died out. 41. A golf ball, of mass 100 g, initially at rest, is given a velocity of 47.0 m/s. Calculate the magnitude of the impulse delivered to the ball. For the system shown, and the coordinate system given, calculate the x-coordinate of the center of mass. m 1 = 4.75 kg m = 3.0 kg m = 8.70 kg 3 Convert 85 ft pounds into the proper SI units. 5 A 1000 kg car has a kinetic energy of J. Calculate the magnitude of its linear momentum. A mass m on the frictionless air track is moving with speed v o. It collides with, and stick to, a mass 3m. Calculate the fraction of the original kinetic energy that is lost as heat. 4. The object shown is bounded by the horizontal x-axis, the line 3 x = x o, and the curve x = ay. It has a thickness t, and a density. Calculate the x-coordinate of the center of mass of this object. 43. A bullet strikes the block M when the string is vertical and the block is motionless. The initial velocity of the bullet is v o, and it emerges from the block at a speed of 1/3 v o. The block is observed to swing to a maximum angle of. M = 1.7 kg L = 1.00 m m = kg Calculate v o. Calculate the mechanical energy lost to heat.

10 44. The spring shown has an unstretched length of m. When m is placed upon it, and all oscillation has died out, the length is m. The mass m is dropped 3.57 m and lands and stays on top of m. m 1 = 1.50 kg m =.50 kg 1 Calculate the length of the spring at its maximum compression. Calculate the energy lost to heat. 45. Calculate the x-coordinate of the center of mass of the spheres shown. A golf ball at rest with a mass of 110 g is acted on with an impulse of 5.0 N s. What is the velocity given to the ball? 6 A car with a mass of 150 kg has a kinetic energy of J. Determine its linear momentum. On a frictionless air track two carts, each with a mass of 1.50 kg, are moving from opposite directions towards each other. Each cart has a speed of 0.30 m/s. They collide and stick together. How much mechanical energy is lost to heat? Convert 4750 J to ft-pounds of energy. 46. In the diagram shown the force, F, is at 15.0 above the horizontal. Calculate the work done by gravity when the block is moved 1.35 m down the plane. What is the work done by the force (F) when the block is moved.35 m down the plane. Find the work done by friction when the block is moved.35 m down the plane. m = 3.0 kg; = 0.75; = 0.55; F = 35.0 N s k 47. On a frictionless horizontal air track there is a collision between a mass (A) with an initial velocity of v o = 1.85 m/s and a mass (B) which is at rest. Mass A is three times the mass of B. Find the velocities of the two masses after a perfect elastic collision. Be sure to indicate which velocity is the final velocity of mass A and which is the final velocity of mass B. Show clearly how you arrive at this conclusion.

11 48. In the drawing, the tabletop is frictionless. Mass 1 is launched by the spring. It strikes and sticks to mass. The two masses swing on a string in a vertical circle. At the exact top the tension in the string is 155 N. Calculate the distance the spring must be squeezed initially to achieve the results described. m 1 = 1.50 kg m = kg 4 k = N/m L =.75 m 49. A car weighing 635 lbs is moving at 65.0 mi/hr. Find the momentum of the car in SI units. For the car in, calculate the energy in Joules. A ball of mass 1.0 kg moving with v = 5.70 m/s hits the wall at an angle of 30 and bounces off. If the collision lasted for 0.01 s, what is the average force exerted on the wall? Find the center of mass for the system shown. An object of M = 3.50 kg is traveling at 1.0 m/s in the positive x direction. A second object of m = 1.5 kg is traveling at 4.5 m/s in the negative direction. The two masses collide in a completely inelastic collision (they stick together). Calculate the energy lost. 50. Find the work done by the force on the path: CA BC AB Is a conservative force?

12 51. A bullet (mass m) hits a pendulum (mass M and length L) and remains in the pendulum. What would the initial velocity v o of the bullet have to be in order for the pendulum to barely complete the circle if: L is a string; L is a massless rod. 5. The block, as shown in the drawing, is launched up the incline with an initial velocity of v o = 4.00 m/s. A very weak spring with k = 0.0 N/m is placed on the top of the incline a distance l.00 m from the block. The block hits the spring, compresses it and stops. (No energy is lost when the block hits the spring.) What is the maximum compression of the spring? The spring launches the block back. What will be the velocity of the block when it is again 1.00 m from the spring (at its initial position)? 53. Find the x-coordinate of the center of mass of the system shown. = 0.4; = 0.30; = 1.00 m; m =.00 kg; k = 0 N/m k s Find the linear momentum of Karl Malone (m = 10 kg) moving at a speed of 6.50 m/s. Find the kinetic energy of Karl Malone as described in. A car that weighs 00 pounds and is moving at 65.0 mi/hr slams on the brakes and comes to rest after 100 feet. Calculate the energy lost to friction (in ft-pounds). A.00 kg mass is pushed up the incline at a constant speed for a distance of 4.00 m. Find the work done on the block by friction if the coefficients of friction are s = 0.78 and = k 54. Calculate the x-coordinate of the center of mass of the object shown. It is a thin sheet of metal of uniform density in the shape given. If the density depends on x as = ox, now recalculate the x-coordinate of the center of mass.

13 55. Calculate the linear momentum, in British units, of Karl Malone (weight 65 pounds) moving at a speed of 15.0 ft/s. Determine the kinetic energy, in joules, of Karl Malone moving as in. In the drawing, m 1 = 7.5 kg, m = 6.00 kg, m 3 = 4.5 kg. Calculate the x-coordinate of the center of mass of this system. Convert 787 ft pounds to joules. A car weighing 300 pounds is moving at 65.0 mi/hr. Calculate its kinetic energy in joules. 56. A massless Hooke's Law spring has an unsqueezed, unstretched length of 1.90 m. When a 1.0 kg mass is placed on it, and slowly lowered until the mass is at rest, the spring is squeezed to 1.75 m in length. Now, with the 1.0 kg mass removed, drop a 10.0 kg mass from a height of 4.5 m above the spring. What is the maximum value of the compression of the spring from its unsqueezed, unstretched length? (Numerical answer.) What is the velocity of the block after the spring has been compressed m from its unsqueezed length. (Numerical answer.) 57. Rock 1 with a mass of 3.5 kg is dropped from the top of the Park Building. At the same instant, rock with a mass of.75 kg, is projected upward with an initial velocity of V o = 1.0 m/s. Find the location above the ground of the center of mass of the system when rock reaches its maximum height. What is the velocity (with sign) of the center of mass the instant rock 1 hits the ground? 58. On a frictionless horizontal air track a cart of mass m and another of mass 3m collide. Initially the cart of mass 3m has a velocity of v o = 1.5 m/s and the smaller cart has an initial velocity of zero. Take m = 1.75 kg. If they collide in a completely inelastic collision, calculate the final velocity. How much mechanical energy is lost in this collision? If they collide in a completely elastic collision, calculate the final velocity of each cart. 59. A mass M 1 rests with zero velocity on an ideal spring. The spring constant is 140 N/m. M 1 has the value.75 kg and the spring has a length of 1.5 m when M 1 is at rest on it. A mass M = 4.75 kg falls from rest a distance of m before striking number 1 and sticking to it. What is the length of the spring when it has its maximum compression? What is the length of the spring when all motion has ceased after the impact? 60. A mass m is attached to massless string that is wrapped around a solid cylinder that is on a frictionless axle. The system starts at rest. Mass m = 1.34 kg, cylinder mass M =.37 kg and radius is 0.15 m, and the coefficients of friction are 0.45 and Calculate the speed of mass m after it has moved 1.5 m down the plane. Calculate the acceleration of mass m down the plane.

14 61. Calculate the moment of inertia of a sphere of mass 7.35 kg and diameter of 0.13 m. The Earth rotates on its axis once every 4.0 hrs. Calculate its angular velocity (in rad/s). The speed of a point on the rim of a rotating wheel is 40.0 m/s. The wheel has a moment of inertia of 0.50 km m and a radius of m. Calculate its rotational kinetic energy. Given a sphere of radius R o and mass M, calculate the moment of inertia about an axis parallel to the diameter at a distance R /3 from the center. o The speed of a point on the equator of a rotating sphere is 17.0 m/s. The sphere has a mass of 1.5 kg and a diameter of 4.5 cm. Calculate its rotational kinetic energy. 6 Given a uniform cylinder of mass 5.0 kg, radius 5.00 cm and length 0.0 cm, calculate its mass moment of inertia for rotation about an axis parallel to the long axis of the cylinder and halfway between the center and the outside of the cylinder. A wheel starting from rest accelerates uniformly. If it undergoes 475 complete revolutions in 6 min 35 s, what is the angular acceleration? A sphere of mass 8.00 kg and diameter 15.0 cm, is rolling without sliding. If its translational velocity is 17.0 m/s, find its rotational kinetic energy. For the sphere in problem, calculate its total kinetic energy. For the sphere in, calculate its angular momentum about the center of mass. 63. Given the Atwood machine shown. Mass 1 is 0.5 kg, and mass is 1.0 kg. The moment of inertia of the pulley is 0.5 kg m. The radius of the pulley is 0.1 m. Calculate the acceleration of the system. The rope does not slip on the pulley. Calculate the tension in each cord. 64. Two cars approach an intersection as shown. Car 1 weighs 4500 lbs and has a speed of 55.0 mi/hr. Car weighs 3750 pounds with a speed of 60.0 mi/hr. They collide in a completely inelastic collision at the intersection. Calculate the direction the wreckage moves after the collision. Express this as an angle measured counterclockwise from the positive x-axis. If the coefficient of kinetic friction is 0.55 for the tires on this road, and the wheels of the car are locked (not rolling), calculate the distance the wreckage slides from the collision point.

15 65. A small mass m 1 slides in a completely frictionless spherical bowl. m 1 starts at rest at height h = ½ R above the bottom of the bowl. When it reaches the bottom of the bowl it strikes a mass m, where m = 3m, in a completely elastic collision. 1 Calculate the height that m moves up the bowl after the collision (measured vertically from the bottom of the bowl). Calculate the height that m 1 moves up the bowl after the collision. 66. A wheel slows from 157 rev/min to 118 rev/min in a time of 43.0 s. Find its angular acceleration. A wheel starts at rest, with an angular acceleration of.50 rad/s. Through what angle has it turned at the end of 100 seconds? Calculate the moment of inertia for the object shown for rotation about an axis through A, 0.50 m from the right end. Calculate the moment of inertia of a sphere of mass m and radius R, for rotation about an axis R/ from the center of the sphere. The angular speed of a bicycle wheel is 5 rad/s. If the radius of the wheel is 0.50 m, find the velocity of a point on the rim. 67. A wheel with radius R = is rolling without sliding. The velocity of the center of mass is 7.5 m/s. Find the angular velocity of the wheel. A wheel on an axle has an angular acceleration of -.10 r/s, and an initial angular velocity of 17.5 r/s. How many revolutions does it undergo in 5.0 s? Two spheres touch each other. Each has a mass of 5.00 kg and a radius of 4.5 cm. Calculate the moment of inertia for rotation about an axis through the point of contact and tangent to the two spheres, as shown. (f) A rotating wheel with initial angular velocity of 175 r/s coasts to a stop in 115 s. What is its TOTAL angular displacement during this time? Calculate the TOTAL kinetic energy of a bowling ball that is rolling without sliding. The ball weighs 16.0 pounds, its radius is 4.00 inches and the translational velocity of its center-of-mass is 35.0 ft/s. An object weighing 19.6 N on Earth is given a velocity on the moon of 5.0 m/s. Calculate its kinetic energy on the moon.

16 68. A block of mass M is on frictionless table. It is attached to the wall with a Hooke's law spring. The system is initially at rest at the equilibrium position of the spring. A bullet with mass m is fired into the block and its stops there. If the initial position of the block is x = 0 and the maximum value of the displacement from equilibrium is -6.5 cm, calculate the initial velocity of the bullet. 69. On the frictionless air track, a completely elastic collision occurs between m 1 and m. Initially, m 1 is moving at 1.35 m/s, and m (m = 4 m 1) is at rest. Calculate the final velocities of m 1 and m with proper signs. 70. A small mass, 3m, slides without friction on the inside of a spherical bowl. It starts at rest at the position shown, and collides with a mass m at the bottom. (3 m has three times the mass of m.) The mass at the bottom is initially at rest. The two masses stick together. Calculate the maximum height, h, that the combined mass reaches. Express this as a fraction of R. Calculate the maximum angle,, that the combined mass reaches. Calculate the mechanical energy lost as heat. Express this as mgr times a number. Remember m is the mass of the small block. 71. The length of the spring when unstretched and unsqueezed is 1.00 m Block A is placed on the plane and the length of the spring is 0.75 m. Block B is launched down the plane with an initial speed v o = 4.00 m/s. The distance d is m. Calculate the speed of block B just before its impact with block A. If block B strikes block A in a completely inelastic collision, what is the energy lost in the collision? Find the length of the spring at its maximum compression. k = 0.40; s = 0.50; m A = 1.75 kg; m B =.45 kg; k = N/m 7. In the drawing shown, A is a cylinder on a fixed axle. The mass B falls freely. The pulley is massless and frictionless. Determine the speed of B after B has fallen 1.65 m from rest using energy methods. Calculate the downward acceleration of B in this system using nd Law methods. m A = 46.0 kg; m B = 9.5 kg; R A = 4.50 cm

17 73. Two pendulums of mass m and m and length are attached to the same point by strings. One of pendulums is moved such that it makes a 90 angle with the vertical and released. Describe the motion (velocities) after collision and calculate the maximum angle (angles are + to the right and to the left) each of these will move with the vertical after the collision if the collision is completely inelastic; perfectly elastic. Find the fraction of the initial energy lost in the collision for both and. 74. On the frictionless air track a collision is arranged between two carts. The collision is completely elastic. The initial speed of each cart is v o with direction, as shown. Cart 1 has a mass of 3 times the mass of cart. Calculate the final velocities of carts 1 and using the positive direction as shown. 75. The incline shown is frictionless. The mass M 1 (M 1 = 4.75 kg) is attached to a rope that is wrapped around a cylinder. The cylinder is on a fixed axle. The cylinder has a mass M (M = 3.0 kg) and a radius R of 6.05 cm. The block is released from rest. Calculate its acceleration down the incline. USE ENERGY METHODS: The block is released from rest. Calculate its velocity after it has traveled 1.39 m along the incline. 76. Calculate the moment of inertia of an uniform cylinder of radius 0.35 m, length 0.75 m and a mass of 4.75 kg. The cylinder rotates about an axis through the center of mass and parallel to the cylinder axis. Given a sphere of radius R o and mass M, find its inertia for rotation about an axis that is a distance R /3 from the center of the sphere. o Find the kinetic energy of a sphere (mass = kg, radius = 0.60 m) rotating at 800 rev/minute about an axis through its center. A rotating wheel slows from 150 rev/minute to a stop in 57.0 sec. Determine its angular acceleration assuming it is uniform. A wheel starts at rest and accelerates uniformly with an angular acceleration of 3.30 rad/s. Calculate its angular velocity after 15.0 s. 77. A mass m is attached to massless string that is wrapped around a solid cylinder that is on a frictionless axle. The system starts at rest. Mass m = 0.97 kg, cylinder mass M = 1.10 kg and radius is 0.0 m, and the coefficients of friction are = 0.45 and = s k Calculate the speed of mass m after it has moved 1.37 m down the plane. (Use energy methods.) Calculate the acceleration of mass m down the plane.

18 78. A small mass m 1 slides in a completely frictionless spherical bowl. m 1 starts at rest at height h = b R above the bottom of the bowl. When it reaches the bottom of the bowl it strikes a mass m, where m = 4m 1, in a completely elastic collision. The positive direction is shown on the drawing. Calculate the velocity of m after the collision. Calculate the velocity of m after the collision. 1 DO NOT USE canned equations from your card for the elastic collision. Use momentum and energy conservation properly to get the velocities. 79. A block, m 1 =.00 kg, is launched by a spring that has a spring constant of 50.0 N/m and an unsqueezed unstretched length of 3.00 m. The block travels a distance d = 1.0 m and hits a second block, m = 4.00 kg, which is at rest. The blocks stick together in a completely inelastic collision. The coefficients of friction are s = 0.75 and k = Calculate how far you would have to compress the spring to move the -mass system m past the collision point before it stops. 80. A yo-yo has a mass of 8 kg and a moment of inertia of 0.01 kg m. The inner radius is 1 cm (0.01 m). Calculate the rotational velocity after the yo-yo has fallen 1 m from rest. Calculate the linear velocity after it has fallen 1 m. The string is longer than 1 m. 81. A cylinder of length 1.0 ft and radius 1.0 in weighs 6.0 lb. Two cords are wrapped around the cylinder, one near each end, and the cord ends are attached to hooks on the ceiling. The cylinder is held horizontally with the two cords exactly vertical and is then released (see figure). Find the tension in the cords as they unwind, and determine the linear acceleration of the cylinder as it falls. 8. A sphere of radius R and mass M has a long string wrapped around its equator. The string is suspended from a fixed point, and the sphere is allowed to fall. Calculate the downward acceleration of the sphere. This should be expressed as a fraction of g. No other quantities are needed. Calculate the angular acceleration of the sphere. Express this as a fraction times g/r. Calculate the tension in the string as a fraction of the weight of the sphere.

19 83. Given an object with the shape of the cross hatched area, thickness t and a density. Constants a and b have appropriate units. Calculate the mass moment of inertia for this object for rotation about the y-axis. Express your answer in terms of a, b and M (the mass of the object) and numbers. 84. Given a right circular cylinder of total mass M whose density varies with radius as = o(1 - r). and o are constants. The density is o/ at the outer radius, which is R o. The cylinder has a length L Find its mass moment of inertia about an axis through the center of mass, and parallel to the long axis of the cylinder, as shown in the figure. Express your answer in terms of M, R and numbers only. o 85. Captain Kirk and the crew of the Enterprise find an artificial planet built by the Klingons. It is in the shape of a right circular cylinder whose radius R o is equal to its height. It rotates about the axis of the 11 cylinder. They measure the mass of this planetoid to be kg. They discover that the density varies with distance from the axis of rotation. Assume the density variation to be of the form = (1 - r). o Find a general expression for the moment of inertia in terms of M, R and. If the radius is 00 meters, find the average density. 3 If the density at R = 0 is 5500 kg/m, find the value of. Using the above, find the density at the outer surface of the cylinder. 86. The cross hatched figure shown has a density and thickness t. The quantities b and a are positive constants. Calculate the moment of inertia of this object for rotation about the y-axis. Be sure to put your answer in proper finished form. 87. A small sphere of radius r rolls without sliding starting from rest at the top of a large sphere of radius R. Calculate the value of the angle at which the sphere is no longer touching the large sphere. (Numerical value. You do have enough information.) 88. The cross-hatched area shown in an object bounded by a curve given by y = A - bx and the x-axis. It is a plate of material of density and thickness t. Calculate the y coordinate of its center of mass. Set up the integral only for part. Calculate its moment of inertia for rotation about the y-axis. Express in proper form. Calculate its moment of inertia for rotation about the line x = 10.

20 89. A sphere of mass M 1 and radius R is supported above a table and rotates on a fixed axis through its center. A string is wrapped around its equator, and is attached to M after passing over a frictionless, massless pulley as shown. Calculate the downward acceleration of M in terms of M 1, M, g, R and as needed. Calculate in terms of the same quantities. If M 1 = 6.75 kg, M = 1.15 kg and R = 5.00 cm, calculate the distance M falls in 3.5 s assuming it starts from rest. 90. A wheel with an axle has a string wrapped around and tied to the axle. A mass M is attached to the string. If the moment of inertia of the wheel plus axle is kg m, the mass M is 1.75 kg, the radius of the axle is.00 cm, and the angular velocity of the wheel is 6 rad/s at t = 0 (in the direction of the arrow). Calculate the linear acceleration of M while it is moving. Find the vertical distance M moves before stopping. The string is as long as is needed so that M never strikes the axle. The total number of revolutions of the wheel until M reaches its highest point. 91. A weight is attached to a massless string. The string is wrapped around a cylinder on a frictionless axle whose radius is 5.00 cm and whose mass M is 1.00 kg. The weight has a mass m of 4.75 kg. The coefficients of friction between the weight and plane are and 0.45 and = S K Calculate the linear acceleration of the weight. Calculate the angular acceleration of the cylinder. Calculate the tension in the string. 9. A uniform steel rod of length 1.0 m and mass of 6.40 kg has attached to each end a small ball of mass 1.06 kg. The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint. At a certain instant it is observed to be rotating with an angular speed of 39.0 rev/s. Because of axle friction it comes to rest 3.0 s later. Assume a constant frictional torque. Find the angular acceleration. Calculate the retarding torque exerted by axle friction. Determine the total work done by the axle friction. Calculate the number of revolutions executed during the 3.0 s. 93. A string is wrapped around a cylinder of radius R. The cylinder is on a horizontal axis without friction. Attached to a string is a mass m. The system is released from rest, and the mass falls a distance h. Calculate the velocity V of the mass after it falls the distance h. Calculate the angular velocity of the cylinder when the mass has fallen the distance h. How long does it take for the mass to fall the distance h?

21 94. A wheel, in the form of a disc of mass = 3.0 kg, radius =.0 cm, and thickness = 4.00 cm, is attached to a car and rolls without sliding. The car has a speed of 17.0 m/s. Calculate the kinetic energy of the wheel. Two cylinders of mass 16.0 kg and radius 4.00 cm are attached as shown. Calculate the mass moment of inertia for this system for rotation about an axis perpendicular to the paper, and through the center of the cylinder (A). If the car in slows to a stop with a deceleration = m/s, calculate how many revolutions of the wheel occur between the beginning of deceleration from 17.0 m/s and stopping. A sphere of radius m and mass kg, rolls down an inclined plane. At the bottom its center of mass is.00 m lower than its starting point. Calculate its angular velocity. A grinding wheel in the shape of a disc of thickness 1.5 inches and radius 14.0 inches, is slowed by friction with the tool being ground. The initial speed is 33.0 rad/s. The weight of the wheel is 65.0 pounds. If it slows to a full stop in 47.0 s, calculate the frictional force exerted by the tool. 4 (f) Calculate the kinetic energy of the rotation of the Earth. Take the mass as 6 10 kg and its radius as 6000 m. It rotates once in 4 hours. Approximate the Earth as a sphere. 95. A yo-yo is mounted on a frictionless plane as shown. The string is fixed at the top. The numerical values of the appropriate quantities are shown. R is the outer radius of the yo-yo, r is the radius of the shaft about which the string is wrapped. Calculate the acceleration of the yo-yo. Find the tension in the string. M = 1.4 kg r = cm -4 I = kg m R = 3.00 cm 96. A cylinder rolls without slipping along an inclined plane. It starts at rest. The mass and radius are M = 1. kg and R = m. Find the translational velocity of the center of the mass of the cylinder after it has traveled.10 m down the plane. What is the angular velocity of the cylinder after it has traveled.10 m down the plane? Calculate the time the cylinder takes to travel.10 m down the plane. 97. Four cylinders are touching each in the arrangement shown. Calculate the moment of inertia for rotation about the axis A, at the exact center, and perpendicular to the paper. Take the mass of each cylinder as M, and the radius of each as R. A wheel is accelerated from rest at an angular acceleration of 3.75 r/s. Calculate the total angular displacement after 8.90 s. For the wheel in, calculate the magnitude of the angular velocity when the total angular displacement is 18.0 rad. A car is traveling at 60 mi/hr. If the wheels have a radius of 15 inches, what is the magnitude of their angular velocity in rad/s? A bowling ball has a weight of 16.0 pounds. If it is rolling without sliding and the radius is 4.00 inches, calculate the rotational kinetic energy if its translational speed is 30.0 ft/s. (f) Calculate magnitude of the angular momentum of a baseball of mass kg if it rotates at 1600 RPM (revolutions per minute). Assume a uniform mass distribution. Take the radius as 4.00 cm.

22 98. A small mass, 3m, slides without friction on the inside of a spherical bowl. It starts at rest at the position shown, and collides with a mass m at the bottom. (3 m has three times the mass of m.) The mass at the bottom is initially at rest. The two masses stick together. Calculate the maximum height, h, that the combined mass reaches. Express this as a fraction of R. Calculate the maximum angle,, that the combined mass reaches. Calculate the mechanical energy lost as heat. Express this as mgr times a number. Remember m is the mass of the small block. 99. The crosshatched area shown is a board with thickness t and density. Calculate the moment of inertia of this object for rotation about the y-axis. Your answer should be expressed in appropriate final form using x o as the only dimension of the object appearing in the answer, with M o, the mass of the object, and numbers Two uniform solid spheres are connected and touch as shown. If the mass of each sphere is M and the radius R, calculate the moment of inertia for rotation about the axis A. A solid cylinder of mass 3.7 kg and radius.50 cm has a string wrapped around it. The string is pulled with a constant force F of 5.60 N. Find the angular velocity after 7.00 seconds have passed. (f) For the system in, calculate the angular velocity after the string has been pulled a distance of 3.00 m. A wheel of radius 14.0 inches rolls without slipping for exactly one mile. Calculate the total angular displacement (in radians). A sphere that weighs 16.1 pounds and has a radius of 4.00 inches rotates on an axis through its center of mass at an angular velocity of 3.7 rad/s. It comes to a stop after 7.5 complete revolutions. Assuming that the acceleration is constant, what is the magnitude of the torque on the sphere? If the distance between the center of the earth and the center of the moon is 00,000 miles (not quite a true value), and the mass of the moon is 1.00% of the mass of the earth (close to the true value), what is the distance from the center of the earth to the center of mass of the earth-moon system? 101. In the drawing shown, the crosshatched area is bounded by the x and y axes and the curve y = B - Ax. Consider this to be a sheet of metal of density and thickness t. Find the x coordinate of the center of mass of this object. (Note that x o and B are related, so express the answer using a number and x o.) Calculate the moment of inertia from this object for rotation about the y axis. (Note that x o and B are related, so express the answer using a number, M and x.) o

23 10. A car with wheels of 6.0 inches diameter is traveling at 55.0 mph. Calculate the angular speed of the wheels in rad/s. Four, solid, uniform spheres, as shown, are touching each other. Find the moment of inertia for this system about axis A which is at the exact center and perpendicular to the paper. Each sphere has a mass m and radius R. A bowling ball with a mass of 15.0 pounds and radius 5.00 inches is rolling without slipping at 30.0 feet/second. Calculate the TOTAL kinetic energy of the ball in Joules. A wheel attached to an axle has an angular acceleration of -.10 rad/s and an initial angular velocity of 1.5 rad/s. How many revolutions does it make in 5.0 s? A wheel of M = 45.0 kg and R = 15.0 cm is rotating at 67.0 rad/s. It slows down and stops in 0.0 s. Find the torque (assumed constant) applied to it. Treat the wheel as a disc. (f) Calculate the angular momentum of a cylinder with R = 10.0 cm and M = 5.0 kg which is rotating at = 10.0 rad/s, about the axis through the center of mass and parallel to the sides of the cylinder The object shown is a thin plate of thickness t and density and has a shape of. Find the x center of mass. Find the y center of mass. Calculate the moment of inertia for rotation about the y axis A block of mass M = 5.00 kg is supported by two string which are wrapped around a cylinder of m = 40.0 kg and R = 0.0 cm. The cylinder is on a horizontal axis without friction. The system is released from rest. Find the velocity v of the block after it falls a distance h = 5.00 m. What is the acceleration of the block? How long does it take the block to fall the distance h = 5.00 m?

24 Data: Use these constants (where it states for example, 1 ft, the 1 is exact for significant figure purposes). 1 ft = 1 in (exact) 1 m = 3.8 ft 1 mile = 580 ft (exact) 1 hour = 3600 sec = 60 min (exact) 1 day = 4 hr (exact) g earth = 9.80 m/s = 3. ft/s g moon = 1.67 m/s = 5.48 ft/s 1 year = days 1 kg = slug 1 N = 0.5 pound 1 horsepower = 550 ft pounds/s (exact)