Energy
1975 B1 - Class
A 2kilogram block is released from rest at the top of a curved incline in the shape of a quarter of a circle of radius R. The block then slides onto a horizontal plane where it finally comes to rest 8 meters from the beginning of the plane. The curved incline is frictionless, but there is an 8newton force of friction on the block while it slides horizontally. Assume g = 10 meters per second2.
a. Determine the magnitude of the acceleration of the block while it slides along the horizontal plane.
b. What time elapses while the block is sliding horizontally?
c. Calculate the radius of the incline in meters.
1997 B1 - Homework
A 0.20 kg object moves along a straight line. The net force acting on the object varies with the object's displacement as shown in the graph above. The object starts from rest at displacement x = 0 and time t = 0 and is displaced a distance of 20 m. Determine each of the following.
a. The acceleration of the particle when its displacement x is 6 m.
b. The time taken for the object to be displaced the first 12 m.
c. The amount of work done by the net force in displacing the object the first 12 m.
d. The speed of the object at displacement x = 12 m.
e. The final speed of the object at displacement x = 20 m.
f. The change in the momentum of the object as it is displaced from x = 12 m to x = 20 m
1979 B1 - Homework
From the top of a cliff 80 meters high, a ball of mass 0.4 kilogram is launched horizontally with a velocity of 30 meters per second at time t = 0 as shown above. The potential energy of the ball is zero at the bottom of the cliff. Use g = 10 meters per second squared. a. Calculate the potential, kinetic, and focal energies of the ball at time t = O.
b. On the axes below, sketch and Label graphs of the potential, kinetic, and total energies of the ball as functions of the distance fallen from the top of the cliff cliff cliff
c. On the axes below sketch and label the kinetic and potential energies of the ball as functions of time until the ball hits
1981 B2 - Class
A massless spring is between a 1kilogram mass and a 3kilogram mass as shown above, but is not attached to either mass. Both masses are on a horizontal frictionless table. In an experiment, the 1kilogram mass is held in place and the spring is compressed by pushing on the 3kilogram mass. The 3kilogram mass is then released and moves off with a speed of 10 meters per second.
a. Determine the minimum work needed to compress the spring in this experiment.
The spring is compressed again exactly as above, but this time both masses are released simultaneously.
b. Determine the final velocity of each mass relative to the cable after the masses are released. 1985 B1 - Class
A 2kilogram block initially hangs at rest at the end of two 1meter strings of negligible mass as shown on the left diagram above. A 0.003kilogram bullet, moving horizontally with a speed of 1000 meters per second, strikes the block and becomes embedded in it. After the collision, the bullet/ block combination swings upward, but does not rotate. a. Calculate the speed v of the bullet/ block combination just after the collision. b. Calculate the ratio of the initial kinetic energy of the bullet to the kinetic energy of the bullet/ block combination immediately after the collision. c. Calculate the maximum vertical height above the initial rest position reached by the bullet/block combination.
1988 B2 - Class
A ball thrown vertically downward strikes a horizontal surface with a speed of 15 meters per second. It then bounces, and reaches a maximum height of 5 meters. Neglect air resistance on the ball.
a. What is the speed of the ball immediately after it rebounds from the surface?
b. What fraction of the ball's initial kinetic energy is apparently lost during the bounce?
c. If the specific heat of the ball is 1,800 J/kg °C, and if all of the lost energy is absorbed by the molecules of the ball, by how much does the temperature of the ball increase?
1992 B1 - Homework
A 0. 10-kilogram solid rubber ball is attached to the end of an 0.80-meter length of light thread. The ball is swung in a vertical circle, as shown in the diagram above. Point P, the lowest point of the circle, is 0.20 meter above the floor. The speed of the ball at the top of the circle is 6.0 meters per second, and the total energy of the ball is kept constant.
a. Determine the total energy of the ball, using the floor as the zero point for gravitational potential energy.
b. Determine the speed of the ball at point P, the lowest point of the circle.
c. Determine the tension in the thread at i. the top of the circle; ii. the bottom of the circle.
The ball only reaches the top of the circle once before the thread breaks when the ball is at the lowest point of the circle.
d. Determine the horizontal distance that the ball travels before hitting the floor.
1994 B2 - Homework
A track consists of a frictionless arc XY, which is a quartercircle of radius R, and a rough horizontal section YZ. Block A of mass M is released from rest at point X, slides down the curved section of the track, and collides instantaneously and inelastically with identical block B at point Y. The two blocks move together to the right, sliding past point P, which is a distance l from point Y. The coefficient of kinetic friction between the blocks and the horizontal part of the track is Express your answers in terms of M, l, , R, and g.
a. Determine the speed of block A just before it hits block B.
b. Determine the speed of the combined blocks immediately after the collision.
c. Determine the amount of kinetic energy lost due to the collision.
d. The specific heat of the material used to make the blocks is c. Determine the temperature rise that results from the collision in terms of c and the other given quantities. (Assume that no energy is transferred to the track or to the air surrounding the blocks.)
e. Determine the additional thermal energy that is generated as the blocks move from Y to P
1999 B1 - Class
The Sojourner rover vehicle shown in the sketch above was used to explore the surface of Mars as part of the Pathfinder mission in 1997. Use the data in the tables below to answer the questions that follow.
Mars Data Sojourner Data
Radius: 0.53 x Earth's radius Mass of Sojourner vehicle: 11.5 kg
Mass: 0.11 x Earth's mass Wheel diameter: 0.13 m Stored energy available: 5.4 x 105 J Power required for driving under average conditions: 10 W Land speed: 6.7 x 103 m/s
a. Determine the acceleration due to gravity at the surface of Mars in terms of g, the acceleration due to gravity at the surface of Earth.
b. Calculate Sojourner's weight on the surface of Mars.
c. Assume that when leaving the Pathfinder spacecraft Sojourner rolls down a ramp inclined at 20° to the horizontal. The ramp must be lightweight but strong enough to support Sojourner. Calculate the minimum normal force that must be supplied by the ramp.
d. What is the net force on Sojourner as it travels across the Martian surface at constant velocity? Justify your answer.
e. Determine the maximum distance that Sojourner can travel on a horizontal Martian surface using its stored energy.
f. Suppose that 0.010% of the power for driving is expended against atmospheric drag as Sojourner travels on the Martian surface. Calculate the magnitude of the drag force.
2002 B-A - Homework
2. (15 points)
A 3.0 kg object subject to a restoring force F is undergoing simple harmonic motion with a small amplitude.
The potential energy U of the object as a function of distance x from its equilibrium position is shown above.
This particular object has a total energy E of 0.4 J. (a) What is the object’s potential energy when its displacement is 44 cm from its equilibrium position? (b) What is the farthest the object moves along the x-axis in the positive direction? Explain your reasoning. (c) Determine the object’s kinetic energy when its displacement is —7 cm. (d) What is the object’s speed at x =0?
(e) Suppose the object undergoes this motion because it is the bob of a simple pendulum as shown above. If the object breaks loose from the string at the instant the pendulum reaches its lowest point and hits the ground at point P shown, what is the horizontal distance d that it travels?
A 2kilogram block is released from rest at the top of a curved incline in the shape of a quarter of a circle of radius R. The block then slides onto a horizontal plane where it finally comes to rest 8 meters from the beginning of the plane. The curved incline is frictionless, but there is an 8newton force of friction on the block while it slides horizontally. Assume g = 10 meters per second2.
a. Determine the magnitude of the acceleration of the block while it slides along the horizontal plane.
b. What time elapses while the block is sliding horizontally?
c. Calculate the radius of the incline in meters.
1997 B1 - Homework
A 0.20 kg object moves along a straight line. The net force acting on the object varies with the object's displacement as shown in the graph above. The object starts from rest at displacement x = 0 and time t = 0 and is displaced a distance of 20 m. Determine each of the following.
a. The acceleration of the particle when its displacement x is 6 m.
b. The time taken for the object to be displaced the first 12 m.
c. The amount of work done by the net force in displacing the object the first 12 m.
d. The speed of the object at displacement x = 12 m.
e. The final speed of the object at displacement x = 20 m.
f. The change in the momentum of the object as it is displaced from x = 12 m to x = 20 m
1979 B1 - Homework
From the top of a cliff 80 meters high, a ball of mass 0.4 kilogram is launched horizontally with a velocity of 30 meters per second at time t = 0 as shown above. The potential energy of the ball is zero at the bottom of the cliff. Use g = 10 meters per second squared. a. Calculate the potential, kinetic, and focal energies of the ball at time t = O.
b. On the axes below, sketch and Label graphs of the potential, kinetic, and total energies of the ball as functions of the distance fallen from the top of the cliff cliff cliff
c. On the axes below sketch and label the kinetic and potential energies of the ball as functions of time until the ball hits
1981 B2 - Class
A massless spring is between a 1kilogram mass and a 3kilogram mass as shown above, but is not attached to either mass. Both masses are on a horizontal frictionless table. In an experiment, the 1kilogram mass is held in place and the spring is compressed by pushing on the 3kilogram mass. The 3kilogram mass is then released and moves off with a speed of 10 meters per second.
a. Determine the minimum work needed to compress the spring in this experiment.
The spring is compressed again exactly as above, but this time both masses are released simultaneously.
b. Determine the final velocity of each mass relative to the cable after the masses are released. 1985 B1 - Class
A 2kilogram block initially hangs at rest at the end of two 1meter strings of negligible mass as shown on the left diagram above. A 0.003kilogram bullet, moving horizontally with a speed of 1000 meters per second, strikes the block and becomes embedded in it. After the collision, the bullet/ block combination swings upward, but does not rotate. a. Calculate the speed v of the bullet/ block combination just after the collision. b. Calculate the ratio of the initial kinetic energy of the bullet to the kinetic energy of the bullet/ block combination immediately after the collision. c. Calculate the maximum vertical height above the initial rest position reached by the bullet/block combination.
1988 B2 - Class
A ball thrown vertically downward strikes a horizontal surface with a speed of 15 meters per second. It then bounces, and reaches a maximum height of 5 meters. Neglect air resistance on the ball.
a. What is the speed of the ball immediately after it rebounds from the surface?
b. What fraction of the ball's initial kinetic energy is apparently lost during the bounce?
c. If the specific heat of the ball is 1,800 J/kg °C, and if all of the lost energy is absorbed by the molecules of the ball, by how much does the temperature of the ball increase?
1992 B1 - Homework
A 0. 10-kilogram solid rubber ball is attached to the end of an 0.80-meter length of light thread. The ball is swung in a vertical circle, as shown in the diagram above. Point P, the lowest point of the circle, is 0.20 meter above the floor. The speed of the ball at the top of the circle is 6.0 meters per second, and the total energy of the ball is kept constant.
a. Determine the total energy of the ball, using the floor as the zero point for gravitational potential energy.
b. Determine the speed of the ball at point P, the lowest point of the circle.
c. Determine the tension in the thread at i. the top of the circle; ii. the bottom of the circle.
The ball only reaches the top of the circle once before the thread breaks when the ball is at the lowest point of the circle.
d. Determine the horizontal distance that the ball travels before hitting the floor.
1994 B2 - Homework
A track consists of a frictionless arc XY, which is a quartercircle of radius R, and a rough horizontal section YZ. Block A of mass M is released from rest at point X, slides down the curved section of the track, and collides instantaneously and inelastically with identical block B at point Y. The two blocks move together to the right, sliding past point P, which is a distance l from point Y. The coefficient of kinetic friction between the blocks and the horizontal part of the track is Express your answers in terms of M, l, , R, and g.
a. Determine the speed of block A just before it hits block B.
b. Determine the speed of the combined blocks immediately after the collision.
c. Determine the amount of kinetic energy lost due to the collision.
d. The specific heat of the material used to make the blocks is c. Determine the temperature rise that results from the collision in terms of c and the other given quantities. (Assume that no energy is transferred to the track or to the air surrounding the blocks.)
e. Determine the additional thermal energy that is generated as the blocks move from Y to P
1999 B1 - Class
The Sojourner rover vehicle shown in the sketch above was used to explore the surface of Mars as part of the Pathfinder mission in 1997. Use the data in the tables below to answer the questions that follow.
Mars Data Sojourner Data
Radius: 0.53 x Earth's radius Mass of Sojourner vehicle: 11.5 kg
Mass: 0.11 x Earth's mass Wheel diameter: 0.13 m Stored energy available: 5.4 x 105 J Power required for driving under average conditions: 10 W Land speed: 6.7 x 103 m/s
a. Determine the acceleration due to gravity at the surface of Mars in terms of g, the acceleration due to gravity at the surface of Earth.
b. Calculate Sojourner's weight on the surface of Mars.
c. Assume that when leaving the Pathfinder spacecraft Sojourner rolls down a ramp inclined at 20° to the horizontal. The ramp must be lightweight but strong enough to support Sojourner. Calculate the minimum normal force that must be supplied by the ramp.
d. What is the net force on Sojourner as it travels across the Martian surface at constant velocity? Justify your answer.
e. Determine the maximum distance that Sojourner can travel on a horizontal Martian surface using its stored energy.
f. Suppose that 0.010% of the power for driving is expended against atmospheric drag as Sojourner travels on the Martian surface. Calculate the magnitude of the drag force.
2002 B-A - Homework
2. (15 points)
A 3.0 kg object subject to a restoring force F is undergoing simple harmonic motion with a small amplitude.
The potential energy U of the object as a function of distance x from its equilibrium position is shown above.
This particular object has a total energy E of 0.4 J. (a) What is the object’s potential energy when its displacement is 44 cm from its equilibrium position? (b) What is the farthest the object moves along the x-axis in the positive direction? Explain your reasoning. (c) Determine the object’s kinetic energy when its displacement is —7 cm. (d) What is the object’s speed at x =0?
(e) Suppose the object undergoes this motion because it is the bob of a simple pendulum as shown above. If the object breaks loose from the string at the instant the pendulum reaches its lowest point and hits the ground at point P shown, what is the horizontal distance d that it travels?