physics 123 final exam
PHYSICS 123 FINAL EXAM
December 10, 2001
1 vx = vox + ax t vx2 = vo2x + 2ax (x − xo ) GIVEN: x = xo + ax t2 + vox t; 2 1 y = yo + ay t2 + voy t; vy = voy + ay t vy2 = vo2y + 2ay (y − yo ) 2 2 F~net = m~a; g = 9.8m/s2 = 32 ft/s2 ; Fcentripetal = mv r GM1 M2 1 Fgravity = P Egravity = mgh; P Espring = kx2 ; 2 r 2 Z ~ = area under plot of force vs displacement. W = F~ · ~s; W = F~ · ds ∆KE = Wnet mv 2 mi xi Fcentripetal = ; xCM = i p~ = m~v ; r Mtotal 1 1 ~ = I~ KEtrans = mv 2 ; KErot = Iω 2 L ω 2 2 1 θ = αt2 + ω0 t; ω = ω0 + αt ω 2 = ω02 + 2α(θ − θ0 ) 2 1 2 1 Irod−about−center = M l2 Isphere = M R2 Ihoop = M R2 Idisk = Isolid−cylinder = M R2 12 5 2 1 d~` ~` = ~r × p~ Irod−about−end = M l2 ~τ = ~r × F~ ~τ = 3 dt d~ p ~ ~ ~τ = I α ~; |~τ | = |F ||~r| sin θ p~ = m~v ; F = ; 2 π radians = 360 degrees dt Parallel Axis Theorem: IP = ICM + M L2 where L is distance from CM to pivot point P . dU (x) ; F (x) = − dx
P
d~ p F~ = ; dt
PART ONE: MULTIPLE CHOICE ( 4 points each) Circle or check the correct response. Use GIVEN data on above if needed. . 1. A baseball is thrown vertically upwards. The acceleration of the ball at its highest point is: (a) zero (b) 9.8 m/s2 upward (c) 9.8 m/s2 downward (d) changing suddenly from 9.8 m/s2 upward to 9.8 m/s2 downward 2. A freely falling body (neglect air resistance) has an acceleration of 9.8 m/s2 downward. This means that: (a) the body falls 9.8 meters during each second (b) the body falls 9.8 meters during the first second only (c) the speed of the body increases by 9.8 m/s during each second (d) the acceleration of the body increases by 9.8 m/s2 in each second (e) the acceleration of the body decreases by 9.8 m/s2 in each second 3. Which of the following is NOT an example of accelerated motion? (a) vertical component of projectile motion (b) circular motion at constant speed (c) the motion of the Earth around the sun (d) a swinging pendulum (e) horizontal component of projectile motion
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4. An object is moving at constant speed in a circular path. The instantaneous velocity and instantaneous acceleration vectors are: (a) both tangent to the circular path (b) perpendicular to each other (c) both perpendicular to the circular path (d) none of the above 5. A boat is traveling upstream at 14 mph with respect to the water in a river. The current in the river moves at a speed of 6 mph with respect to the ground. A man on the boat runs across the boat, perpendicular to the river, from one side to the other at 6 mph. The speed of the man with respect to the ground is: (a) 10 mph (b) 14 mph (c) 18.5 mph (d) 21 mph (e) 26 mph 6. A 1000 kg elevator is moving downward with an downward acceleration of 5 m/s2 . The tension in the elevator cable is: (It might be helpful if you draw a FBD for the elevator): (a) 6800 N (b) 1000 N (c) 3000 N (d) 9800 N (e) 4800 N
7. A 50 kg woman is ice skating towards the east on a frozen lake when she collides with a 2000 sport utility vehicle (SUV) which is driving on the ice towards the west.The maximum force exerted on the woman by the SUV is 800 N, westward. What is the force that the woman exerts on the truck? (a) 800 N eastward (b) 50 N westward (c) 50 N eastward (d) 2000 N eastward 8. A crate rests on a horizontal surface and a woman pulls on it with a 10 N force. No matter what the orientation of the force, the crate does not move. Rank the situations shown below according to the magnitude of the friction force exerted by the surface on the crate, smallest first:
(a) 1, 2, 3
(b) 2,1,3
(c) 2,3,1
(d) 1,2,3
(e) 3,2,1
9. A physics textbook is on a spring scale in an elevator. Of the following, the scale shows the highest reading when the elevator: (a) moves downward with decreasing speed (b) moves downward with increasing speed (c) moves downward with constant speed (d) moves upward with constant speed 10. A car of mass m travels around a flat circular track of radius r, with speed v. Which of the following is true? (a) The net force on the car is zero. 2 (b) the gravity force on the car equals mv r 2 (c) The friction force on the car equals mv r 2 (d) The normal force on the car equals mv r
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11. A pile driver is pounding a steel beam into the ground. The hammer (the falling object) has mass 500 kg and falls from a height of 2 m onto the beam, starting at rest. The kinetic energy of the hammer just before it hits the beam is:
(a) 9800 Joules
(b) 4900 Joules
(c) 1000 Joules
(d) 500 Joules
12. In the situation described in Question 11, the beam then penetrates 10 cm into the ground. What is the average force the ground exerts on the beam as it is forced downward? (a) 5000 N (b) 98000 N (c) 4900 N (d) 49000 N 13. A roller coaster is pulled to the top of its track and allowed to roll down. For a bigger thrill you want to re-design the track so the car is going twice as fast at the bottom of the run. How much higher should you make the track?
(a) the same height
(b) four times as high
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(c) half as high
(d) twice as high
14. A person attempts to knock down a large wooden bowling pin by throwing a ball at it. The person has one ball made of putty and a ball of the same mass made of rubber. The rubber ball bounces back while the ball made of putty sticks to the pin. Which ball is more likely to topple the pin? (a) the rubber ball (b) the putty ball (c) both are equally likely 15. A compact car and a large truck collide and stick together. Which undergoes the larger momentum change? (a) the car (b) the truck (c) both have the same momentum change (d) impossible to predict 16. A 60 kg hunter gets a rope around a 300 kg polar bear. They are at rest on frictionless ice 30 meters apart. The hunter pulls the polar bear towards him. When they meet the polar bear will move: (a) 30 m (b) 25 m (c) 15 m (d) 10 m (e) 5 m 17. A rider in a “barrel of fun” at the amusement park finds herself stuck with her back to the wall. As the floor is lowered and the ride rotates she remains stuck. Which diagram correctly shows the forces acting on her?
18. The fan shown below has been turned on and is speeding up as it rotates clockwise. The direction of the acceleration of the point X on the fan tip is best represented by which arrow below?
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19. A meter stick shown below rotates about an axis through the dot at the 20 cm point, Five forces act on the meter stick as shown. Rank the forces according to the magnitudes of the torque they produce about the pivot point, greatest to least.
(a) F~1 > F~2 > F~3 > F~4 > F~5 (b) F~1 = F~3 > F~2 > F~5 > F~4 (c)F~3 = F~1 > F~4 > F~5 = F~2 (d) F~1 = F~2 > F~4 > F~2 = F~5 20. The object below consists of four identical particles of very small radius and mass m which are connected by very light sticks to form a rigid body. The rotational inertia of the object about the y axis shown below is:
(a) 3ma2
(b) 4ma2
(c) 2ma2
(d) ma2
21. Two disks are mounted on low friction bearings on a common shaft. THe first disc has rotational inertia I and is spinning with angular velocity ω. The second disk has rotational inertia 2I and is spinning in the same direction as the first disk with angular velocity 2ω as shown. The disks are slowly forced together along the shaft until they couple and have a final angular velocity ωf . What is ωf ?
(a) 5ω/3
p
(b) ω/3
(c) ω/ (3)
5
(d) 5ω/2
22. An astronaut in an orbiting spacecraft feels “weightless” because she: (a) has no acceleration. (b) is beyond the gravity pull of the Earth. (c) is pulled outwards by centrifugal force. (d) is in the same free-fall orbit as the spacecraft. 23. An artificial Earth satellite is moved from a circular orbit of radius R to a circular orbit of radius 2R. During the move: (a) the gravitational force does positive work, the kinetic energy of the satellite increases, and the potential energy of the Earth-satellite system increases. (b) the gravitational force does positive work, the kinetic energy of the satellite increases, and the potential energy of the Earth-satellite system decreases. (c) the gravitational force does negative work, the kinetic energy of the satellite decreases, and the potential energy of the Earth-satellite system increases. (d) the gravitational force does negative work, the kinetic energy of the satellite decreases, and the potential energy of the Earth-satellite system decreases. 24. A particle is placed, in turn, a distance d from four objects, each of mass m: a uniform solid sphere, a large uniform spherical shell, a small solid sphere, and a small spherical shell. Which object exerts the largest gravitational force on the particle? (a) the large solid sphere (b) the small solid sphere (c) the large spherical shell (d) the small spherical shell (e) all exert the same force 25. The figure below shows a uniform beam of mass m which is acted on by a force F~ and is held vertical by a cable. Draw a free body diagram for the beam, clearly labeling all of the forces that act on it.
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PART TWO: PROBLEMS (25 points each) Work each problem on the exam page. Be sure all answers are clearly labeled, with units. Show all work clearly. Answers with no clear reasoning will receive little or no credit. 1. In the figure below, a stone is projected at a cliff of height h with an initial speed or 42.0 m/s, directed 60◦ above the horizontal. The stone hits point A 4.5 s after it is launched.
(a) Find the height h of the cliff. (b) Find the speed of the stone just before impact. (c) Find the maximum height H of the stone.
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2. A car moves around a circular banked track as shown below.
(a) Draw a free body diagram for the car, showing all forces. (b) If there is no friction between the car and the track, what is the speed of the car? Express your answer algebraically in terms of the banking angle θ, the mass M of the car, the radius R of the track, and g.
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3. A 40 kg child and her father simultaneously dive from a 100 kg boat that is initially motionless. Ths child dives horizontally with a speed of 2.0 m/s toward the east. Her father dives toward the south with a velocity of 1.5 m/s at an angle of 37◦ above the horizontal. Find the magnitude and direction of the velocity of the boat immediately after the dives. Assume the vertical component of the father’s dive velocity does not affect the motion of the boat along the surface of the water.
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4. In the figure below a wheel of radius 0.20 m is mounted on frictionless bearings on a horizontal axis. The rotational inertia of the wheel about the axis is 0.40 kg-m2 . A massless cord is wrapped around the wheel and is attached to a 6.0 kg box. A short time after being released from rest, the box has a kinetic energy of 6.0 J. The string does not slip.
(a) What is the kinetic energy of the pulley at the time the box has kinetic energy 6.0 J? (b) What is the distance the box has fallen at that instant?
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December 10, 2001
1 vx = vox + ax t vx2 = vo2x + 2ax (x − xo ) GIVEN: x = xo + ax t2 + vox t; 2 1 y = yo + ay t2 + voy t; vy = voy + ay t vy2 = vo2y + 2ay (y − yo ) 2 2 F~net = m~a; g = 9.8m/s2 = 32 ft/s2 ; Fcentripetal = mv r GM1 M2 1 Fgravity = P Egravity = mgh; P Espring = kx2 ; 2 r 2 Z ~ = area under plot of force vs displacement. W = F~ · ~s; W = F~ · ds ∆KE = Wnet mv 2 mi xi Fcentripetal = ; xCM = i p~ = m~v ; r Mtotal 1 1 ~ = I~ KEtrans = mv 2 ; KErot = Iω 2 L ω 2 2 1 θ = αt2 + ω0 t; ω = ω0 + αt ω 2 = ω02 + 2α(θ − θ0 ) 2 1 2 1 Irod−about−center = M l2 Isphere = M R2 Ihoop = M R2 Idisk = Isolid−cylinder = M R2 12 5 2 1 d~` ~` = ~r × p~ Irod−about−end = M l2 ~τ = ~r × F~ ~τ = 3 dt d~ p ~ ~ ~τ = I α ~; |~τ | = |F ||~r| sin θ p~ = m~v ; F = ; 2 π radians = 360 degrees dt Parallel Axis Theorem: IP = ICM + M L2 where L is distance from CM to pivot point P . dU (x) ; F (x) = − dx
P
d~ p F~ = ; dt
PART ONE: MULTIPLE CHOICE ( 4 points each) Circle or check the correct response. Use GIVEN data on above if needed. . 1. A baseball is thrown vertically upwards. The acceleration of the ball at its highest point is: (a) zero (b) 9.8 m/s2 upward (c) 9.8 m/s2 downward (d) changing suddenly from 9.8 m/s2 upward to 9.8 m/s2 downward 2. A freely falling body (neglect air resistance) has an acceleration of 9.8 m/s2 downward. This means that: (a) the body falls 9.8 meters during each second (b) the body falls 9.8 meters during the first second only (c) the speed of the body increases by 9.8 m/s during each second (d) the acceleration of the body increases by 9.8 m/s2 in each second (e) the acceleration of the body decreases by 9.8 m/s2 in each second 3. Which of the following is NOT an example of accelerated motion? (a) vertical component of projectile motion (b) circular motion at constant speed (c) the motion of the Earth around the sun (d) a swinging pendulum (e) horizontal component of projectile motion
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4. An object is moving at constant speed in a circular path. The instantaneous velocity and instantaneous acceleration vectors are: (a) both tangent to the circular path (b) perpendicular to each other (c) both perpendicular to the circular path (d) none of the above 5. A boat is traveling upstream at 14 mph with respect to the water in a river. The current in the river moves at a speed of 6 mph with respect to the ground. A man on the boat runs across the boat, perpendicular to the river, from one side to the other at 6 mph. The speed of the man with respect to the ground is: (a) 10 mph (b) 14 mph (c) 18.5 mph (d) 21 mph (e) 26 mph 6. A 1000 kg elevator is moving downward with an downward acceleration of 5 m/s2 . The tension in the elevator cable is: (It might be helpful if you draw a FBD for the elevator): (a) 6800 N (b) 1000 N (c) 3000 N (d) 9800 N (e) 4800 N
7. A 50 kg woman is ice skating towards the east on a frozen lake when she collides with a 2000 sport utility vehicle (SUV) which is driving on the ice towards the west.The maximum force exerted on the woman by the SUV is 800 N, westward. What is the force that the woman exerts on the truck? (a) 800 N eastward (b) 50 N westward (c) 50 N eastward (d) 2000 N eastward 8. A crate rests on a horizontal surface and a woman pulls on it with a 10 N force. No matter what the orientation of the force, the crate does not move. Rank the situations shown below according to the magnitude of the friction force exerted by the surface on the crate, smallest first:
(a) 1, 2, 3
(b) 2,1,3
(c) 2,3,1
(d) 1,2,3
(e) 3,2,1
9. A physics textbook is on a spring scale in an elevator. Of the following, the scale shows the highest reading when the elevator: (a) moves downward with decreasing speed (b) moves downward with increasing speed (c) moves downward with constant speed (d) moves upward with constant speed 10. A car of mass m travels around a flat circular track of radius r, with speed v. Which of the following is true? (a) The net force on the car is zero. 2 (b) the gravity force on the car equals mv r 2 (c) The friction force on the car equals mv r 2 (d) The normal force on the car equals mv r
2
11. A pile driver is pounding a steel beam into the ground. The hammer (the falling object) has mass 500 kg and falls from a height of 2 m onto the beam, starting at rest. The kinetic energy of the hammer just before it hits the beam is:
(a) 9800 Joules
(b) 4900 Joules
(c) 1000 Joules
(d) 500 Joules
12. In the situation described in Question 11, the beam then penetrates 10 cm into the ground. What is the average force the ground exerts on the beam as it is forced downward? (a) 5000 N (b) 98000 N (c) 4900 N (d) 49000 N 13. A roller coaster is pulled to the top of its track and allowed to roll down. For a bigger thrill you want to re-design the track so the car is going twice as fast at the bottom of the run. How much higher should you make the track?
(a) the same height
(b) four times as high
3
(c) half as high
(d) twice as high
14. A person attempts to knock down a large wooden bowling pin by throwing a ball at it. The person has one ball made of putty and a ball of the same mass made of rubber. The rubber ball bounces back while the ball made of putty sticks to the pin. Which ball is more likely to topple the pin? (a) the rubber ball (b) the putty ball (c) both are equally likely 15. A compact car and a large truck collide and stick together. Which undergoes the larger momentum change? (a) the car (b) the truck (c) both have the same momentum change (d) impossible to predict 16. A 60 kg hunter gets a rope around a 300 kg polar bear. They are at rest on frictionless ice 30 meters apart. The hunter pulls the polar bear towards him. When they meet the polar bear will move: (a) 30 m (b) 25 m (c) 15 m (d) 10 m (e) 5 m 17. A rider in a “barrel of fun” at the amusement park finds herself stuck with her back to the wall. As the floor is lowered and the ride rotates she remains stuck. Which diagram correctly shows the forces acting on her?
18. The fan shown below has been turned on and is speeding up as it rotates clockwise. The direction of the acceleration of the point X on the fan tip is best represented by which arrow below?
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19. A meter stick shown below rotates about an axis through the dot at the 20 cm point, Five forces act on the meter stick as shown. Rank the forces according to the magnitudes of the torque they produce about the pivot point, greatest to least.
(a) F~1 > F~2 > F~3 > F~4 > F~5 (b) F~1 = F~3 > F~2 > F~5 > F~4 (c)F~3 = F~1 > F~4 > F~5 = F~2 (d) F~1 = F~2 > F~4 > F~2 = F~5 20. The object below consists of four identical particles of very small radius and mass m which are connected by very light sticks to form a rigid body. The rotational inertia of the object about the y axis shown below is:
(a) 3ma2
(b) 4ma2
(c) 2ma2
(d) ma2
21. Two disks are mounted on low friction bearings on a common shaft. THe first disc has rotational inertia I and is spinning with angular velocity ω. The second disk has rotational inertia 2I and is spinning in the same direction as the first disk with angular velocity 2ω as shown. The disks are slowly forced together along the shaft until they couple and have a final angular velocity ωf . What is ωf ?
(a) 5ω/3
p
(b) ω/3
(c) ω/ (3)
5
(d) 5ω/2
22. An astronaut in an orbiting spacecraft feels “weightless” because she: (a) has no acceleration. (b) is beyond the gravity pull of the Earth. (c) is pulled outwards by centrifugal force. (d) is in the same free-fall orbit as the spacecraft. 23. An artificial Earth satellite is moved from a circular orbit of radius R to a circular orbit of radius 2R. During the move: (a) the gravitational force does positive work, the kinetic energy of the satellite increases, and the potential energy of the Earth-satellite system increases. (b) the gravitational force does positive work, the kinetic energy of the satellite increases, and the potential energy of the Earth-satellite system decreases. (c) the gravitational force does negative work, the kinetic energy of the satellite decreases, and the potential energy of the Earth-satellite system increases. (d) the gravitational force does negative work, the kinetic energy of the satellite decreases, and the potential energy of the Earth-satellite system decreases. 24. A particle is placed, in turn, a distance d from four objects, each of mass m: a uniform solid sphere, a large uniform spherical shell, a small solid sphere, and a small spherical shell. Which object exerts the largest gravitational force on the particle? (a) the large solid sphere (b) the small solid sphere (c) the large spherical shell (d) the small spherical shell (e) all exert the same force 25. The figure below shows a uniform beam of mass m which is acted on by a force F~ and is held vertical by a cable. Draw a free body diagram for the beam, clearly labeling all of the forces that act on it.
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PART TWO: PROBLEMS (25 points each) Work each problem on the exam page. Be sure all answers are clearly labeled, with units. Show all work clearly. Answers with no clear reasoning will receive little or no credit. 1. In the figure below, a stone is projected at a cliff of height h with an initial speed or 42.0 m/s, directed 60◦ above the horizontal. The stone hits point A 4.5 s after it is launched.
(a) Find the height h of the cliff. (b) Find the speed of the stone just before impact. (c) Find the maximum height H of the stone.
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2. A car moves around a circular banked track as shown below.
(a) Draw a free body diagram for the car, showing all forces. (b) If there is no friction between the car and the track, what is the speed of the car? Express your answer algebraically in terms of the banking angle θ, the mass M of the car, the radius R of the track, and g.
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3. A 40 kg child and her father simultaneously dive from a 100 kg boat that is initially motionless. Ths child dives horizontally with a speed of 2.0 m/s toward the east. Her father dives toward the south with a velocity of 1.5 m/s at an angle of 37◦ above the horizontal. Find the magnitude and direction of the velocity of the boat immediately after the dives. Assume the vertical component of the father’s dive velocity does not affect the motion of the boat along the surface of the water.
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4. In the figure below a wheel of radius 0.20 m is mounted on frictionless bearings on a horizontal axis. The rotational inertia of the wheel about the axis is 0.40 kg-m2 . A massless cord is wrapped around the wheel and is attached to a 6.0 kg box. A short time after being released from rest, the box has a kinetic energy of 6.0 J. The string does not slip.
(a) What is the kinetic energy of the pulley at the time the box has kinetic energy 6.0 J? (b) What is the distance the box has fallen at that instant?
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