Deck 8: Conservation of Energy

Question 1

A 3.2-kg mass is placed at the top of a rough incline( $\mu = 0.1$ ). The incline makes an angle of 32º with the horizontal, and the height of the incline is 2.5 m. The gravitational potential energy of the mass at the top of the incline is

A) 55 J.
B) 78 J.
C) 46 J.
D) 36 J.

Answer 1

78 J.

Question 2

A 3.2-kg mass is placed at the top of a rough incline( $\mu = 0.1$ ). The incline makes an angle of 32º with the horizontal, and the height of the incline is 2.5 m. The kinetic energy of the mass at the top of the incline is

A) 0 J.
B) 39 J.
C) 78 J.
D) 44 J.

Answer 2

0 J.

Question 3

A 3.2-kg mass is placed at the top of a rough incline ( $\mu = 0.1$ ). The incline makes an angle of 32º with the horizontal, and the height of the incline is 2.5 m. The mass is released and slides down to the bottom. The velocity of the mass at the bottom is

A) 7.0 m/s.
B) 6.8 m/s.
C) 6.6 m/s.
D) 6.4 m/s.

Answer 3

6.4 m/s.

Question 4

A 3.2-kg mass is placed at the top of a rough incline ( $\mu = 0.1$ ). The incline makes an angle of 32º with the horizontal, and the height of the incline is 2.5 m. The mass is released and slides down to the bottom. The amount of mechanical energy lost is

A) 0 J.
B) 13 J.
C) 4.5 J.
D) 9.8 J.

Question 5

A 3.2-kg mass is placed at the top of a rough incline ( $\mu = 0.1$ ). The incline makes an angle of 32º with the horizontal, and the height of the incline is 2.5 m. The mass is released and slides down to the bottom. The kinetic energy of the mass at the bottom is

A) 78 J.
B) 13 J.
C) 66 J.
D) 23 J.

Question 6

A 5.6-kg mass is held 5 cm above the end of a vertically oriented linear spring ( $k = 1200$ N/m). The mass is released. The maximum distance that the spring is compressed by the mass is

A) 3.6 cm.
B) 7.5 cm.
C) 13 cm.
D) 11 cm.

Question 7

A 5.6-kg mass is held 5 cm above the end of a vertically oriented linear spring ( $k = 1200$ N/m). The mass is released; it falls, compressing the spring. The spring then expands and does work on the mass. The velocity of the mass as it leaves the spring (in the upward direction) is

A) 1 m/s.
B) 3 m/s.
C) 4 m/s.
D) 2 m/s.

Question 8

A nonlinear spring stores potential energy according to $U ( x ) = \alpha x ^ { 4 } + \beta x ^ { 2 }$ , where $\alpha$ and $\beta$ are constants and $x$ is the distance that the spring is either stretched or compressed. The force that would be required to compress this spring a distance $d$ is

A) $\alpha d ^ { 5 } / 5 + \beta d ^ { 3 } / 3$
B) $\alpha d ^ { 4 } + \beta d ^ { 2 }$
C) $2 d \left( 2 \alpha d ^ { 2 } + \beta \right)$
D) $4 \alpha d ^ { 4 } + 3 \beta d ^ { 3 }$

Question 9

The force to compress a nonlinear spring is given by $F ( x ) = \gamma x ^ { 2 }$ , where $\gamma = 45,000 \mathrm {~N} / \mathrm { m } ^ { 2 }$ . The amount of energy stored in this spring when it is compressed $10 \mathrm {~cm}$ is

A) 450 J.
B) 220 J.
C) 45 J.
D) 15 J.

Question 10

A 1.2-kg mass is attached to the end of a 4.5-m string; the other end of the string is attached to the ceiling (forming a pendulum). The mass is lifted 15 cm (above its lowest point) and then released. The mass freely swings back and forth. The maximum velocity of the mass is

A) 1.7 m/s.
B) 1.2 m/s.
C) 2.2 m/s.
D) 2.6 m/s.

Question 11

A steel spring having spring constant $k _ { s }$ and length $L _ { s }$ is placed in series with a brass spring having spring constant $k _ { b }$ and length $L _ {b }$ . The two springs are then inserted into a vise (in a coaxial arrangement), and the vise squeezes the springs a distance $d$ . The entire length of the compressed steel spring is

A) $L _ { s } - k _ { b } d / \left( k _ { b } + k _ { s } \right)$
B) $L _ { s } - k _ { b } L _ { b } / \left( k _ { b } + k _ { s } \right)$
C) $L _ { b } - k _ { s } L _ { s } / \left( k _ { b } + k _ { s } \right)$
D) $L _ { s } - k _ { s } d / \left( k _ { b } + k _ { s } \right)$

Question 12

A solid ball having a mass of 2.6 kg and a radius of 0.2 m is placed at the top of an incline. The height of the incline is 3.0 m. The ball is then released and rolls without slipping down the incline. The translational velocity of the ball at the bottom of the incline is

A) 7.7 m/s.
B) 6.5 m/s.
C) 8.2 m/s.
D) 6.9 m/s.

Question 13

A solid ball having a mass of 2.6 kg and a radius of 0.2 m is placed at the top of an incline. The height of the incline is 3.0 m. The ball is then released and rolls without slipping down the incline. The rotational velocity of the ball at the bottom of the incline is

A) 32 rad/s.
B) 26 rad/s.
C) 19 rad/s.
D) 12 rad/s.

Question 14

A solid ball having a mass of 2.6 kg and a radius of 0.2 m is placed at the top of an incline. The incline makes an angle of $\theta$ = 51º with the horizontal. The ball is then released and rolls without slipping down the incline. The acceleration of the ball's center of mass along the incline is

A) 6.9 m/s $2$
B) 6.3 m/s $2$
C) 4.5 m/s $2$
D) 5.4 m/s $2$

Question 15

A solid ball having a mass of 2.6 kg and a radius of 0.2 m is placed at the top of an incline. The incline makes an angle of $\theta$ = 51º with the horizontal. The ball is then released and rolls without slipping down the incline. The frictional force that the incline exerts on the (rolling) ball is

A) 20 N.
B) 5.7 N.
C) 7.9 N.
D) 0.0 N.

Question 16

Two objects of differing masses $\left( m _ { a } , m _ { b } \right)$ are released from the top of two frictionless inclines having the same height. Mass $m _ { a }$ is on an incline that makes an angle of $15 ^ { \circ }$ with the horizontal; mass $m _ { b }$ is on an incline that makes an angle of $75 ^ { \circ }$ with the horizontal. At the bottom, the mass that has the greater speed is

A) $m _ { a }$
B) $m _ { b }$
C) Both masses have the same speed.
D) This cannot be determined unless both masses ( $m _ { a } , m _ { b }$ ) are known.

Question 17

Two objects of differing masses $\left( m _ { a } , m _ { b } \right)$ are placed at the top of two frictionless inclines having the same height. Mass $m _ { a }$ is on an incline that makes an angle of $15 ^ { \circ }$ with the horizontal; mass $m _ { b }$ is on an incline that makes an angle of $75 ^ { \circ }$ with the horizontal. The masses are released at the same time. The mass that has the greater acceleration along the incline is:

A) $m _ { a }$
B) $m _ { b }$
C) Both masses have the same acceleration.
D) This cannot be determined unless both masses ( $m _ { a } , m _ { b }$ ) are known.

Question 18

Two objects of differing masses $\left( m _ { a } , m _ { b } \right)$ are placed at the top of two frictionless inclines having the same height. Mass $m _ { a }$ is on an incline that makes an angle of $15 ^ { \circ }$ with the horizontal; mass $m _ { b }$ is on an incline that makes an angle of $75 ^ { \circ }$ with the horizontal. The masses are released at the same time. The mass that reaches the bottom first is

A) $m _ { a }$
B) $m _ { b }$
C) Both masses reach the bottom at the same time.
D) This cannot be determined unless both the masses ( $m _ { a } , m _ { b }$ ) are known.

Question 19

A solid ball $\mathbf { A }$ of mass $m$ and radius $r$ is placed at the top of a smooth, frictionless incline. Another solid ball $\mathbf { B }$ of equal mass and equal radius is placed at the top of a rough incline. Both inclines have the same height $h$ and the same angle. The balls are released. At the bottom, the ball that has the greater total kinetic energy is

A) ball A.
B) ball B.
C) Both have the same.
D) This cannot be determined unless $\mu$ of the rough incline is known.

Question 20

A solid ball $\mathbf { A }$ of mass $m$ and radius $r$ is placed at the top of a smooth, frictionless incline. Another solid ball $\mathbf { B }$ of equal mass and equal radius is placed at the top of a rough incline. Both inclines have the same height $h$ and the same angle. The balls are released. At the bottom, the ball that has the greater translational kinetic energy is

A) ball A.
B) ball B.
C) Both have the same.
D) This cannot be determined unless $\mu$ of the rough incline is known.

Question 21

A solid ball $\mathbf { A }$ of mass $m$ and radius $r$ is placed at the top of a smooth, frictionless incline. Another solid ball $\mathbf { B }$ of equal mass and equal radius is placed at the top of a rough incline. Both inclines have the same height $h$ and the same angle. The balls are released. At the bottom, the ball that has the greater rotational kinetic energy is

A) ball A.
B) ball B.
C) Both have the same.
D) This cannot be determined unless $\mu$ of the rough incline is known.

Question 22

A solid ball $\mathbf { A }$ of mass $m$ and radius $r$ is placed at the top of a smooth, frictionless incline. Another solid ball $\mathbf { B }$ of equal mass and equal radius is placed at the top of a rough incline. Both inclines have the same height $h$ and the same angle. The balls are released at the same time. The ball that reaches the bottom first is

A) ball A.
B) ball B.
C) Both reach the bottom at the same time.
D) This cannot be determined unless $\mu$ of the rough incline is known.

Question 23

An object of mass $m$ initially at rest is acted upon by a single force $f$ over a distance of $d$ . The final velocity of the mass is

A) $f d$
B) $\sqrt { 2 f d / m }$
C) $\frac { 1 } { 2 } m v ^ { 2 }$
D) $m g h$

Question 24

An object of mass $m$ initially at rest is acted upon by a single force $f$ over a distance of $d$ . The final kinetic energy of the object is

A) $f d$
B) $\sqrt { 2 f d / m }$
C) $0$
D) $m g h$

Question 25

An object of mass $m$ initially at rest is acted upon by a single force $f$ over a distance of $d$ . If the single force is the weight of the object, and the distance that the object moved was its original height $( d = h )$ , the final kinetic energy of the object is

A) $\mathrm { fh } / \mathrm { m }$
B) $\sqrt { 2 f h / m }$
C) $0$
D) $m g h$

Question 26

A conservative force

A) may do net work on a body that moves no net distance during the application of the force.
B) may change the speed of a body that moves no net distance during the application of the force.
C) may do net work on a body undergoing motion confined to a direction perpendicular to the direction of the force.
D) may change the direction of motion of a body that moves no net distance during the application of the force.

Question 27

For a conservative force F, all of the following are correct except

A) the line integral of F around a closed path is zero.
B) the line integral of F between points A and B by way of point C is independent of point C.
C) the line integral of F with limits A and B is independent of the order of the limits.
D) there exists an associated potential energy function.

Question 28

Forms of energy include all of the following except

A) gravitational.
B) kinetic.
C) normal.
D) none of the above.

Question 29

All of the following are acceptable dimensions for energy except

A) kilowatt-hours.
B) kilocalories.
C) kilograms.
D) electron-volts.

Question 30

Power has the same dimensions as all of the following except

A) [force] x [speed].
B) [work] x [acceleration].
C) [energy] / [time].
D) none of the above.

Question 31

Force has the same dimensions as all of the following except

A) BTUs/foot.
B) kilograms-meters/second².
C) watt-feet.
D) horsepower/(feet/second).

Question 32

For a small body in contact with and sliding down a frictionless surface of a large stationary sphere, all of the following are dependent on the location of the body on the sphere except

A) the radius of the path.
B) the angle the surface of contact makes with horizontal.
C) the height above the bottom point of the sphere.
D) none of the above.

Question 33

For a small body in contact with and sliding down a frictionless surface of a large stationary sphere, all of the following depend on the location of the body on the sphere except

A) the gravitational potential energy of the small body.
B) the speed of the body.
C) the velocity of the body.
D) none of the above.

Question 34

For a small body in contact with and sliding down the surface of a large stationary sphere, if friction is introduced, the distance along the sphere before the small body loses contact is

A) greater than the distance for the frictionless case.
B) equal to the distance for the frictionless case.
C) less than the distance for the frictionless case.
D) unknown; insufficient information is given to determine the answer.

Question 35

The potential energy U is related to the (conservative) force F by an equation of the form

A) U $\propto$ F · dr.
B) U $\propto$ 11ef2d7c_8600_0f4c_a6da_1197495b9a7b_TBN1003_11 F_r dt.
C) U $\propto$ dF_r/dr.
D) U $\propto$ dF_r/dt.

Question 36

The (conservative) force F_Y is related to the potential energy U by an equation of the form

A) F_Y $\propto$ U dy.
B) F_Y $\propto$ 11ef2d7c_8600_0f4c_a6da_1197495b9a7b_TBN1003_11U dt.
C) F_Y $\propto$ dU/dy.
D) F_Y $\propto$ dU/dt.

Question 37

If the force exerted by a spring is given by F_z = -kz, then the potential energy associated with the spring is given by U $\propto$

A) kz².
B) k + a constant.
C) -kz + a constant.
D) kz³.

Question 38

If a force of 1 newton is required to compress a spring 1 meter, the potential energy stored in a spring compressed 2 meters from its uncompressed length is (in joules)

A) 0.5.
B) 1.0.
C) 2.0.
D) 4.0.

Question 39

If the total potential energy stored in a spring compressed 2 centimeters from its uncompressed length is 20.0 joules, the total potential energy stored in a spring compressed an additional 1 centimeter is (in joules)

A) 10.0.
B) 22.5.
C) 45.0.
D) 67.5.

Question 40

The magnitude of the additional potential energy stored in a spring stretched an additional 2 centimeters from some designated reference length

A) increases as that reference length increases beyond the unstretched length.
B) increases as that reference length increases beyond any length.
C) remains unchanged as that reference length increases beyond the unstretched length.
D) remains unchanged as that reference length increases beyond any length.

Question 41

The justification for your previous response depends on the fact that

A) the force required to stretch a spring increases as the length of stretching increases.
B) the force required to stretch a spring increases as the absolute length of the spring increases.
C) the force required to stretch a spring remains unchanged as the length of stretching increases.
D) the force required to stretch a spring remains unchanged as the absolute length of the spring increases.

Question 42

When a mass hanging vertically from a spring attached to the ceiling is in stable equilibrium, the two forms of energy for the system that are at a minimum are

A) the total energy and the kinetic energy.
B) the kinetic energy and the gravitational potential energy.
C) the gravitational potential energy and the spring potential energy.
D) none of the above.

Question 43

When a mass hanging vertically from a spring attached to the ceiling is in stable equilibrium, the two forms of energy for the system that are at a minimum are

A) the total energy and the gravitational potential energy.
B) the kinetic energy and the spring potential energy.
C) the spring potential energy and the total energy.
D) none of the above.

Question 44

When nonconservative forces are included in a system, energy conservation equations

A) hold just as before.
B) hold, with modification by including thermal energy changes.
C) no longer hold, no matter how modified.
D) Hold it! Nonconservative forces cannot (by definition) be included in the analysis of a system.

Question 45

Two common forms of energy that cannot be negative are

A) total energy and kinetic energy.
B) kinetic energy and gravitational potential energy.
C) gravitational potential energy and spring potential energy.
D) none of the above.

Question 46

Two common forms of energy that cannot be negative are

A) total energy and gravitational potential energy.
B) kinetic energy and spring potential energy.
C) spring potential energy and total energy.
D) none of the above.

Question 47

The mechanical power delivered by a force F that moves a body at a velocity v is given by

A) F $\times$ v.
B) F · v.
C) F/v.
D) v/F.

Question 48

The proper relationship between power P and work W is

A) P = dW/dt.
B) W = dP/dt.
C) Either of the first two responses is valid, depending on the circumstances.
D) Neither of the first two responses is valid.

Question 49

The relationship between kilowatt-hours and joules is

A) 3.6 x 10⁶ kilowatt-hours = 1 joule.
B) 3.6 x 10⁶ joules = 1 kilowatt-hour.
C) 1/3600 kilowatt-hours = 1 joule.
D) 1/3600 joules = 1 kilowatt-hour.

Question 50

The equation E = mc² implies all of the following except that

A) energy must travel at the speed of light.
B) any mass has energy.
C) any energy has mass.
D) none of the above.