## Mastering Physics Solutions Chapter 8 Potential Energy And Conservation Of Energy

**Chapter 8 Potential Energy And Conservation Of Energy Q.1CQ**

Is it possible for the kinetic energy of an object to be negative? Is it possible for the gravitational potential energy of an object to be negative? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.1P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.2CQ**

An avalanche occurs when a mass of snow slides down a steep mountain slope. Discuss the energy conversions responsible for water vapor rising to form clouds, falling as snow on a mountain, and then sliding down a slope as an avalanche.

**Solution:**

As water vapor rises, there is an increase in the gravitational potential energy of the system. Part of this potential energy is released as snow and falls onto the mountain. If an avalanche occurs, the snow on the mountain accelerates down the slope, converting more gravitational potential energy into kinetic energy.

**Chapter 8 Potential Energy And Conservation Of Energy Q.2P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.3CQ**

If the stretch of a spring is doubled, the force it exerts is also doubled. By what factor does the spring’s potential energy increase?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.3P**

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**Chapter 8 Potential Energy And Conservation Of Energy Q.4CQ**

When a mass is placed on top of a vertical spring, the spring compresses and the mass moves downward. Analyze this system in terms of its mechanical energy.

**Solution:**

The initial mechanical energy of the system is the gravitational potential energy of the mass-Earth system. As the mass moves downward, the gravitational potential energy of the system decreases.

At the same time, the potential energy of the spring increases because it is compressed. Initially, the decrease in gravitational potential energy is greater than the increase in the spring’s potential energy, which means that the mass gains kinetic energy. Eventually, the increase in the spring’s energy equals the decrease in the gravitational energy, and the mass comes to rest.

**Chapter 8 Potential Energy And Conservation Of Energy Q.4P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.5CQ**

If a spring is stretched so far that it is permanently deformed, its force is no longer conservative. Why?

**Solution:**

We know that the external force must be equal to the restoring force, and its direction is opposite to the direction of the restoring force. If the external force is greater than the restoring force, then the spring gets permanently deformed. In this situation, the work that was done to stretch the spring is not fully recovered. Some of the work is converted into the energy of the deformation. For this reason, the spring force is not conservative during deformation.

**Chapter 8 Potential Energy And Conservation Of Energy Q.5P**

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**Chapter 8 Potential Energy And Conservation Of Energy Q.6CQ**

An object is thrown upward to a person on a roof. At what point is the object’s kinetic energy at maximum? At what point is the potential energy of the system at maximum? At what locations do these energies have their minimum values?

**Solution:**

When the object is first thrown upward, its speed and its kinetic energy are at a maximum. Its potential energy is zero at that moment.

(i) The potential energy of the system is at a maximum at the highest point of the object’s flight and is at a minimum at the starting point of its journey (when it has just been released).

(ii) The kinetic energy of the system is at maximum when the object has just been thrown up and is at a minimum when it reaches its highest point of flight.

**Chapter 8 Potential Energy And Conservation Of Energy Q.6P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.7CQ**

It is a law of nature that the total energy of the universe is conserved. What do politicians mean, then, when they urge “energy conservation”?

**Solution:**

When the term “energy conservation” is used in everyday language, it doesn’t refer to the total amount of energy in the universe. Instead it refers to using energy wisely, especially when a particular source of energy like oil or natural gas is finite and non-renewable.

**Chapter 8 Potential Energy And Conservation Of Energy Q.7P**

Predict/Explain Ball 1 is thrown to the ground with an initial downward speed; ball 2 is dropped to the ground from rest. Assuming the balls have the same mass and are released from the same height, is the change in gravitational potential energy of ball 1 greater than, less than, or equal to the change in gravitational potential energy of ball 2? (b) Choose the best explanation from among the following:

I. Ball 1 has the greater total energy, and therefore more energy can go into gravitational potential energy.

II. The gravitational potential energy depends only on the mass of the ball and the drop height.

III. All of the initial energy of ball 2 is gravitational potential energy.

**Solution:**

(a) The change in gravitational potential energy of the ball 1 is equal to the change in gravitational potential energy of the ball 2.

(b) This is because the change in gravitational potential energy depends only on the mass of the ball and the height from which the ball is dropped. Therefore option II is the best explanation.

**Chapter 8 Potential Energy And Conservation Of Energy Q.8CQ**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.8P**

A mass is attached to the bottom of a vertical spring. This causes the spring to stretch and the mass to move downward. (a) Does the potential energy of the spring increase, decrease, or stay the same during this process? Explain. (b) Does the gravitational potential energy of the Earth-mass system increase, decrease, or stay the same during this process? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.9CQ**

**Solution:**

We have a photograph which shows an earthmover digging the earth and lifting it and loading it on the truck. In this situation, there are both conservative and non-conservative works involved.

Initially, the engine of the earthmover does positive, non-conservative work as it digs out and lifts a load of rocks. At the same time, gravity does negative, conservative work on the rocks as the gravitational potential energy of the system increases. As the rock is transported to the truck, the earthmover does positive, non-conservative work. When the rocks are released, gravity does positive, conservative work as the gravitational potential energy of the system is converted into kinetic energy. The kinetic energy of the rocks is converted into sound and heat when they are loaded into the truck.

As the truck is loaded, the spring is compressed. The potential energy is stored in the form of the spring’s potential energy, and the work done by the truck is conservative.

**Chapter 8 Potential Energy And Conservation Of Energy Q.9P**

As an Acapulco cliff diver drops to the water from a height of 46 m, his gravitational potential energy decreases by 25,000 J. What is the diver’s weight in newtons?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.10CQ**

A toy frog consists of a suction cup and a spring. When the suction cup is pressed against a smooth surface, the frog is held down. When the suction cup lets go, the frog leaps into the air. Discuss the behavior of the frog in terms of energy conversions.

**Solution:**

The toy frog consists of a suction cup and a spring. When the suction cup is pressed, the energy of the toy is stored in the form of the spring as potential energy. When the suction cup lets go, then all its potential energy is converted into kinetic energy. As a result, the frog leaps into the air. The total energy of the system is conserved.

**Chapter 8 Potential Energy And Conservation Of Energy Q.10P**

Find the gravitational potential energy of an 88-kg person standing atop Mt. Everest at an altitude of 8848 m. Use sea level as the location for y = 0.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.11CQ**

If the force on an object is zero, does that mean the potential energy of the system is zero? If the potential energy of a system is zero, is the force zero?

**Solution:**

No

Zero force implies a zero rate of change in the potential energy. However, the value of the potential energy can be anything at all. Similarly, if the potential energy is zero, it does not mean that the force is zero. Again, what matters is the rate of change of the potential energy.

**Chapter 8 Potential Energy And Conservation Of Energy Q.11P**

Jeopardy! Contestante on the game show Jeopardy! depress spring-loaded buttons to “buzz in” and provide the question corresponding to the revealed answer. The force constant on these buttons is about 130 N/m. Estimate the amount of energy it takes—at a minimum—to buzz in.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.12CQ**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.12**

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**Chapter 8 Potential Energy And Conservation Of Energy Q.13CQ**

When a ball is thrown upward, it spends the same amount of time on the way up as on the way down—as long as air resistance can be ignored. If air resistance is taken into account, is the time on the way down the same as, greater than, or less than the time on the way up? Explain.

**Solution:**

If the air resistance is taken into account, then the total mechanical energy of the system decreases. The distance covered by the ball is the same on the way down as it is on the way up, and so the amount of time will be determined by the average speed of the ball on the two portions of its trip. Note that air resistance does negative, non-conservative work continuously on the ball as it moves. Therefore, its total mechanical energy is less on the way down than it is on the way up, which means that its speed at any given elevation is less on the way down. It follows that more time is required for the downward portion of the trip.

**Chapter 8 Potential Energy And Conservation Of Energy Q.13P**

A vertical spring stores 0.962 J in spring potential energy when a 3.5-kg mass is suspended from it. (a) By what multiplicative factor does the spring potential energy change if the mass attached to the spring is doubled? (b) Verify your answer to part (a) by calculating the spring potential energy when a 7.0-kg mass is attached to the spring.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.14P**

Pushing on the pump of a soap dispenser compresses a small spring. When the spring is compressed 0.50 cm, its potential energy is 0.0025 J. (a) What is the force constant of the spring? (b) What compression is required for the spring potential energy to equal 0.0084 J?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.15P**

A force of 4.1 N is required to stretch a certain spring by 1.4 cm. (a) How far must this spring be stretched for its potential energy to be 0.020 J? (b) How much stretch is required for the spring potential energy to be 0.080 J?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.16P**

The work required to stretch a certain spring from an elongation of 4.00 cm to an elongation of 5.00 cm is 30.5 J. (a) Is the work required to increase the elongation of the spring from 5.00 an to 6.00 cm greater than, less than, or equal to 30.5 J? Explain. (b) Verify your answer to part (a) by calculating the required work.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.17P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.18P**

Predict/Explain You throw a ball upward and let it fall to the ground. Your friend drops an identical ball straight down to the ground from the same height. Is the change in kinetic energy of your ball greater than, less than, or equal to the change in kinetic energy of your friend’s ball? (b) Choose the best explanation from among the following:

I. Your friend’s ball converts all its initial energy into kinetic energy.

II. Your ball is in the air longer, which results in a greater change in kinetic energy.

III. The change in gravitational potential energy is the same for each ball, which means the change in kinetic energy must be the same also.

**Solution:**

(a) The Change in kinetic energy of your ball is equal to the change in kinetic energy of your friend’s ball.

(b) As both of you are at the same height, so when the balls come to the ground both balls lose same amount of potential energy. Therefore change in potential energy of the two balls is same. Then from conservation of energy the change in kinetic energy is also same.

So, option III is the best explanation.

**Chapter 8 Potential Energy And Conservation Of Energy Q.19P**

Suppose the situation described in Conceptual Checkpoint 8-2 is repeated on the fictional planet Epsilon, where the acceleration due to gravity is less than it is on the Earth. (a) Would the height of a hill on Epsilon that causes a reduction in speed from 4 m/s to 0 be greater than, less than, or equal to the height of the corresponding hill on Earth? Explain. (b) Consider the hill on Epsilon discussed in part (a). If the initial speed at the bottom of the hill is 5 m/s, will the final speed at the top of the hill be greater than, less than, or equal to 3 m/s? Explain.

**Solution:**

Solution:

(a) As we are considering on a planet Epsilon where the acceleration due to gravity is less than that on the earth and as the gravitational potential energy is mgh so larger height would be needed to gain the same level of gravitational potential energy. Thus, a higher hill would be needed.

(b) The initial speed at the bottom of the Epsilon is 5m/s which are the same on the earth. Since the values of the initial kinetic energy & final potential energies are the same on the Epsilon.

So therefore the final speed at the top of the hill on Epsilon is the same which is 3m/s

**Chapter 8 Potential Energy And Conservation Of Energy Q.20P**

Predict/Explain When a ball of mass m is dropped from. rest from a height h, its kinetic energy just before landing is K. Now, suppose a second ball of mass Am is dropped from rest from a height h/4. (a) Just before ball 2 lands, is its kinetic energy 4K, 2K, K, K/2, or K/4? (b) Choose the best explanation from among the following:

I. The two balls have the same initial energy.

II. The more massive ball will have the greater kinetic energy.

III. The reduced drop height results in a reduced kinetic energy.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.21P**

Predict/Explain When a ball of mass m is dropped from rest from a height h, its speed just before landing is v. Now, suppose a second ball of mass 4m is dropped from rest from a height h/4. (a) Just before ball 2 lands, is its speed 4v, 2v, v, v/2, or v/4? (b) Choose the best explanation from among the following:

I. The factors of 4 cancel; therefore, the landing speed is the same.

II. The two balls land with the same kinetic energy; therefore, the ball of mass 4m has the speed v/2.

III. Reducing the height by a factor of 4 reduces the speed by a factor of 4.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.22P**

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**Chapter 8 Potential Energy And Conservation Of Energy Q.23P**

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**Chapter 8 Potential Energy And Conservation Of Energy Q.24P**

At an amusement park, a swimmer uses a water slide to enter the main pool. If the swimmer starts at rest, slides without friction, and descends through a vertical height of 2.31 m, what is her speed at the bottom of the slide?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.25P**

In the previous problem, find the swimmer’s speed at the bottom of the slide if she starts with an initial speed of 0.840 m/s.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.26P**

A player passes a 0.600-kg basket ball downcourt for a fast break. The ball leaves the player’s hands with a speed of 8.30 m/s and slows down to 7.10 m/s at its highest point. (a) Ignoring air resistance, how high above the release point is the ball when it is at its maximum height? (b) How would doubling the ball’s mass affect the result in part (a)? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.27P**

**Solution:**

Here the all the three balls are at same height when they reached to the dashed line. And all the balls started from same height. Therefore the gain in potential energy of the three balls is same. Therefore the loss in kinetic energy of each ball is same. Also all the three balls have same initial speed. So all the three balls started with same speed and lost equal amount of kinetic energy. Therefore all the three balls have same speed at the dashes line. So, option (C) is correct.

**Chapter 8 Potential Energy And Conservation Of Energy Q.28P**

In a tennis match, a player wins a point by hitting the ball sharply to the ground on the opponent’s side of the net. (a) If the ball bounces upward from the ground with a speed of 16 m/s, and is caught by a fan in the stands with a speed of 12 m/s, how high above the court is the fan? Ignore air resistance. (b) Explain why it is not necessary to know the mass of the tennis ball.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.29P**

**Solution:**

Here the all the three balls are at same height when they reached to the dashed line. And all the balls started from same height. Therefore the gain in potential energy of the three balls is same. Therefore the loss in kinetic energy of each ball is same. Also all the three balls have same initial speed. So all the three balls started with same speed and lost equal amount of kinetic energy. Therefore all the three balls have same speed at the dashes line. So, option (C) is correct.

**Chapter 8 Potential Energy And Conservation Of Energy Q.30P**

A 2.9-kg block slides with a speed of 1.6 m/s on a frictionless horizontal surface until it encounters a spring. (a) If the block compresses the spring 4.8 cm before coming to rest, what is the force constant of the spring? (b) What initial speed should the block have to compress the spring by 1.2 cm?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.31P**

A 0.26-kg rock is thrown vertically upward from the top of a cliff that is 32 m high. When it hits the ground at the base of the cliff, the rock has a speed of 29 m/s. Assuming that air resistance can be ignored, find (a) the initial speed of the rock and (b) the greatest height of the rock as measured from the base of the cliff.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.32P**

A 1.40-kg block slides with a speed of 0.950 m/s on a frictionless horizontal surface until it encounters a spring with a force constant of 734 N/m. The block comes to rest after compressing the spring 4.15 cm. Find the spring potential energy, u, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for compressions of (a) 0 cm, (b) 1.00 cm, (c) 2.00 cm, (d) 3.00 cm, and (e) 4.00 cm.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.33P**

A 5.76-kg rock is dropped and allowed to fall freely. Find the initial kinetic energy, the final kinetic energy, and the change in kinetic energy for (a) the first 2.00 m of fall and (b) the second 2.00 m of fall.

**Solution:**

If the mass of the bob increases, then the first part in the above equation decreases hence the speed of bob at point A increases. Therefore, the answer to part (b) increases as the mass increases.

**Chapter 8 Potential Energy And Conservation Of Energy Q.34P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.35P**

In the previous problem, (a) what is the bob’s kinetic energy at point B? (b) At some point the bob will come to rest momentarily. Without doing an additional calculation, determine the change in the system’s gravitational potential energy between point B and the point where the bob comes to rest. (c) Find the maximum angle the string makes with the vertical as the bob swings back and forth. Ignore air resistance.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.36P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.37P**

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**Chapter 8 Potential Energy And Conservation Of Energy Q.38P**

You coast up a hill on your bicycle with decreasing speed. Your friend pedals up the hill with constant speed. (a) Ignoring friction, does the mechanical energy of the you-bike-Earth system increase, decrease, or stay the same? Explain. (b) Does the mechanical energy of the friend-bike-Earth system increase, decrease, or stay the same? Explain.

**Solution:**

(A) When a person coasts up a hill on his bicycle with decreasing speed, no non-conservative work is done on his bicycle. Therefore, his mechanical energy is conserved even though his speed decreases.

(B)

To maintain a constant speed, his friend will have to do positive non-conservative work while going uphill. Thus, his friend’s mechanical energy will increase.

**Chapter 8 Potential Energy And Conservation Of Energy Q.39P**

Predict/Explain On reentry, the space shuttle’s protective heat tiles become extremely hot. (a) Is the mechanical energy of the shuttle-Earth system when the shuttle lands greater than, less than, or the same as when it is in orbit? (b) Choose the best explanation from among the following:

I. Dropping out of orbit increases the mechanical energy of the shuttle.

II. Gravity is a conservative force.

III. A portion of the mechanical energy has been converted to heat energy.

**Solution:**

The mechanical energy E of a system is the sum of potential energy U and kinetic energy K.

Non conservative forces might decrease the mechanical energy by converting it to heat energy, or increase it by converting muscular work to kinetic or potential energy.

(a)

The space shuttle’s protective heat tiles become extremely hot, means that some amount of mechanical energy is converted to heat energy due to some non-conservative forces, and total mechanical energy of the system will decrease.

Thus, the mechanical energy of shuttle-Earth system when the shuttle lands is that of the mechanical energy when the shuttle is in orbit.

(b)

The space shuttle’s protective heat tiles become extremely hot, means that some amount of mechanical energy has been converted to heat energy.

Thus, the correct option is .

**Chapter 8 Potential Energy And Conservation Of Energy Q.40P**

Catching a wave, a 77-kg surfer starts with a speed of 1.3 m/s, drops through a height of 1.65 m, and ends with a speed of 8.2 m/s. How much nonconservative work was done on the surfer?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.41P**

At a playground, a 19-kg child plays on a slide that drops through a height of 2.3 m. The child starts at rest at the top of the slide. On the way down, the slide does a nonconservative work of —361 J on the child. What is the child’s speed at the bottom of the slide?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.42P**

Starting at rest at the edge of a swimming pool, a 72.0-kg athlete swims along the surface of the water and reaches a speed of 1.20 m/s by doing the work Wnc1 = + 161 J. Find the nonconservative work, Wnc2, done by the water on the athlete.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.43P**

A 17,000-kg airplane lands with a speed of 82 m/s on a stationary aircraft carrier deck that is 115 m long. Find the work done by nonconservative forces in stopping the plane.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.44P**

The driver of a 1300-kg car moving at 17 m/s brakes quickly to 11 m/s when he spots a local garage sale. (a) Find the change in the car’s kinetic energy. (b) Explain where the “missing” kinetic energy has gone.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.45P**

You ride your bicycle down a hill, maintaining a constant speed the entire time. (a) As you ride, does the gravitational potential energy of the you-bike-Earth system increase, decrease, or stay the same? Explain. (b) Does the kinetic energy of you and your bike increase, decrease, or stay the same? Explain. (c) Does the mechanical energy of the you-bike-Earth system increase, decrease, or stay the same? Explain.

**Solution:**

Solution:

(a) The gravitational potential energy of the earth-bike-rider system should decrease.

Since the position of the biker could eventually drop to ground level height, the potential energy of the system could go down to zero.

(b) Since the speed is constant throughout the entire time. The kinetic energy of you & your bike remains the same

(c) The mechanical energy of the system decrease. Since the total mechanical energy is equal to the sum of potential energy and kinetic energy. As the potential energy decreases & kinetic energy remains the same. So therefore the mechanical energy of the system decreases.

**Chapter 8 Potential Energy And Conservation Of Energy Q.46P**

Suppose the system in Example starts with m2 moving downward with a speed of 1.3 m/s. What speed do the masses have just before m2 lands?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.47P**

A 42.0-kg seal at an amusement park slides from rest down a ramp into the pool below. The top of the ramp is 1.75 m higher than the surface of the water, and the ramp is inclined at an angle of 35.0° above the horizontal. If the seal reaches the water with a speed of 4.40 m/s, what are (a) the work done by kinetic friction and (b) the coefficient of kinetic friction between the seal and the ramp?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.48P**

A 1.9-kg rock is released from rest at the surface of a pond 1.8 m deep. As the rock falls, a constant upward force of 4.6 N is exerted on it by water resistance. Calculate the nonconservative work, Wnc, done by water resistance on the rock, the gravitational potential energy of the system, u, the kinetic energy of the rock, K, and the total mechanical energy of the system, E, when the depth of the rock below the water’s surface is (a) 0 m, (b) 0.50 m, and (c) 1.0 m. Let y = 0 be at the bottom of the pond.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.49P**

A 1250-kg car drives up a hill that is 16.2 m high. During the drive, two nonconservative forces do work on the car: (i) the force of friction, and (ii) the force generated by the car’s engine. The work done by friction is −3.11 × 105 J; the work done by the engine is + 6.44 × 105 J. Find the change in the car’s kinetic energy from the bottom of the hill to the top of the hill.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.50P**

An 81.0-kg in-line skater does + 3420 J of nonconservative work by pushing against the ground with his skates. In addition, friction does −715 J of nonconservative work on the skater. The skater’s initial and final speeds are 2.50 m/s and 1.22 m/s, respectively. (a) Has the skater gone uphill, downhill, or remained at the same level? Explain. (b) Calculate the change in height of the skater.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.51P**

In Example, suppose the two masses start from rest and are moving with a speed of 2.05 m/s just before m2 hits the floor. (a) If the coefficient of kinetic friction is µk = 0.350, what is the distance of travel, d, for the masses? (b) How much conservative work was done on this system? (c) How much non-conservative work was done on this system? (d) Verify the three work relations given in Equations 8-10.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.52P**

A 15,800-kg truck is moving at 12.0 m/s when it starts down a 6.00° incline in the Canadian Rockies. At the start of the descent the driver notices that the altitude is 1630 m. When she reaches an altitude of 1440 m, her speed is 29.0 m/s. Find the change in (a) the gravitational potential energy of the system and (b) the truck’s kinetic energy. (c) Is the total mechanical energy of the system conserved? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.53P**

A 1.80-kg block slides on a rough horizontal surface. The block hits a spring with a speed of 2.00 m/s and compresses it a distance of 11.0 cm before coming to rest. If the coefficient of kinetic friction between the block and the surface is µk = 0.560, what is the force constant of the spring?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.54P**

**Solution:**

The above figure shows that, the object is at rest at initial point A. As the object moves from A to B, some of its potential energy is converted into kinetic energy, and the speed of the object increases at B. So the kinetic energy of the object increases when potential energy decreases.

Now the object moves from point B to point C. In this process, some of its kinetic energy is again converted into potential energy, thus, the speed of the object decreases as it moves from B to C. The object’s speed again increases as it moves from point C to point D, so its potential energy decreases while its kinetic energy increases.

Finally from point D to point E, the kinetic energy of the object decreases as curve rises. Thus, the speed of the object decreases. Therefore, at point E, the speed of the object momentarily becomes zero.

**Chapter 8 Potential Energy And Conservation Of Energy Q.55P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.56P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.57P**

A 23-kg child swings back and forth on a swing suspended by 2.5-m-long ropes. Plot the gravitational potential energy of this system as a function of the angle the ropes make with the vertical, assuming the potential energy is zero when the ropes are vertical. Consider angles up to 90° on either side of the vertical.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.58P**

Find the turning-point angles in the previous problem if the child has a speed of 0.89 m/s when the ropes are vertical. Indicate the turning points on a plot of the system’s potential energy.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.59P**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.60P**

A block of mass m = 0.95 kg is connected to a spring of force constant k = 775 N/m on a smooth, horizontal surface. (a) Plot the potential energy of the spring from x = −5.00 cm to x = 5.00 cm. (b) Determine the turning points of the block if its speed at x = 0 is 1.3 m/s.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.61P**

A ball of mass m = 0.75 kg is thrown straight upward with an initial speed of 8.9 m/s. (a) Plot the gravitational potential energy of the block from its launch height, y = 0, to the height y = 5.0 m. Let u = 0 correspond to y = 0. (b) Determine the turning point (maximum height) of this mass.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.62P**

Two blocks, the of mass m, are connected on a frictionless horizontal table by a spring of force constant k and equilibrium length L. Find the maximum and minimum separation between the two blocks in terms of their maximum speed, vmax,relative to the table. (The two blocks always move in opposite directions as they oscillate back and forth about a fixed position.)

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.63GP**

You and a friend both solve a problem involving a skier going down a slope. When comparing solutions, you notice that your choice for the y = 0 level is different than the y = 0 level chosen by your friend. Will your answers agree or disagree on the following quantities: (a) the skier’s potential energy; (b) the skier’s change in potential energy; (c) the skier’s kinetic energy?

**Solution:**

Answer:

Your answers will disagree on (a), but agree on (b) and (c)

The gravitational potential energy depends upon the reference level but not the change in potential energy. The work done by gravity must be the same in the two solutions so change in potential energy and change in kinetic energy should be same.

**Chapter 8 Potential Energy And Conservation Of Energy Q.64GP**

A particle moves under the influence of a conservative force. At point A the particle has a kinetic energy of 12 J; at point 13 the particle is momentarily at rest, and the potential energy of the system is 25 J; at point C the potential energy of the system is 5 J. (a) What is the potential energy of the system when the particle is at point A? (b) What is the kinetic energy of the particle at point C?

**Solution:**

(a) In the given data the particle is at rest at point B having potential energy 25J. Therefore the total energy of the particle is 25J.

Now at point A the particle has kinetic energy of 12J.

Therefore the potential energy of the system at point A is 25J-12J=13J

(b) Now at point C the particle has potential energy of 5J.

Therefore the kinetic energy of the system at point C is 25J-5J=20J

**Chapter 8 Potential Energy And Conservation Of Energy Q.65GP**

A leaf falls to the ground with constant speed. Is ui + Ki for this system greater than, less than, or the same as uf + Kf for this system? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.66GP**

Consider the two-block system shown in Example. (a) As block 2 descends through the distance d, does its mechanical energy increase, decrease, or stay the same? Explain. (b) Is the nonconservative work done on block 2 by the tension in the rope positive, negative, or zero? Explain.

**Solution:**

Solution:

(a) Since there is no conservative work with this system (the friction from mass 1) the mechanical energy of the system will change. Specifically, the mechanical energy of the system will decrease due to energy being converted into heat by the friction forces.

(b) The tension in the rope deals with the weight of the block that’s dropping down. The non conservative force comes from the friction of the block on the table. The tension in the rope simply deals with mass and gravitational acceleration thus, the tension in the rope is involved in potential/kinetic energy relations, and the amount of non conservative work it does is zero.

**Chapter 8 Potential Energy And Conservation Of Energy Q.67GP**

Taking a leap of faith, a bungee jumper steps off a plat-form and falls until the cord brings her to rest. Suppose you analyze this system by choosing y = 0 at the platform level, and your friend chooses y = 0 at ground level. (a) Is the jumper’s initial potential energy in your calculation greater than, less than, or equal to the same quantity in your friend’s calculation? Explain. (b) Is the change in the jumper’s potential energy in your calculation greater than, less than, or equal to the same quantity in your friend’s calculation? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.68GP**

A sled slides without friction down a small, ice-covered hill. If the sled starts from rest at the top of the hill, its speed at the bottom is 7.50 m/s. (a) On a second run, the sled starts with a speed of 1.50 m/s at the top. When it reaches the bottom of the hill, is its speed 9.00 m/s, more than 9.00 m/s, or less than 9.00 m/s? Explain. (b) Find the speed of the sled at the bottom of the hill after the second run.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.69GP**

In the previous problem, what is the height of the hill?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.70GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.71GP**

Running Shoes The soles of a popular make of running shoe have a force constant of 2.0 × 105 N/m. Treat the soles as ideal springs for the following questions. (a) If a 62-kg person stands in a pair of these shoes, with her weight distributed equally on both feet, how much does she compress the soles? (b) How much energy is stored in the soles of her shoes when she’s standing?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.72GP**

Nasal Strips The force required to flex a nasal strip and apply it to the nose is 0.25 N; the energy stored in the strip when flexed is 0.0022 J. Assume the strip to be an ideal spring for the following calculations. Find (a) the distance through which the strip is flexed and (b) the force constant of the strip.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.73GP**

A pendulum bob with a mass of 0.13 kg is attached to a string with a length of 0.95 m. We choose the potential energy to be zero when the string makes an angle of 90° with the vertical. (a) Find the potential energy of this system when the string makes an angle of 45° with the vertical. (b) Is the magnitude of the change in potential energy from an angle of 90° to 45° greater than, less than, or the same as the magnitude of the change from 45° to 0°? Explain. (c) Calculate the potential energy of the system when the string is vertical.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.74GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.75GP**

An 1865-kg airplane starts at rest on an airport runway at sea level. (a) What is the change in mechanical energy of the airplane if it climbs to a cruising altitude of 2420 m and maintains a constant speed of 96.5 m/s? (b) What cruising speed would the plane need at this altitude if its increase in kinetic energy is to be equal to its increase in potential energy?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.76GP**

At the local playground a child on a swing has a speed of 2.02 m/s when the swing is at its lowest-point. (a) To what maximum vertical height does the child rise, assuming he sits still and “coasts”? Ignore air resistance. (b) How do your results change if the initial speed of the child is halved?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.77GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.78GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.79GP**

A person is to be released from rest on a swing pulled away from the vertical by an angle of 20.0°. The two frayed ropes of the swing are 2.75 m long, and will break if the tension in either of them exceeds 355 N. (a) What is the maximum weight the person can have and not break the ropes? (b) If the person is released at an angle greater than 20.0°, does the maximum weight increase, decrease, or stay the same? Explain.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.80GP**

A car is coasting without friction toward a hill of height h and radius of curvature r. (a) What initial speed, v0, will result in the car’s wheels just losing contact with the roadway as the car crests the hill? (b) What happens if the initial speed of the car is greater than the value found in part (a)?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.81GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.82GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.83GP**

An 8.70-kg block slides with an initial speed of 1.56 m/s up a ramp inclined at an angle of 28.4° with the horizontal. The coefficient of kinetic friction between the block and the ramp is 0.62. Use energy conservation to find the distance the block slides before coming to rest.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.84GP**

Repeat the previous problem for the case of an 8.70-kg block sliding down the ramp, with an initial speed of 1.56 m/s.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.85GP**

Jeff of the Jungle swings on a 7.6-m vine that initially makes an angle of 37° with the vertical. If Jeff starts at restand has a mass of 78 kg, what is the tension in the vine at the lowest point of the swing?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.86GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.87GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.88GP**

Compressing the Ground A running track at Harvard University uses a surface with a force constant of 2.5 × 105 N/m This surface is compressed slightly every time a runner’s foot lands on it. The force exerted by the foot, according to the Saucony shoe company, has a magnitude of 2700 N for a typical runner. Treating the track’s surface as an ideal spring, find (a) the amount of compression caused by a foot hitting the track and (b) the energy stored briefly in the track every time a foot lands.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.89GP**

A Flea’s Jump The resilin in the upper leg (coxa) of a flea has a force constant of about 26 N/m, and when the Flea cocks its jumping legs, the resilin in the leg is stretched by approximately 0.10 mm. Given that the flea has a mass of 0.50 mg, and that two legs are used in a jump, estimate the maximum height a flea can attain by using the energy stored in the resilin. (Assume the resilin to be an ideal spring.)

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.90GP**

A trapeze artist of mass m swings on a rope of length L. Initially, the trapeze artist is at rest and the rope makes an angle θ with the vertical. (a) Find the tension in the rope when it is vertical. (b) Explain why your result for part (a) depends on L in the way it does.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.91GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.92GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.93GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.94GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.95GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.96GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.97GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.98GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.99GP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.100PP**

The Flight of the Dragonflies

Of all the animals you’re likely to see on a summer’s day, the most ancient is the dragonfly. In fact, the fossil record for dragonflies extends back over 250 million years, more than twice as long as for birds. Ancient dragonflies could be as large as a hawk, and were surely buzzing around the heads of both T. Rex and Triceratops.

Dragonflies belong to the order Odonata (“toothed jaws”) and the suborder Anisoptera (“different wings”), a reference to the fact that their hindwings are wider front-to-back than their forewings. (Damselflies, in contrast, have forewings and hind-wings that are the same.) Although ancient in their lineage, dragonflies are the fastest flying and most acrobatic of all insects; some of their maneuvers subject them to accelerations as great as 20g.

The properties of dragonfly wings, and how they account for such speed and mobility, have been of great interest to biologists. Figure shows an experimental setup designed to measure the force constant of Plexiglas models of wings, which are used in wind tunnel tests. A downward force is applied to the model wing at the tip (1 for hindwing, 2 for forewing) or at two-thirds the distance to the tip (3 for hindwing, 4 for forewing). As the force is varied in magnitude, the resulting deflection of the wing is measured. The results are shown in Figure. Notice that significant differences are seen between the hindwings and forewings, as one might expect from their different shapes.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.101PP**

The Flight of the Dragonflies

Of all the animals you’re likely to see on a summer’s day, the most ancient is the dragonfly. In fact, the fossil record for dragonflies extends back over 250 million years, more than twice as long as for birds. Ancient dragonflies could be as large as a hawk, and were surely buzzing around the heads of both T. Rex and Triceratops.

Dragonflies belong to the order Odonata (“toothed jaws”) and the suborder Anisoptera (“different wings”), a reference to the fact that their hindwings are wider front-to-back than their forewings. (Damselflies, in contrast, have forewings and hind-wings that are the same.) Although ancient in their lineage, dragonflies are the fastest flying and most acrobatic of all insects; some of their maneuvers subject them to accelerations as great as 20g.

The properties of dragonfly wings, and how they account for such speed and mobility, have been of great interest to biologists. Figure shows an experimental setup designed to measure the force constant of Plexiglas models of wings, which are used in wind tunnel tests. A downward force is applied to the model wing at the tip (1 for hindwing, 2 for forewing) or at two-thirds the distance to the tip (3 for hindwing, 4 for forewing). As the force is varied in magnitude, the resulting deflection of the wing is measured. The results are shown in Figure. Notice that significant differences are seen between the hindwings and forewings, as one might expect from their different shapes.

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.102PP**

The Flight of the Dragonflies

Of all the animals you’re likely to see on a summer’s day, the most ancient is the dragonfly. In fact, the fossil record for dragonflies extends back over 250 million years, more than twice as long as for birds. Ancient dragonflies could be as large as a hawk, and were surely buzzing around the heads of both T. Rex and Triceratops.

Dragonflies belong to the order Odonata (“toothed jaws”) and the suborder Anisoptera (“different wings”), a reference to the fact that their hindwings are wider front-to-back than their forewings. (Damselflies, in contrast, have forewings and hind-wings that are the same.) Although ancient in their lineage, dragonflies are the fastest flying and most acrobatic of all insects; some of their maneuvers subject them to accelerations as great as 20g.

The properties of dragonfly wings, and how they account for such speed and mobility, have been of great interest to biologists. Figure shows an experimental setup designed to measure the force constant of Plexiglas models of wings, which are used in wind tunnel tests. A downward force is applied to the model wing at the tip (1 for hindwing, 2 for forewing) or at two-thirds the distance to the tip (3 for hindwing, 4 for forewing). As the force is varied in magnitude, the resulting deflection of the wing is measured. The results are shown in Figure. Notice that significant differences are seen between the hindwings and forewings, as one might expect from their different shapes.

**Solution:**

From the graph it is clear that by the application of same force the deflection of wing is more for hindwing than the forewing.

Therefore force constant of forewing is greater than the force constant of hindwing.

So forewing is stiffer than the hindwing.

Therefore option B. is correct.

**Chapter 8 Potential Energy And Conservation Of Energy Q.103PP**

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.104IP**

Referring to Example Consider a spring with a force constant of 955 N/m. (a) Suppose the mass of the block is 1.70 kg, but its initial speed can be varied. What initial speed is required to give a maximum spring compression of 4.00 cm? (b) Suppose the initial speed of the block is 1.09 m/s, but its mass can be varied. What mass is required to give a maximum spring compression of 4.00 cm?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.105IP**

Referring to Example Suppose the block is released from rest with the spring compressed 5.00 cm. The mass of the block is 1.70 kg and the force constant of the spring is 955 N/m. (a) What is the speed of the block when the spring expands to a compression of only 2.50 cm? (b) What is the speed of the block after it leaves the spring?

**Solution:**

**Chapter 8 Potential Energy And Conservation Of Energy Q.106IP**

Referring to Example Suppose we would like the landing speed of block 2 to be increased to 1.50 m/s. (a) Should the coefficient of kinetic friction between block 1 and the table-top be increased or decreased? (b) Find the required coefficient of kinetic friction for a landing speed of 1.50 m/s. Note that m1 = 2.40 kg, m2 = 1.80 kg, and d = 0.500 m.

**Solution:**