Chapter 6

A block of mass \(5.0 \mathrm{~kg}\) slides without friction at a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) on a horizontal table surface until it strikes and sticks to a mass of \(4.0 \mathrm{~kg}\) attached to a horizontal spring (with spring constant of \(k=2000.0 \mathrm{~N} / \mathrm{m}\) ), which in turn is attached to a wall. How far is the spring compressed before the masses come to rest? a) \(0.40 \mathrm{~m}\) b) \(0.54 \mathrm{~m}\) c) \(0.30 \mathrm{~m}\) d) \(0.020 \mathrm{~m}\) e) \(0.67 \mathrm{~m}\)

A pendulum swings in a vertical plane. At the bottom of the swing, the kinetic energy is \(8 \mathrm{~J}\) and the gravitational potential energy is 4 J. At the highest position of its swing, the kinetic and gravitational potential energies are a) kinetic energy \(=0 \mathrm{~J}\) and gravitational potential energy \(=4 \mathrm{~J}\) b) kinetic energy \(=12 \mathrm{~J}\) and gravitational potential energy \(=0 \mathrm{~J}\) c) kinetic energy \(=0 \mathrm{~J}\) and gravitational potential energy \(=12 \mathrm{~J}\) d) kinetic energy \(=4\) J and gravitational potential energy \(=8 \mathrm{~J}\) e) kinetic energy \(=8 \mathrm{~J}\) and gravitational potential energy \(=4\) J.

A child throws three identical marbles from the same height above the ground so that they land on the flat roof of a building. The marbles are launched with the same initial speed. The first marble, marble \(\mathrm{A}\), is thrown at an angle of \(75^{\circ}\) above horizontal, while marbles \(\mathrm{B}\) and \(\mathrm{C}\) are thrown with launch angles of \(60^{\circ}\) and \(45^{\circ}\), respectively. Neglecting air resistance, rank the marbles according to the speeds with which they hit the roof. a) \(A

Which of the following is not a valid potential energy function for the spring force \(F=-k x ?\) a) \(\left(\frac{1}{2}\right) k x^{2}\) b) \(\left(\frac{1}{2}\right) k x^{2}+10 \mathrm{~J}\) c) \(\left(\frac{1}{2}\right) k x^{2}-10 \mathrm{~J}\) d) \(-\left(\frac{1}{2}\right) k x^{2}\) e) None of the above is valid.

You use your hand to stretch a spring to a displacement \(x\) from its equilibrium position and then slowly bring it back to that position. Which is true? a) The spring's \(\Delta U\) is positive. b) The spring's \(\Delta U\) is negative. c) The hand's \(\Delta U\) is positive. d) The hand's \(\Delta U\) is negative. e) None of the above statements is true.

Which of the following is not a unit of energy? a) newton-meter b) joule c) kilowatt-hour d) \(\operatorname{kg} \mathrm{m}^{2} / \mathrm{s}^{2}\) e) all of the above

A spring has a spring constant of \(80 \mathrm{~N} / \mathrm{m}\). How much potential energy does it store when stretched by \(1.0 \mathrm{~cm} ?\) a) \(4.0 \cdot 10^{-3}\) J b) \(0.40 \mathrm{~J}\) c) 80 d) \(800 \mathrm{~J}\) e) \(0.8 \mathrm{~J}\)

Can the kinetic energy of an object be negative? Can the potential energy of an object be negative?

a) If you jump off a table onto the floor, is your mechanical energy conserved? If not, where does it go? b) A car moving down the road smashes into a tree. Is the mechanical energy of the car conserved? If not, where does it go?

An arrow is placed on a bow, the bowstring is pulled back, and the arrow is shot straight up into the air; the arrow then comes back down and sticks into the ground. Describe all of the changes in work and energy that occur.

A girl of mass \(49.0 \mathrm{~kg}\) is on a swing, which has a mass of \(1.0 \mathrm{~kg} .\) Suppose you pull her back until her center of mass is \(2.0 \mathrm{~m}\) above the ground. Then you let her \(\mathrm{go},\) and she swings out and returns to the same point. Are all forces acting on the girl and swing conservative?

Can a potential energy function be defined for the force of friction?

Can the potential energy of a spring be negative?

One end of a rubber band is tied down and you pull on the other end to trace a complicated closed trajectory. If you were to measure the elastic force \(F\) at every point and took its scalar product with the local displacements, \(\vec{F} \cdot \Delta \vec{r},\) and then summed all of these, what would you get?

Can a unique potential energy function be identified with a particular conservative force?

A projectile of mass \(m\) is launched from the ground at \(t=0\) with a speed \(v_{0}\) and at an angle \(\theta_{0}\) above the horizontal. Assuming that air resistance is negligible, write the kinetic, potential, and total energies of the projectile as explicit functions of time.

The energy height, \(H\), of an aircraft of mass \(m\) at altitude \(h\) and with speed \(v\) is defined as its total energy (with the zero of the potential energy taken at ground level) divided by its weight. Thus, the energy height is a quantity with units of length. a) Derive an expression for the energy height, \(H\), in terms of the quantities \(m, h\), and \(v\). b) A Boeing 747 jet with mass \(3.5 \cdot 10^{5} \mathrm{~kg}\) is cruising in level flight at \(250.0 \mathrm{~m} / \mathrm{s}\) at an altitude of \(10.0 \mathrm{~km} .\) Calculate the value of its energy height. Note: The energy height is the maximum altitude an aircraft can reach by "zooming" (pulling into a vertical climb without changing the engine thrust). This maneuver is not recommended for a 747 , however.

A body of mass \(m\) moves in one dimension under the influence of a force, \(F(x)\), which depends only on the body's position. a) Prove that Newton's Second Law and the law of conservation of energy for this body are exactly equivalent. b) Explain, then, why the law of conservation of energy is considered to be of greater significance than Newton's Second Law.

The molecular bonding in a diatomic molecule such as the nitrogen \(\left(\mathrm{N}_{2}\right)\) molecule can be modeled by the Lennard Jones potential, which has the form $$ U(x)=4 U_{0}\left(\left(\frac{x_{0}}{x}\right)^{12}-\left(\frac{x_{0}}{x}\right)^{6}\right) $$ where \(x\) is the separation distance between the two nuclei and \(x_{0}\), and \(U_{0}\) are constants. Determine, in terms of these constants, the following: a) the corresponding force function; b) the equilibrium separation \(x_{0}\), which is the value of \(x\) for which the two atoms experience zero force from each other; and c) the nature of the interaction (repulsive or attractive) for separations larger and smaller than \(x_{0}\).

What is the gravitational potential energy of a \(2.0-\mathrm{kg}\) book \(1.5 \mathrm{~m}\) above the floor?

a) If the gravitational potential energy of a 40.0 -kg rock is 500 . J relative to a value of zero on the ground, how high is the rock above the ground? b) If the rock were lifted to twice its original height, how would the value of its gravitational potential energy change?

A 20.0 -kg child is on a swing attached to ropes that are \(L=1.50 \mathrm{~m}\) long. Take the zero of the gravitational potential energy to be at the position of the child when the ropes are horizontal. a) Determine the child's gravitational potential energy when the child is at the lowest point of the circular trajectory. b) Determine the child's gravitational potential energy when the ropes make an angle of \(45.0^{\circ}\) relative to the vertical. c) Based on these results, which position has the higher potential energy?

A \(1.50 \cdot 10^{3}-\mathrm{kg}\) car travels \(2.50 \mathrm{~km}\) up an incline at constant velocity. The incline has an angle of \(3.00^{\circ}\) with respect to the horizontal. What is the change in the car's potential energy? What is the net work done on the car?

A particle is moving along the \(x\) -axis subject to the potential energy function \(U(x)=1 / x+x^{2}+x-1\) a) Express the force felt by the particle as a function of \(x\). b) Plot this force and the potential energy function. c) Determine the net force on the particle at the coordinate \(x=2.00 \mathrm{~m}\)

Calculate the force \(F(y)\) associated with each of the following potential energies: a) \(U=a y^{3}-b y^{2}\) b) \(U=U_{0} \sin (c y)\)

The potential energy of a certain particle is given by \(U=10 x^{2}+35 z^{3}\). Find the force vector exerted on the particle.

A ball is thrown up in the air, reaching a height of \(5.00 \mathrm{~m}\). Using energy conservation considerations, determine its initial speed.

A cannonball of mass \(5.99 \mathrm{~kg}\) is shot from a cannon at an angle of \(50.21^{\circ}\) relative to the horizontal and with an initial speed of \(52.61 \mathrm{~m} / \mathrm{s}\). As the cannonball reaches the highest point of its trajectory, what is the gain in its potential energy relative to the point from which it was shot?

A basketball of mass \(0.624 \mathrm{~kg}\) is shot from a vertical height of \(1.2 \mathrm{~m}\) and at a speed of \(20.0 \mathrm{~m} / \mathrm{s}\). After reaching its maximum height, the ball moves into the hoop on its downward path, at \(3.05 \mathrm{~m}\) above the ground. Using the principle of energy conservation, determine how fast the ball is moving just before it enters the hoop.

A classmate throws a \(1.0-\mathrm{kg}\) book from a height of \(1.0 \mathrm{~m}\) above the ground straight up into the air. The book reaches a maximum height of \(3.0 \mathrm{~m}\) above the ground and begins to fall back. Assume that \(1.0 \mathrm{~m}\) above the ground is the reference level for zero gravitational potential energy. Determine a) the gravitational potential energy of the book when it hits the ground. b) the velocity of the book just before hitting the ground.

Suppose you throw a 0.052 -kg ball with a speed of \(10.0 \mathrm{~m} / \mathrm{s}\) and at an angle of \(30.0^{\circ}\) above the horizontal from a building \(12.0 \mathrm{~m}\) high. a) What will be its kinetic energy when it hits the ground? b) What will be its speed when it hits the ground?

A uniform chain of total mass \(m\) is laid out straight on a frictionless table and held stationary so that one-third of its length, \(L=1.00 \mathrm{~m},\) is hanging vertically over the edge of the table. The chain is then released. Determine the speed of the chain at the instant when only one-third of its length remains on the table.

a) If you are at the top of a toboggan run that is \(40.0 \mathrm{~m}\) high, how fast will you be going at the bottom, provided you can ignore friction between the sled and the track? b) Does the steepness of the run affect how fast you will be going at the bottom? c) If you do not ignore the small friction force, does the steepness of the track affect the value of the speed at the bottom?

A block of mass \(0.773 \mathrm{~kg}\) on a spring with spring constant \(239.5 \mathrm{~N} / \mathrm{m}\) oscillates vertically with amplitude \(0.551 \mathrm{~m}\). What is the speed of this block at a distance of \(0.331 \mathrm{~m}\) from the equilibrium position?

A spring with \(k=10.0 \mathrm{~N} / \mathrm{cm}\) is initially stretched \(1.00 \mathrm{~cm}\) from its equilibrium length. a) How much more energy is needed to further stretch the spring to \(5.00 \mathrm{~cm}\) beyond its equilibrium length? b) From this new position, how much energy is needed to compress the spring to \(5.00 \mathrm{~cm}\) shorter than its equilibrium position?

A 5.00 -kg ball of clay is thrown downward from a height of \(3.00 \mathrm{~m}\) with a speed of \(5.00 \mathrm{~m} / \mathrm{s}\) onto a spring with \(k=\) \(1600 . \mathrm{N} / \mathrm{m} .\) The clay compresses the spring a certain maximum amount before momentarily stopping a) Find the maximum compression of the spring. b) Find the total work done on the clay during the spring's compression.

A horizontal slingshot consists of two light, identical springs (with spring constants of \(30.0 \mathrm{~N} / \mathrm{m}\) ) and a light cup that holds a 1.00 -kg stone. Each spring has an equilibrium length of \(50.0 \mathrm{~cm}\). When the springs are in equilibrium, they line up vertically. Suppose that the cup containing the mass is pulled to \(x=70.0 \mathrm{~cm}\) to the left of the vertical and then released. Determine a) the system's total mechanical energy. b) the speed of the stone at \(x=0\)

A 80.0 -kg fireman slides down a 3.00 -m pole by applying a frictional force of \(400 .\) N against the pole with his hands. If he slides from rest, how fast is he moving once he reaches the ground?

A large air-filled 0.100 -kg plastic ball is thrown up into the air with an initial speed of \(10.0 \mathrm{~m} / \mathrm{s}\). At a height of \(3.00 \mathrm{~m}\) the ball's speed is \(3.00 \mathrm{~m} / \mathrm{s}\). What fraction of its original energy has been lost to air friction?

How much mechanical energy is lost to friction if a 55.0-kg skier slides down a ski slope at constant speed of \(14.4 \mathrm{~m} / \mathrm{s}\) ? The slope is \(123.5 \mathrm{~m}\) long and makes an angle of \(14.7^{\circ}\) with respect to the horizontal.

A truck of mass 10,212 kg moving at a speed of \(61.2 \mathrm{mph}\) has lost its brakes. Fortunately, the driver finds a runaway lane, a gravel-covered incline that uses friction to stop a truck in such a situation; see the figure. In this case, the incline makes an angle of \(\theta=40.15^{\circ}\) with the horizontal, and the gravel has a coefficient of friction of 0.634 with the tires of the truck. How far along the incline \((\Delta x)\) does the truck travel before it stops?

A snowboarder of mass \(70.1 \mathrm{~kg}\) (including gear and clothing), starting with a speed of \(5.1 \mathrm{~m} / \mathrm{s}\), slides down a slope at an angle \(\theta=37.1^{\circ}\) with the horizontal. The coefficient of kinetic friction is \(0.116 .\) What is the net work done on the snowboarder in the first 5.72 s of descent?

The greenskeepers of golf courses use a stimpmeter to determine how "fast" their greens are. A stimpmeter is a straight aluminum bar with a V-shaped groove on which a golf ball can roll. It is designed to release the golf ball once the angle of the bar with the ground reaches a value of \(\theta=20.0^{\circ} .\) The golf ball \((\) mass \(=1.62 \mathrm{oz}=0.0459 \mathrm{~kg})\) rolls 30.0 in down the bar and then continues to roll along the green for several feet. This distance is called the "reading." The test is done on a level part of the green, and stimpmeter readings between 7 and \(12 \mathrm{ft}\) are considered acceptable. For a stimpmeter reading of \(11.1 \mathrm{ft},\) what is the coefficient of friction between the ball and the green? (The ball is rolling and not sliding, as we usually assume when considering friction, but this does not change the result in this case.)

A 1.00 -kg block is pushed up and down a rough plank of length \(L=2.00 \mathrm{~m},\) inclined at \(30.0^{\circ}\) above the horizontal. From the bottom, it is pushed a distance \(L / 2\) up the plank, then pushed back down a distance \(L / 4,\) and finally pushed back up the plank until it reaches the top end. If the coefficient of kinetic friction between the block and plank is \(0.300,\) determine the work done by the block against friction.

A 1.00 -kg block initially at rest at the top of a 4.00 -m incline with a slope of \(45.0^{\circ}\) begins to slide down the incline. The upper half of the incline is frictionless, while the lower half is rough, with a coefficient of kinetic friction \(\mu_{\mathrm{k}}=0.300\). a) How fast is the block moving midway along the incline, before entering the rough section? b) How fast is the block moving at the bottom of the incline?

A spring with a spring constant of \(500 . \mathrm{N} / \mathrm{m}\) is used to propel a 0.500 -kg mass up an inclined plane. The spring is compressed \(30.0 \mathrm{~cm}\) from its equilibrium position and launches the mass from rest across a horizontal surface and onto the plane. The plane has a length of \(4.00 \mathrm{~m}\) and is inclined at \(30.0^{\circ} .\) Both the plane and the horizontal surface have a coefficient of kinetic friction with the mass of \(0.350 .\) When the spring is compressed, the mass is \(1.50 \mathrm{~m}\) from the bottom of the plane. a) What is the speed of the mass as it reaches the bottom of the plane? b) What is the speed of the mass as it reaches the top of the plane? c) What is the total work done by friction from the beginning to the end of the mass's motion?

A 70.0 -kg skier moving horizontally at \(4.50 \mathrm{~m} / \mathrm{s}\) encounters a \(20.0^{\circ}\) incline. a) How far up the incline will the skier move before she momentarily stops, ignoring friction? b) How far up the incline will the skier move if the coefficient of kinetic friction between the skies and snow is \(0.100 ?\)

A ball of mass \(1.84 \mathrm{~kg}\) is dropped from a height \(y_{1}=$$1.49 \mathrm{~m}\) and then bounces back up to a height of \(y_{2}=0.87 \mathrm{~m}\) How much mechanical energy is lost in the bounce? The effect of air resistance has been experimentally found to be negligible in this case, and you can ignore it.

A car of mass \(987 \mathrm{~kg}\) is traveling on a horizontal segment of a freeway with a speed of \(64.5 \mathrm{mph}\). Suddenly, the driver has to hit the brakes hard to try to avoid an accident up ahead. The car does not have an ABS (antilock braking system), and the wheels lock, causing the car to slide some distance before it is brought to a stop by the friction force between the car's tires and the road surface. The coefficient of kinetic friction is \(0.301 .\) How much mechanical energy is lost to heat in this process?

Two masses are connected by a light string that goes over a light, frictionless pulley, as shown in the figure. The 10.0 -kg mass is released and falls through a vertical distance of \(1.00 \mathrm{~m}\) before hitting the ground. Use conservation of mechanical energy to determine: a) how fast the 5.00 -kg mass is moving just before the 10.0 -kg mass hits the ground; and b) the maximum height attained by the 5.00 -kg mass.