| A 50 g mass is attached to a light, rigid, 75-cm-long rod. The other end of the rod is pivoted so that the mass can rotate in a vertical circle. What speed does the mass need at the bottom of the circle to barely make it over the top of the circle?

|| A $50\text{g}$ ice cube can slide up and down a frictionless $30\text{°}$ slope. At the bottom, a spring with spring constant $25\text{N}\text{/}\text{m}$ is compressed $10\text{cm}$ and used to launch the ice cube up the slope. How high does it go above its starting point?

|| You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 m to the track’s lowest point. You’ve determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top.

1. What spring constant should you specify?

2. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

|| INT A block of mass $m$ slides down a frictionless track, then around the inside of a circular loop-the-loop of radius $R$. From what minimum height $h$ must the block start to make it around without falling off? Give your answer as a multiple of $R$.

|| INT It’s been a great day of new, frictionless snow. Julie starts at the top of the $60\text{°}$ slope shown in FIGURE P10.45. At the bottom, a circular arc carries her through a $90\text{°}$ turn, and she then launches off a 3.0-m-high ramp. How far horizontally is her touchdown point from the end of the ramp?

FIGURE P10.45

|| A 1000 kg safe is 2.0 m above a heavy-duty spring when the rope holding the safe breaks. The safe hits the spring and compresses it 50 cm. What is the spring constant of the spring?

|| You have a ball of unknown mass, a spring with spring constant $950\text{N}\text{/}\text{m}$, and a meter stick. You use various compressions of the spring to launch the ball vertically, then use the meter stick to measure the ball’s maximum height above the point at which it is released. Your data are as follows:

Use an appropriate graph of the data to determine the ball’s mass.

|| A horizontal spring with spring constant $100\text{N}\text{/}\text{m}$ is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

|| Two blocks with masses $m_{\text{A}}$ and $m_{\text{B}}$ are connected by a massless string over a massless, frictionless pulley. Block B, which is more massive than block A, is released from height $h$ and falls.

1. Write an expression for the speed of the blocks just as block B reaches the ground.

2. A 1.0 kg block and a 2.0 kg block are connected by a massless string over a massless, frictionless pulley. The impact speed of the heavier block, after falling, is 1.8 m/s. From how high did it fall?

|| A 75 kg student dives off a 30-m-high bridge with an 18-m-long bungee cord tied to his feet and to the bridge. A bungee cord is essentially a flexible spring. What spring constant must the bungee cord have for the student’s lowest point to be 2.0 m above the water?

|| Use work and energy to find an expression for the speed of the block in FIGURE P10.51 just before it hits the floor if (a) the coefficient of kinetic friction for the block on the table is ${\mu }_{\text{k}}$ and (b) the table is frictionless.

FIGURE P10.51

|| A freight company uses a compressed spring to shoot 2.0 kg packages up a 1.0-m-high frictionless ramp into a truck, as FIGURE P10.52 shows. The spring constant is $500\text{N}\text{/}\text{m}$ and the spring is compressed 30 cm.

1. What is the speed of the package when it reaches the truck?

2. A careless worker spills his soda on the ramp. This creates a 50-cm-long sticky spot with a coefficient of kinetic friction 0.30. Will the next package make it into the truck?

FIGURE P10.52

1. || A 50 g ice cube can slide without friction up and down a $30\text{°}$ slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 10 cm. The spring constant is $25\text{N}\text{/}\text{m}$. When the ice cube is released, what total distance will it travel up the slope before reversing direction?

2. The ice cube is replaced by a 50 g plastic cube whose coefficient of kinetic friction is 0.20. How far will the plastic cube travel up the slope? Use work and energy.

|| The spring shown in FIGURE P10.54 is compressed 50 cm and used to launch a 100 kg physics student. The track is frictionless until it starts up the incline. The student’s coefficient of kinetic friction on the $30\text{°}$ incline is 0.15.

1. What is the student’s speed just after losing contact with the spring?

2. How far up the incline does the student go?

FIGURE P10.54

|| FIGURE 10.23 showed the potential-energy curve for the $\text{O}{}_2$ molecule. Consider a molecule with the energy $E_1$ shown in the figure.

1. What is the maximum speed of an oxygen atom as it oscillates back and forth? Don’t forget that the kinetic energy is the total kinetic energy of the system. The mass of an oxygen atom is 16 u, where $1\text{u}=1$ atomic mass unit $=1.66\times {10}^{-27}\text{ kg}$.

2. The atom’s speed varies as it oscillates, but you can make the reasonable assumption that the average speed is half the maximum speed. With that assumption, what is the oscillation frequency of an oxygen atom in a molecule that has energy $E_1?$ Think carefully about how far each atom travels during one cycle. Give your answer in THz, where $\text{1 THz}=\text{1 terahertz}={10}^{12}\text{Hz and 1 Hz}=\text{1 hertz}=\text{1 cycle/s}$ is the unit of frequency.

||| A sled starts from rest at the top of the frictionless, hemispherical, snow-covered hill shown in FIGURE P10.56.

1. Find an expression for the sled’s speed when it is at angle $\varphi$.

2. Use Newton’s laws to find the maximum speed the sled can have at angle $\varphi$ without leaving the surface.

3. At what angle ${\varphi }_{\text{max}}$ does the sled “fly off” the hill?

FIGURE P10.56

|| CALC A 2.6 kg block is attached to a horizontal rope that exerts a variable force $F_x=(20-5x)\text{N,}$ where $x$ is in m. The coefficient of kinetic friction between the block and the floor is 0.25. Initially the block is at rest at $x=0\text{m}$. What is the block’s speed when it has been pulled to $x=4.0\text{m}?$

|| A 1.0 kg mass that can move along the $x$-axis experiences the potential energy $U=(x^2-x)\text{J,}$ where $x$ is in m. The mass has velocity $v_x=3.0\text{m}\text{/}\text{s}$ at position $x=1.0\text{m}$. At what position has it slowed to $1.0\text{m}\text{/}\text{s}?$

|| CALC A system has potential energy

as a particle moves over the range $0\text{m}\le x\le 3\text{m}$.

1. Where are the equilibrium positions in this range?

2. For each, is it a point of stable or unstable equilibrium?

|| A 100 g particle experiences the one-dimensional, conservative force $F_x$ shown in FIGURE P10.60.

1. Let the zero of potential energy be at $x=0\text{m}$. What is the potential energy at $x=\text{1}\text{.0, 2}\text{.0, 3}\text{.0, and 4}\text{.0 m?}$

   Hint: Use the definition of potential energy and the geometric interpretation of work.

2. Suppose the particle is shot to the right from $x=1.0\text{m}$ with a speed of 25 m/s. Where is its turning point?

FIGURE P10.60

|| CALC A particle that can move along the $x$-axis is part of a system with potential energy

where $A$ and $B$ are positive constants.

1. Where are the particle’s equilibrium positions?

2. For each, is it a point of stable or unstable equilibrium?

|| CALC A clever engineer designs a “sprong” that obeys the force law $F_x=-q(x-x_{\text{eq}}{)}^3$, where $x_{\text{eq}}$ is the equilibrium position of the end of the sprong and $q$ is the sprong constant. For simplicity, we’ll let $x_{\text{eq}}=0\text{m}$. Then $F_x=-q{x}^3$.

1. What are the units of $q$?

2. Find an expression for the potential energy of a stretched or compressed sprong.

3. A sprong-loaded toy gun shoots a 20 g plastic ball. What is the launch speed if the sprong constant is 40,000, with the units you found in part a, and the sprong is compressed 10 cm? Assume the barrel is frictionless.

|| CALC The potential energy for a particle that can move along the $x$-axis is $U=A{x}^2+B\text{sin}(\pi x/L)$, where $A$, $B$, and $L$ are constants. What is the force on the particle at (a) $x=0$, (b) $x=L/2$, and (c) $x=L?$

|| CALC A particle that can move along the $x$-axis experiences an interaction force $F_x=(3x^2-5x)\text{N,}$ where $x$ is in m. Find an expression for the system’s potential energy.

|| CALC An object moving in the xy-plane is subjected to the force $\vec{F}=(2xyî+x^2ĵ)\text{N,}$ where $x$ and $y$ are in m.

1. The particle moves from the origin to the point with coordinates $(a,b)$ by moving first along the $x$-axis to $(a,0)$, then parallel to the $y$-axis. How much work does the force do?

2. The particle moves from the origin to the point with coordinates $(a,b)$ by moving first along the $y$-axis to $(0,b)$, then parallel to the $x$-axis. How much work does the force do?

3. Is this a conservative force?

Write a realistic problem for which the energy bar chart shown in FIGURE P10.66 correctly shows the energy at the beginning and end of the problem.

FIGURE P10.66

$\frac{1}{2}(1500\text{kg})(5.0{\text{m/s)}}^2+(1500\text{kg})(9.80{\text{m/s}}^2)(10\text{m})=\frac{1}{2}(1500\text{kg})v_{\text{i}}{}^2+(1500\text{kg})(9.80{\text{m/s}}^2)(0\text{m})$

$\frac{1}{2}(0.20\text{kg})(2.0{\text{m/s)}}^2+\frac{1}{2}k(0{\text{m)}}^2=\frac{1}{2}(0.20\text{kg})(0{\text{m/s)}}^2+\frac{1}{2}k(-0.15{\text{m)}}^2$

$\frac{1}{2}(0.50\text{ kg})v_{\text{f}}{}^2+(0.50\text{ kg})(9.80{\text{ m/s}}^2)(0\text{ m})$

$+\frac{1}{2}\left(400\text{N/m}\right){\left(0\text{m}\right)}^2=\frac{1}{2}\left(0.50\text{kg}\right){\left(0\text{m/s}\right)}^2$

$+(0.50\text{ kg})(9.80{\text{ m/s}}^2)((-0.10\text{ m) sin }30\text{°})$

$+\frac{1}{2}(400\text{ N/m)(}\text{−0.10}{\text{ m)}}^2$

||| A pendulum is formed from a small ball of mass $m$ on a string of length $L$. As FIGURE CP10.70 shows, a peg is height $h=L\text{/}3$ above the pendulum’s lowest point. From what minimum angle $\theta$ must the pendulum be released in order for the ball to go over the top of the peg without the string going slack?

FIGURE CP10.70

||| In a physics lab experiment, a compressed spring launches a 20 g metal ball at a 30° angle. Compressing the spring 20 cm causes the ball to hit the floor 1.5 m below the point at which it leaves the spring after traveling 5.0 m horizontally. What is the spring constant?

||| It’s your birthday, and to celebrate you’re going to make your first bungee jump. You stand on a bridge 100 m above a raging river and attach a 30-m-long bungee cord to your harness. A bungee cord, for practical purposes, is just a long spring, and this cord has a spring constant of $40\text{N}\text{/}\text{m}$. Assume that your mass is 80 kg. After a long hesitation, you dive off the bridge. How far are you above the water when the cord reaches its maximum elongation?

||| CALC A 10 kg box slides 4.0 m down the frictionless ramp shown in FIGURE CP10.73, then collides with a spring whose spring constant is $250\text{N}\text{/}\text{m}$.

1. What is the maximum compression of the spring?

2. At what compression of the spring does the box have its maximum speed?

FIGURE CP10.73

||| The spring in FIGURE CP10.74 has a spring constant of $1000\text{N}\text{/}\text{m}$. It is compressed 15 cm, then launches a 200 g block. The horizontal surface is frictionless, but the block’s coefficient of kinetic friction on the incline is 0.20. What distance $d$ does the block sail through the air?

FIGURE CP10.74

||| CALC Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons—hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is

where $x$ is the distance between the centers of the two nucleons, $x_0$ is a constant having the value $x_0=2.0\times {10}^{-15}\text{m,}$ and $U_0=6.0\times {10}^{-11}\text{J}$.

Quantum effects are essential for a proper understanding of nucleons, but let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass $1.67\times {10}^{-27}\text{kg}$ and diameter $1.0\times {10}^{-15}\text{m}$. Suppose you hold two neutrons $5.0\times {10}^{-15}\text{m}$ apart, measured between their centers, then release them. What is the speed of each neutron as they crash together? Keep in mind that both neutrons are moving.