|| A 1000 kg elevator accelerates upward at $1.0\text{m}\text{/}{\text{s}}^2$ for 10 m, starting from rest.

1. How much work does gravity do on the elevator?

2. How much work does the tension in the elevator cable do on the elevator?

3. What is the elevator’s kinetic energy after traveling 10 m?

||

1. Starting from rest, a crate of mass $m$ is pushed up a frictionless slope of angle $\theta$ by a horizontal force of magnitude $F$. Use work and energy to find an expression for the crate’s speed $v$ when it is at height $h$ above the bottom of the slope.

2. Doug uses a $25\text{N}$ horizontal force to push a $5.0\text{k}\text{g}$ crate up a $2.0\text{-}\text{m}\text{-}\text{high}\text{,}20\text{°}$ frictionless slope. What is the speed of the crate at the top of the slope?

||| CALC A 150 g particle at $x=0$ is moving at $2.00\text{m}\text{/}\text{s}$ in the $+$x-direction. As it moves, it experiences a force given by $F_x=(0.250\text{ N})\text{ sin }(x\text{/}2.00\text{ m})$. What is the particle’s speed when it reaches $x=3.14\text{m}?$

||| Susan’s 10 kg baby brother Paul sits on a mat. Susan pulls the mat across the floor using a rope that is angled $30\text{°}$ above the floor. The tension is a constant 30 N and the coefficient of friction is 0.20. Use work and energy to find Paul’s speed after being pulled 3.0 m.

|| A pile driver lifts a 250 kg weight and then lets it fall onto the end of a steel pipe that needs to be driven into the ground. A fall from an initial height of 1.5 m drives the pipe in 35 cm. What is the average force that the weight exerts on the pipe?

| A ball shot straight up with kinetic energy $K_0$ reaches height h. What height will it reach if the initial kinetic energy is doubled?

|| INT A 50 kg ice skater is gliding along the ice, heading due north at $4.0\text{m}\text{/}\text{s}$. The ice has a small coefficient of static friction, to prevent the skater from slipping sideways, but ${\mu }_{\text{k}}=0$. Suddenly, a wind from the northeast exerts a force of 4.0 N on the skater.

1. Use work and energy to find the skater’s speed after gliding 100 m in this wind.

2. What is the minimum value of ${\mu }_{\text{s}}$ that allows her to continue moving straight north?

|| A 737-800 jet airliner has twin engines, each with 105 kN thrust. A 78,000 kg jet reaches a takeoff speed of $70\text{m}\text{/}\text{s}$ in a distance of 1100 m.

1. How much work is done by the jet engines?

2. What is the increase in thermal energy due to rolling friction and air drag?

||| CALC Hooke’s law describes an ideal spring. Many real springs are better described by the restoring force $(F_{\text{Sp}}){}_s=-k\text{Δ}s-q{\left(\text{Δ}s\right)}^3$, where q is a constant. Consider a spring with $k=250\text{N}\text{/}\text{m}$ and $q=800{\text{N/m}}^3$.

1. How much work must you do to compress this spring 15 cm? Note that, by Newton’s third law, the work you do on the spring is the negative of the work done by the spring.

   Hint: Let the spring lie along the s-axis with the equilibrium position of the end of the spring at $s=0$. Then $\text{Δ}s=s$.

2. By what percent has the cubic term increased the work over what would be needed to compress an ideal spring?

|| CALC The force acting on a particle is $F_x=F_0{e}^{-x/L}$. How much work does this force do as the particle moves along the x-axis from $x=0$ to $x=L?$

||| CALC The gravitational attraction between two objects with masses $m_{\text{A}}$ and $m_{\text{B}}$, separated by distance x, is $F=G{m}_{\text{A}}{m}_{\text{B}}/x^2$, where G is the gravitational constant.

1. How much work is done by gravity when the separation changes from $x_1$ to $x_2?$ Assume $x_2<x_1$.

2. If one mass is much greater than the other, the larger mass stays essentially at rest while the smaller mass moves toward it. Suppose a $1.5\times {10}^{13}\text{kg}$ comet is passing the orbit of Mars, heading straight for the sun at a speed of $3.5\times {10}^4\text{m}\text{/}\text{s}$. What will its speed be when it crosses the orbit of Mercury? Astronomical data are given in the tables at the back of the book, and $G=6.67\times {10}^{-11}\text{N}{\text{m}}^2/{\text{kg}}^2$.

|| A red ball has a mass of 250 g. A constant force pushes the red ball horizontally and launches it at a speed of $15\text{m}\text{/}\text{s}$. The same force pushes a green ball through the same distance, launching it at $25\text{m}\text{/}\text{s}$. What is the mass of the green ball?

|| INT A 30 g mass is attached to one end of a 10-cm-long spring. The other end of the spring is connected to a frictionless pivot on a frictionless, horizontal surface. Spinning the mass around in a circle at 90 rpm causes the spring to stretch to a length of 12 cm. What is the value of the spring constant?

|| A 50 g rock is placed in a slingshot and the rubber band is stretched. The magnitude of the force of the rubber band on the rock is shown by the graph in FIGURE P9.56. The rubber band is stretched 30 cm and then released. What is the speed of the rock?

FIGURE P9.56

|| CALC

1. FIGURE P9.57 shows a rope holding a mass m in place at angle $\theta$ on the side of a frictionless circular slope of radius R. What is the tension in the rope?

2. How much work does tension do to pull the mass from the bottom of the hill $(\theta =0)$ to the top at constant speed? To answer this question, write an expression for the work done when the mass moves through a very small distance ds while it has angle $\theta$, replace ds with an equivalent expression involving R and $d\theta$, then integrate.

|| A horizontal spring with spring constant 250 N/m is compressed by 12 cm and then used to launch a 250 g box across the floor. The coefficient of kinetic friction between the box and the floor is 0.23. What is the box’s launch speed?

|| A spring of equilibrium length $L_1$ and spring constant $k_1$ hangs from the ceiling. Mass $m_1$ is suspended from its lower end. Then a second spring, with equilibrium length $L_2$ and spring constant $k_2$, is hung from the bottom of $m_1$. Mass $m_2$ is suspended from this second spring. How far is $m_2$ below the ceiling?

| A 90 kg firefighter needs to climb the stairs of a 20-m-tall building while carrying a 40 kg backpack filled with gear. How much power does he need to reach the top in 55 s?

||| A hydroelectric power plant uses spinning turbines to transform the kinetic energy of moving water into electric energy with 80% efficiency. That is, 80% of the kinetic energy becomes electric energy. A small hydroelectric plant at the base of a dam generates 50 MW of electric power when the falling water has a speed of $18\text{m/s}$. What is the water flow rate—kilograms of water per second—through the turbines?

|| BIO INT When you ride a bicycle at constant speed, nearly all the energy you expend goes into the work you do against the drag force of the air. Model a cyclist as having cross-section area $0.45{\text{m}}^2$ and, because the human body is not aerodynamically shaped, a drag coefficient of $0.90$. Use $1.2{\text{kg/m}}^3$ as the density of air at room temperature.

1. What is the cyclist’s power output while riding at a steady $7.3\text{m}\text{/}\text{s}\left(16\text{mph}\right)?$

2. Metabolic power is the rate at which your body “burns” fuel to power your activities. For many activities, your body is roughly 25% efficient at converting the chemical energy of food into mechanical energy. What is the cyclist’s metabolic power while cycling at $7.3\text{m}\text{/}\text{s}?$

3. The food calorie is equivalent to $4190\text{J}$. How many calories does the cyclist burn if he rides over level ground at $7.3\text{m}\text{/}\text{s}$ for $1\text{h}?$

|| INT A Porsche 944 Turbo has a rated engine power of 217 hp. 30% of the power is lost in the engine and the drive train, and 70% reaches the wheels. The total mass of the car and driver is 1480 kg, and two-thirds of the weight is over the drive wheels.

1. What is the maximum acceleration of the Porsche on a concrete surface where ${\mu }_{\text{s}}=1.00?$

   Hint: What force pushes the car forward?

2. If the Porsche accelerates at $a_{\text{max}}$, what is its speed when it reaches maximum power output?

3. How long does it take the Porsche to reach the maximum power output?

|| INT A farmer uses a tractor to pull a 150 kg bale of hay up a $\text{15}°$ incline to the barn at a steady $\text{5}\text{.0 km/h}$. The coefficient of kinetic friction between the bale and the ramp is 0.45. What is the tractor’s power output?

|| Astronomers using a 2.0-m-diameter telescope observe a distant supernova—an exploding star. The telescope’s detector records $9.1\times {10}^{-11}\text{J}$ of light energy during the first 10 s. It’s known that this type of supernova has a visible-light power output of $5.0\times {10}^{37}\text{W}$ for the first 10 s of the explosion. How distant is the supernova? Give your answer in light years, where one light year is the distance light travels in one year. The speed of light is $3.0\times {10}^8\text{m}\text{/}\text{s}$.

|| BIO INT A paramecium is an elongated unicellular organism approximately $50\mu \text{m}$ in diameter and $150\mu \text{m}$ in length. It swims through water by whip-like movements of cilia, small hairs on the outside of its body. Because it moves “head first” through the water, drag is determined primarily by its diameter and only secondarily by its length, so it’s reasonable to model the paramecium as a $70\text{-}\mu \text{m-diameter}$ sphere. A paramecium uses 1.5 pW of locomotive power to propel itself through $\text{20}°\text{C}$ water, where $1\text{pW}=1\text{picowatt}={10}^{-12}\text{W}$. What is its swimming speed in $\mu \text{m}\text{/}\text{s}?$