|| Derive Equations 8.3 for the acceleration of a projectile subject to drag.

|| CALC A 100 g bead slides along a frictionless wire with the parabolic shape $y=(2\text{m}{}^{-1})x^2$.

1. Find an expression for $a_y$, the vertical component of acceleration, in terms of x, $v_x$, and $a_x$.

   Hint: Use the basic definitions of velocity and acceleration.

2. Suppose the bead is released at some negative value of x and has a speed of 2.3 m/s as it passes through the lowest point of the parabola. What is the net force on the bead at this instant? Write your answer in component form.

|| A 250 g ball is launched with a speed of 35 m/s at a 30° angle. A strong headwind exerts a constant horizontal drag force on the ball. What is the magnitude of the drag force if the wind reduces the ball’s travel distance by 20%?

|| A charged particle in an electric field experiences an electric force $\vec{F}{}_{\text{E}}=q\vec{E}$, where q is the charge in coulombs (C) and $\vec{E}$ is the electric field in N/C. This is analogous to the gravitational force mg on mass m. A small 5.0 mg particle charged to $3.0\times {10}^{-10}\text{C}$ is fired horizontally at $\vec{v}{}_0=0.75î\text{m}\text{/s}$ into a horizontal electric field $\vec{E}=1.0\times {10}^5î\text{N}\text{/}\text{C}$ from a height of 15 cm. How far has the particle traveled horizontally when it hits the ground? Assume that the motion takes place in a vacuum. Be careful converting mg to the appropriate SI units!

|| INT A motorcycle daredevil plans to ride up a 2.0-m-high, $20\text{°}$ ramp, sail across a 10-m-wide pool filled with hungry crocodiles, and land at ground level on the other side. He has done this stunt many times and approaches it with confidence. Unfortunately, the motorcycle engine dies just as he starts up the ramp. He is going $11\text{m}\text{/s}$ at that instant, and the rolling friction of his rubber tires (coefficient 0.02) is not negligible. Does he survive, or does he become crocodile food? Justify your answer by calculating the distance he travels through the air after leaving the end of the ramp.

||| A rocket-powered hockey puck has a thrust of 2.0 N and a total mass of 1.0 kg. It is released from rest on a frictionless table, 4.0 m from the edge of a 2.0 m drop. The front of the rocket is pointed directly toward the edge. How far does the puck land from the base of the table?

| A car can just barely turn a corner on an unbanked road at 45 km/h on a dry sunny day. What is the car’s maximum cornering speed on a rainy day when the coefficient of static friction has been reduced by 50%?

|| CALC A 2.0 kg projectile with initial velocity $\vec{v}=8.0î\text{m}\text{/s}$ experiences the variable force $\vec{F}=-2.0tî+4.0t^2ĵ\text{N,}$ where t is in s.

1. What is the projectile’s speed at $t=2.0\text{s}?$

2. At what instant of time is the projectile moving parallel to the y-axis?

|| A 75 kg man weighs himself at the north pole and at the equator. Which scale reading is higher? By how much? Assume the earth is spherical.

||

1. An object of mass m swings in a horizontal circle on a string of length L that tilts downward at angle $\theta$. Find an expression for the angular velocity $\omega$.

2. A student ties a 500 g rock to a 1.0-m-long string and swings it around her head in a horizontal circle. At what angular speed, in rpm, does the string tilt down at a $10\text{°}$ angle?

|| A concrete highway curve of radius 70 m is banked at a $15\text{°}$ angle. What is the maximum speed with which a 1500 kg rubber-tired car can take this curve without sliding?

||| You’ve taken your neighbor’s young child to the carnival to ride the rides. She wants to ride The Rocket. Eight rocket-shaped cars hang by chains from the outside edge of a large steel disk. A vertical axle through the center of the ride turns the disk, causing the cars to revolve in a circle. You’ve just finished taking physics, so you decide to figure out the speed of the cars while you wait. You estimate that the disk is 5 m in diameter and the chains are 6 m long. The ride takes 10 s to reach full speed, then the cars swing out until the chains are $20\text{°}$ from vertical. What is the cars’ speed?

||| INT A 4.4-cm-diameter, 24 g plastic ball is attached to a 1.2-m-long string and swung in a vertical circle. The ball’s speed is 6.1 m/s at the point where it is moving straight up. What is the magnitude of the net force on the ball? Air resistance is not negligible.

|| A 2.00-m-diameter horizontal turntable rotates on a laboratory table. A 1.00-m-tall vertical post is attached to the rim of the turntable. A 75.0-cm-long string with a 50.0 g mass at the end is tied to the top of the post. What is the turntable’s angular velocity in rpm if the string makes a 7.00° angle with vertical?

Hint: For angles smaller than 10°, you can use the small-angle approximation $\text{tan}\theta \approx \text{sin}\theta$.

|| Two wires are tied to the $2.0\text{kg}$ sphere shown in FIGURE P8.45. The sphere revolves in a horizontal circle at constant speed.

1. For what speed is the tension the same in both wires?

2. What is the tension?

FIGURE P8.45

|| A 25 kg child is on a swing that has 3.0-m-long massless ropes. Each rope’s tension at the bottom of the swing’s arc is 300 N. What is the child’s speed at the bottom of the arc?

|| A 2.0 kg pendulum bob swings on a 2.0-m-long string. The bob’s speed is 1.5 m/s when the string makes a 15° angle with vertical and the bob is moving toward the bottom of the arc. At this instant, what are the magnitudes of (a) the bob’s tangential acceleration, (b) the bob’s radial acceleration, and (c) the tension in the string?

|| A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. FIGURE P8.48 shows that the string traces out the surface of a cone, hence the name.

1. Find an expression for the tension T in the string.

2. Find an expression for the ball’s angular speed $\omega$.

3. What are the tension and angular speed (in rpm) for a 500 g ball swinging in a 20-cm-radius circle at the end of a 1.0-m-long string?

FIGURE P8.48

|| BIO The ultracentrifuge is an important tool for separating and analyzing proteins. Because of the enormous centripetal accelerations, the centrifuge must be carefully balanced, with each sample matched by a sample of identical mass on the opposite side. Any difference in the masses of opposing samples creates a net force on the shaft of the rotor, potentially leading to a catastrophic failure of the apparatus. Suppose a scientist makes a slight error in sample preparation and one sample has a mass 10 mg larger than the opposing sample. If the samples are 12 cm from the axis of the rotor and the ultracentrifuge spins at 70,000 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples?

||| In an old-fashioned amusement park ride, passengers stand inside a 5.0-m-diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about a vertical axis. Then the floor on which the passengers are standing suddenly drops away! If all goes well, the passengers will “stick” to the wall and not slide. Clothing has a static coefficient of friction against steel in the range 0.60 to 1.0 and a kinetic coefficient in the range 0.40 to 0.70. A sign next to the entrance says “No children under 30 kg allowed.” What is the minimum angular speed, in rpm, for which the ride is safe?

|| In an amusement park ride called The Roundup, passengers stand inside a 16-m-diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in FIGURE P8.51.

1. Suppose the ring rotates once every 4.5 s. If a rider’s mass is 55 kg, with how much force does the ring push on her at the top of the ride? At the bottom?

2. What is the longest rotation period of the wheel that will prevent the riders from falling off at the top?

FIGURE P8.51

|| Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity ${\omega }_{\text{min}}$ you must maintain if you want the ball to complete the full circle without the string going slack at the top.

1. Find an expression for ${\omega }_{\text{min}}$.

2. Evaluate ${\omega }_{\text{min}}$ in rpm for a 65 g ball tied to a 1.0-m-long string.

||| A 30 g ball rolls around a 40-cm-diameter L-shaped track, shown in FIGURE P8.53, at 60 rpm. What is the magnitude of the net force that the track exerts on the ball? Rolling friction can be neglected.

Hint: The track exerts more than one force on the ball.

FIGURE P8.53

|| FIGURE P8.54 shows two small 1.0 kg masses connected by massless but rigid 1.0-m-long rods. What is the tension in the rod that connects to the pivot if the masses rotate at 30 rpm in a horizontal circle?

FIGURE P8.54

|| BIO The physics of circular motion sets an upper limit to the speed of human walking. (If you need to go faster, your gait changes from a walk to a run.) If you take a few steps and watch what’s happening, you’ll see that your body pivots in circular motion over your forward foot as you bring your rear foot forward for the next step. As you do so, the normal force of the ground on your foot decreases and your body tries to “lift off” from the ground.

1. A person’s center of mass is very near the hips, at the top of the legs. Model a person as a particle of mass m at the top of a leg of length L. Find an expression for the person’s maximum walking speed $v_{\text{max}}$.

2. Evaluate your expression for the maximum walking speed of a 70 kg person with a typical leg length of 70 cm. Give your answer in both m/s and mph, then comment, based on your experience, as to whether this is a reasonable result. A “normal” walking speed is about 3 mph.

|| An airplane feels a lift force $\vec{L}$ perpendicular to its wings. In level flight, the lift force points straight up and is equal in magnitude to the gravitational force on the plane. When an airplane turns, it banks by tilting its wings, as seen from behind, by an angle $\theta$ from horizontal. This causes the lift to have a radial component, similar to a car on a banked curve. If the lift had constant magnitude, the vertical component of $\vec{L}$ would now be smaller than the gravitational force, and the plane would lose altitude while turning. However, you can assume that the pilot uses small adjustments to the plane’s control surfaces so that the vertical component of $\vec{L}$ continues to balance the gravitational force throughout the turn.

1. Find an expression for the banking angle $\theta$ needed to turn in a circle of radius r while flying at constant speed v.

2. An 80,000 kg commercial jet flies at $850\text{km}\text{/}\text{h}$. The standard rate of turning requires 60 s to complete a 90° turn. What is the proper banking angle for this turn?

|| INT A 60 g ball is tied to the end of a 50-cm-long string and swung in a vertical circle. The center of the circle, as shown in Figure  P8.57, is 150 cm above the floor. The ball is swung at the minimum speed necessary to make it over the top without the string going slack. If the string is released at the instant the ball is at the top of the loop, how far to the right does the ball hit the ground?

FIGURE P8.57

|| INT A 100 g ball on a 60-cm-long string is swung in a vertical circle about a point 200 cm above the floor. The string suddenly breaks when it is parallel to the ground and the ball is moving upward. The ball reaches a height 600 cm above the floor. What was the tension in the string an instant before it broke?

|| Elm Street has a pronounced dip at the bottom of a steep hill before going back uphill on the other side. Your science teacher has asked everyone in the class to measure the radius of curvature of the dip. Some of your classmates are using surveying equipment, but you decide to base your measurement on what you’ve learned in physics. To do so, you sit on a spring scale, drive through the dip at different speeds, and for each speed record the scale’s reading as you pass through the bottom of the dip. Your data are as follows:

Sitting on the scale while the car is parked gives a reading of 588 N. Analyze your data, using a graph, to determine the dip’s radius of curvature.

||| CALC Scientists design a new particle accelerator in which protons $($mass $1.7\times {10}^{-27}\text{kg}$) follow a circular trajectory given by $\vec{r}=c\text{cos}\left(k{t}^2\right)î+c\text{sin}\left(k{t}^2\right)ĵ$, where $c=5.0\text{m}$ and $k=8.0\times {10}^4{\text{rad/s}}^2$ are constants and t is the elapsed time.

1. What is the radius of the circle?

2. What is the proton’s speed at $t=3.0\text{s}?$

3. What is the force on the proton at $t=3.0\text{s}?$ Give your answer in component form.

|| A 1500 kg car starts from rest and drives around a flat 50-m-diameter circular track. The forward force provided by the car’s drive wheels is a constant 1000 N.

1. What are the magnitude and direction of the car’s acceleration at $t=10\text{s}?$ Give the direction as an angle from the r-axis.

2. If the car has rubber tires and the track is concrete, at what time does the car begin to slide out of the circle?

|| A 2.0 kg ball swings in a vertical circle on the end of an 80-cm-long string. The tension in the string is 20 N when its angle from the highest point on the circle is $\theta =30\text{°}$.

1. What is the ball’s speed when $\theta =30\text{°}?$

2. What are the magnitude and direction of the ball’s acceleration when $\theta =30\text{°}?$

||| For safety, elevators have a rotational governor, a device that is attached to and rotates with one of the elevator’s pulleys. The governor, shown in FIGURE P8.63, is a disk with two hollow channels holding springs with metal blocks of mass m attached to their free ends. The faster the governor spins, the more the springs stretch. At a critical angular velocity ${\omega }_{\text{c}}$, the metal blocks contact the housing, which completes a circuit and activates an emergency brake. The spring force on a mass, which we will explore more thoroughly in Chapter 9, is $F_{\text{Sp}}=k(r-L)$, where k is the spring constant measured in N/m, and L is the relaxed (unstretched) length of the spring. Suppose a rotational governor has $L=0.80R$ and the emergency brake activates when the metal blocks reach $r=R$. What is the critical angular velocity in rpm if $R=15\text{cm,}$ $k=20\text{N}\text{/}\text{m}$, and $m=25\text{g}?$ Ignore gravity.

FIGURE P8.63

||| A 500 g steel block rotates on a steel table while attached to a 2.0-m-long massless rod. Compressed air fed through the rod is ejected from a nozzle on the back of the block, exerting a thrust force of 4.2 N. The nozzle is $70\text{°}$ from the radial line, as shown in FIGURE P8.64. The block starts from rest.

1. What is the block’s angular velocity after 10 rev?

2. What is the tension in the rod after 10 rev?

FIGURE P8.64

$60\text{N}=(0.30\text{kg}){\omega }^2(0.50\text{m})$

$(1500\text{kg})\left(9.8\text{m/}{\text{s}}^2\right)-11\text{,}760\text{N}=\left(1500\text{kg}\right)v^2\text{/}(200\text{m})$

||| Sam (75 kg) takes off up a 50-m-high, $10\text{°}$ frictionless slope on his jet-powered skis. The skis have a thrust of 200 N. He keeps his skis tilted at $10\text{°}$ after becoming airborne, as shown in Figure CP8.67. How far does Sam land from the base of the cliff?

FIGURE CP8.67

||| CALC In the absence of air resistance, a projectile that lands at the elevation from which it was launched achieves maximum range when launched at a $45\text{°}$ angle. Suppose a projectile of mass m is launched with speed $v_0$ into a headwind that exerts a constant, horizontal retarding force $\vec{F}{}_{\text{wind}}=-F_{\text{wind}}î$.

1. Find an expression for the angle at which the range is maximum.

2. By what percentage is the maximum range of a 0.50 kg ball reduced if $F_{\text{wind}}=0.60\text{N}?$

||| The 10 mg bead in FIGURE  CP8.69 is free to slide on a frictionless wire loop. The loop rotates about a vertical axis with angular velocity $\omega$. If $\omega$ is less than some critical value ${\omega }_{\text{c}}$, the bead sits at the bottom of the spinning loop. When $\omega >{\omega }_{\text{c}}$, the bead moves out to some angle $\theta$.

1. What is ${\omega }_{\text{c}}$ in rpm for the loop shown in the figure?

2. At what value of $\omega$, in rpm, is $\theta =30\text{°}$?

FIGURE CP8.69

||| A 500 g steel block rotates on a steel table while attached to a 1.2-m-long hollow tube as shown in FIGURE CP8.70. Compressed air fed through the tube and ejected from a nozzle on the back of the block exerts a thrust force of 4.0 N perpendicular to the tube. The maximum tension the tube can withstand without breaking is 50 N. If the block starts from rest, how many revolutions does it make before the tube breaks?

FIGURE CP8.70

||| A small bead slides around a horizontal circle at height y inside the cone shown in FIGURE CP8.71. Find an expression for the bead’s speed in terms of a, h, y, and g.

FIGURE CP8.71

||| If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.72, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equation $z=\left({\omega }^2\text{/}2g\right)r^2$. Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.

FIGURE CP8.72