|| A 2.0 kg object initially at rest at the origin is subjected to the time-varying force shown in FIGURE P6.38. What is the object’s velocity at $t=4\text{s}?$

FIGURE P6.38

|| The 1000 kg steel beam in FIGURE P6.39 is supported by two ropes. What is the tension in each?

FIGURE P6.39

|| BIO An accident victim with a broken leg is being placed in traction. The patient wears a special boot with a pulley attached to the sole. The foot and boot together have a mass of 4.0 kg, and the doctor has decided to hang a 6.0 kg mass from the rope. The boot is held suspended by the ropes, as shown in FIGURE P6.40, and does not touch the bed.

1. Determine the amount of tension in the rope by using Newton’s laws to analyze the hanging mass.

   Hint: If the pulleys are frictionless, which we will assume, the tension in the rope is constant from one end to the other.

2. The net traction force needs to pull straight out on the leg. What is the proper angle $\theta$ for the upper rope?

3. What is the net traction force pulling on the leg?

FIGURE P6.40

|| Henry, whose mass is 95 kg, stands on a bathroom scale in an elevator. The scale reads 830 N for the first 3.0 s after the elevator starts moving, then 930 N for the next 3.0 s. What is the elevator’s velocity 6.0 s after starting?

|| BIO Seat belts and air bags save lives by reducing the forces exerted on the driver and passengers in an automobile collision. Cars are designed with a “crumple zone” in the front of the car. In the event of an impact, the passenger compartment decelerates over a distance of about 1 m as the front of the car crumples. An occupant restrained by seat belts and air bags decelerates with the car. By contrast, an unrestrained occupant keeps moving forward with no loss of speed (Newton’s first law!) until hitting the dashboard or windshield. These are unyielding surfaces, and the unfortunate occupant then decelerates over a distance of only about 5 mm.

1. A 60 kg person is in a head-on collision. The car’s speed at impact is $15\text{m}\text{/}\text{s}$. Estimate the net force on the person if he or she is wearing a seat belt and if the air bag deploys.

2. Estimate the net force that ultimately stops the person if he or she is not restrained by a seat belt or air bag.

|| The piston of a machine exerts a constant force on a ball as it moves horizontally through a distance of 15 cm. You use a motion detector to measure the speed of five different balls as they come off the piston; the data are shown in the table. Use theory to find two quantities that, when graphed, should give a straight line. Then use the graph to find the size of the piston’s force.

|| Compressed air is used to fire a 50 g ball vertically upward from a 1.0-m-tall tube. The air exerts an upward force of 2.0 N on the ball as long as it is in the tube. How high does the ball go above the top of the tube? Neglect air resistance.

|| A rifle with a barrel length of 60 cm fires a 10 g bullet with a horizontal speed of $400\text{m}\text{/}\text{s}$. The bullet strikes a block of wood and penetrates to a depth of 12 cm.

1. What resistive force (assumed to be constant) does the wood exert on the bullet?

2. How long does it take the bullet to come to rest?

||| A truck with a heavy load has a total mass of 7500 kg. It is climbing a $15\text{°}$ incline at a steady $15\text{m}\text{/}\text{s}$ when, unfortunately, the poorly secured load falls off! Immediately after losing the load, the truck begins to accelerate at $1.5\text{m}\text{/}{\text{s}}^2$. What was the mass of the load? Ignore rolling friction.

1. || A rocket of mass m is launched straight up with thrust $\vec{F}{}_{\text{thrust}}$. Find an expression for the rocket’s speed at height h if air resistance is neglected.

2. The motor of a 350 g model rocket generates 9.5 N thrust. If air resistance can be neglected, what will be the rocket’s speed as it reaches a height of 85 m?

|| An object of mass m is at rest at the top of a smooth slope of height h and length L. The coefficient of kinetic friction between the object and the surface, ${\mu }_{\text{k}}$, is small enough that the object will slide down the slope after being given a very small push to get it started. Find an expression for the object’s speed at the bottom of the slope.

|| Sam, whose mass is 75 kg, takes off across level snow on his jet-powered skis. The skis have a thrust of 200 N and a coefficient of kinetic friction on snow of 0.10. Unfortunately, the skis run out of fuel after only 10 s.

1. What is Sam’s top speed?

2. How far has Sam traveled when he finally coasts to a stop?

|| A baggage handler drops your 10 kg suitcase onto a conveyor belt running at $2.0\text{m}\text{/}\text{s}$. The materials are such that ${\mu }_{\text{s}}=0.50$ and ${\mu }_{\text{k}}=0.30$. How far is your suitcase dragged before it is riding smoothly on the belt?

|| A 5.0 kg wooden sled is launched up a 25° snow-covered slope with an initial speed of 10 m/s.

1. What vertical height does the sled reach above its starting point?

2. What is the sled’s speed when it slides back down to its starting point?

|| It’s a snowy day and you’re pulling a friend along a level road on a sled. You’ve both been taking physics, so she asks what you think the coefficient of friction between the sled and the snow is. You’ve been walking at a steady $1.5\text{m}\text{/}\text{s}$, and the rope pulls up on the sled at a $30\text{°}$ angle. You estimate that the mass of the sled, with your friend on it, is 60 kg and that you’re pulling with a force of 75 N. What answer will you give?

|| While driving to work one day, I was holding my coffee mug in my left hand while changing the radio station with my right hand. Then my cell phone rang, so I placed the mug on the flat part of my dashboard. Then, believe it or not, a deer ran out of the woods and onto the road right in front of me. Fortunately, my reaction time was zero, and I was able to stop from a speed of 22 m/s in a mere 50 m, just barely avoiding the deer, without the coffee mug sliding. What is the minimum possible value of the coefficient of static friction between the mug and the dashboard? The mug (with coffee) had a mass of 550 g, and the mass of the deer was 150 kg.

|| A large box of mass M is pulled across a horizontal, frictionless surface by a horizontal rope with tension T. A small box of mass m sits on top of the large box. The coefficients of static and kinetic friction between the two boxes are ${\mu }_{\text{s}}$ and ${\mu }_{\text{k}}$, respectively. Find an expression for the maximum tension $T_{\text{max}}$ for which the small box rides on top of the large box without slipping.

|| A large box of mass M is moving on a horizontal surface at speed $v_0$. A small box of mass m sits on top of the large box. The coefficients of static and kinetic friction between the two boxes are ${\mu }_{\text{s}}$ and ${\mu }_{\text{k}}$, respectively. Find an expression for the shortest distance $d_{\text{min}}$ in which the large box can stop without the small box slipping.

|| A person with compromised pinch strength in his fingers can exert a force of only 6.0 N to either side of a pinch-held object, such as the book shown in FIGURE P6.56. What is the weight of the heaviest book he can hold vertically before it slips out of his fingers? The coefficient of static friction between his fingers and the book cover is 0.80.

FIGURE P6.56

|| A 1.0 kg wood block is pressed against a vertical wood wall by the 12 N force shown in FIGURE P6.57. If the block is initially at rest, will it move upward, move downward, or stay at rest?

FIGURE P6.57

|| The 2.0 kg wood box in FIGURE P6.58 slides down a vertical wood wall while you push on it at a $45\text{°}$ angle. What magnitude of force should you apply to cause the box to slide down at a constant speed?

FIGURE P6.58

|| CALC Astronauts in space “weigh” themselves by oscillating on a spring. Suppose the position of an oscillating 75 kg astronaut is given by $x=(0.30\text{m})\text{s}\text{i}\text{n}((\pi \text{rad}\text{/}\text{s})\times t)$, where t is in s. What force does the spring exert on the astronaut at (a) $t=1.0\text{s}$ and (b) 1.5 s? Note that the angle of the sine function is in radians.

|| CALC A particle of mass m moving along the x-axis experiences the net force $F_x=ct$, where c is a constant. The particle has velocity $v_{0x}$ at $t=0$. Find an algebraic expression for the particle’s velocity $v_x$ at a later time t.

|| CALC At $t=0$, an object of mass m is at rest at $x=0$ on a horizontal, frictionless surface. A horizontal force $F_x=F_0(1-t\text{/}T)$, which decreases from $F_0$ at $t=0$ to zero at $t=T$, is exerted on the object. Find an expression for the object’s (a) velocity and (b) position at time T.

|| CALC A 500 g ball moves horizontally with velocity $v_x=(15\text{m})\text{/}(t+1\text{s})$ for $t>0\text{s}$. What is the net force on the ball at $t=1\text{s}?$

|| A ball is shot from a compressed-air gun at twice its terminal speed.

1. What is the ball’s initial acceleration, as a multiple of g, if it is shot straight up?

2. What is the ball’s initial acceleration, as a multiple of g, if it is shot straight down?

|| An oceanographic research ship uses a cable to tow a submerged sonar system that maps the ocean floor. The submersible travels horizontally because the vertical component of the cable tension and an upward buoyant force from the water balance the gravitational force. One submersible has a cross-section area of $1.3{\text{m}}^2$ and a drag coefficient of 0.85. The tow cable makes a 30° angle with the horizontal when pulling the submersible at 5.1 m/s. What is the magnitude of the tension in the cable? The density of seawater is $1020{\text{kg/m}}^3$.

||| A 6.5-cm-diameter ball has a terminal speed of 26 m/s. What is the ball’s mass? Use the properties of 20°C air at sea level.

|| What is the magnitude of the acceleration of a skydiver at the instant she is falling at one-half her terminal speed?

|| A 70 kg bicyclist is coasting down a long hill that has a 5.0° slope. His cross-section area is $0.35{\text{m}}^2$, his drag coefficient is 0.85, his bicycle’s coefficient of rolling friction is 0.02, and the air temperature is 20°C. What speed does he eventually reach?

|| Electrostatic precipitation is used to remove small particulates from the emissions of coal-fired power plants. The particles are given a small electric charge that causes them to move toward a large, oppositely charged plate where they stick. A typical particulate is a $1.0\text{-}\mu \text{m}\text{-}\text{diameter}$ sphere with a density of $1300{\text{kg/m}}^3$. With what speed in mm/s does it drift toward the collector plate if the electric force is $1.2\times {10}^{-12}\text{N}?$ Gravity can be neglected because the electric force is much larger. The viscosity of air in the 200°C exhaust is $2.5\times {10}^{-5}\text{Pa s}$.

$-0.80n=(1500\text{kg})a_x$

$n-(1500\text{kg})\left(9.80\text{m}\text{/}{\text{s}}^2\right)=0$

$T-0.20n-(20\text{kg})\left(9.80\text{m}\text{/}{\text{s}}^2\right)\text{s}\text{i}\text{n}20\text{°}$

$=(20\text{kg})\left(2.0\text{m}\text{/}{\text{s}}^2\right)$

$n-(20\text{kg})\left(9.80\text{m}\text{/}{\text{s}}^2\right)\text{c}\text{o}\text{s}20\text{°}=0$

$(100\text{N) cos }30\text{°}-f_{\text{k}}=\left(20\text{kg}\right)a_x$

$n+(100\text{N})\text{s}\text{i}\text{n}30\text{°}-\left(20\text{kg}\right)\left(9.80\text{m}\text{/}{\text{s}}^2\right)=0$

$f_{\text{k}}=0.20n$

||| CALC A block of mass m is at rest at the origin at $t=0$. It is pushed with constant force $F_0$ from $x=0$ to $x=L$ across a horizontal surface whose coefficient of kinetic friction is ${\mu }_{\text{k}}={\mu }_0(1-x\text{/}L)$. That is, the coefficient of friction decreases from ${\mu }_0$ at $x=0$ to zero at $x=L$.

1. Use what you’ve learned in calculus to prove that

   $$a_x=v_x{\displaystyle \frac{d{v}_x}{dx}}$$

2. Find an expression for the block’s speed as it reaches position L.

||| CALC A spring-loaded toy gun exerts a variable force on a plastic ball as the spring expands. Consider a horizontal spring and a ball of mass m whose position when barely touching a fully expanded spring is $x=0$. The ball is pushed to the left, compressing the spring. You’ll learn in Chapter 9 that the spring force on the ball, when the ball is at position x (which is negative), can be written as $(F_{\text{Sp}}){}_x=-kx$, where k is called the spring constant. The minus sign is needed to make the x-component of the force positive. Suppose the ball is initially pushed to $x_0=-L$, then released and shot to the right.

1. Use what you’ve learned in calculus to prove that

   $$a_x=v_x{\displaystyle \frac{d{v}_x}{dx}}$$

2. Find an expression, in terms of m, k, and L, for the speed of the ball as it comes off the spring at $x=0$.

||| CALC A smaller holding tension $T_{\text{hold}}$ can support a larger load tension $T_{\text{load}}$ if a rope is wrapped partially or completely around a cylinder. In old sailing ships, the enormous tension in a rope holding a sail could be controlled by only a few sailors when the rope was wrapped around a capstan. Similarly, belaying rock climbers can support their weight in case of a fall. FIGURE CP6.76 shows a rope wrapped through angle $\varphi$ around a cylinder of radius R whose coefficient of static friction is ${\mu }_{\text{s}}$. The minimum tension needed to hold the load occurs when the rope is on the verge of slipping. The inset shows a small segment of rope spanning angle $d\varphi$. Assume that the tension on the load side is infinitesimally larger than on the holding side. The static friction force is tangent to the rope and thus perpendicular to $\vec{n}$. However, the small curvature of this segment means that the two tension forces act at angles ${\displaystyle \frac{1}{2}}d\varphi$ to the tangent line.

1. Apply Newton’s second law to this segment of the rope to get an equation relating $dT$, the increase in tension across this little segment, to $d\varphi$, then integrate from $T=T_{\text{hold}}$ at $\varphi =0$ to $T=T_{\text{load}}$ at angle $\varphi$. You will need to use the small-angle approximations $\text{sin }\theta \approx \theta \text{and cos}\theta \approx 1$ when $\theta$, in radians, is very small. Give your result as an equation for $T_{\text{hold}}$, the minimum tension force needed to support load $T_{\text{load}}$. This result is known as the capstan equation.

2. A rock climber who weighs 750 N is dangling from a rope. His partner uses a belaying device in which the rope is wrapped around a cylinder with ${\mu }_{\text{s}}=0.50$. What is the magnitude of the tension needed to support the climber if the rope is wrapped through an angle of (i) 90°, (ii) 180°, and (iii) 360°?

FIGURE CP6.76

1. ||| CALC A spherical particle of mass m is shot horizontally with initial speed $v_0$ into a viscous fluid. Use Stokes’ law to find an expression for $v_x(t)$, the horizontal velocity as a function of time. Vertical motion due to gravity can be ignored.

2. A 4.0-cm-diameter, 55 g ball is shot horizontally into a tank of 40°C honey. How long will it take for the horizontal speed to decrease to 10% of its initial value?

1. ||| CALC Use what you’ve learned in calculus to prove that

   $$a_x=v_x{\displaystyle \frac{d{v}_x}{dx}}$$

2. A spherical particle of mass m is shot horizontally with initial speed $v_0$ into a viscous fluid. Use Stokes’ law to find an expression for $v_x(x)$, the horizontal velocity as a function of distance traveled. Vertical motion due to gravity can be ignored.

3. A 1.0-cm-diameter, 2.0 g marble is shot horizontally into a tank of 20°C olive oil at 10 cm/s. How far in cm will it travel before stopping?

||| CALC An object with cross section A is shot horizontally across frictionless ice. Its initial velocity is $v_{0x}$ at $t_0=0\text{s}$. Air resistance is not negligible.

1. Show that the velocity at time t is given by the expression

   $$v_x={\displaystyle \frac{v_{0x}}{1+C_{\text{d}}\rho A{v}_{0x}t\text{/}2m}}$$

2. A 1.6-m-wide, 1.4-m-high, 1500 kg car with a drag coefficient of 0.35 hits a very slick patch of ice while going $20\text{m}\text{/}\text{s}$. If friction is neglected, how long will it take until the car’s speed drops to $10\text{m}\text{/}\text{s}?$ To $5\text{m}\text{/}\text{s}?$