Chapter 6

CALC A balky cow is leaving the barn as you try harder and harder to push her back in. In coordinates with the origin at the barn door, the cow walks from \(x=0\) to \(x=6.9 \mathrm{m}\) as you apply a force with \(x\) -component \(F_{x}=-[20.0 \mathrm{N}+(3.0 \mathrm{N} / \mathrm{m}) x] .\) How much work does the force you apply do on the cow during this displacement?

CALC Rotating Bar. A thin, uniform 12.0 -kg bar that is 2.00 \(\mathrm{m}\) long rotates uniformly about a pivot at one end, making 5.00 complete revolutions every 3.00 seconds. What is the kinetic energy of this bar? (Hint. Different points in the bar have different speeds. Break the bar up into infinitesimal segments of mass dm and integrate to add up the kinetic energies of all these segments.)

A Near-Earth Asteroid. On April \(13,2029\) (Friday the 13th!), the asteroid 99942 Apophis will pass within \(18,600\) mi of the earth- about \(\frac{1}{13}\) the distance to the moon! It has a density of \(2600 \mathrm{kg} / \mathrm{m}^{3},\) can be modeled as a sphere 320 \(\mathrm{m}\) in diameter, and will be traveling at 12.6 \(\mathrm{km} / \mathrm{s}\) . (a) If, due to a small disturbance in its orbit, the asteroid were to hit the earth, how much kinetic energy would it deliver? (b) The largest nuclear bomb ever tested by the United States was the "Castle/Bravo" bomb, having a yield of 15 megatons of TNT. (A megaton of TNT releases \(4.184 \times 10^{15}\) J of energy.) How many Castle/Bravo bombs would be equivalent to the energy of Apophis?

A luggage handler pulls a 20.0 -kg suitcase up a ramp inclined at \(25.0^{\circ}\) above the horizontal by a force \(\vec{\boldsymbol{F}}\) of magnitude 140 \(\mathrm{N}\) that acts parallel to the ramp. The coefficient of kinetic friction between the ramp and the incline is \(\mu_{\mathrm{k}}=0.300\) . If the suitcase travels 3.80 \(\mathrm{m}\) along the ramp, calculate (a) the work done on the suitcase by the force \(\vec{\boldsymbol{F}} ;\) (b) the work done on the suitcase by the gravitational force; (c) the work done on the suitcase by the normal force; (d) the work done on the suitcase by the friction force; (e) the total work done on the suitcase. (f ) If the speed of the suitcase is zero at the bottom of the ramp, what is its speed after it has traveled 3.80 \(\mathrm{m}\) along the ramp?

BIO Chin-Ups, While doing a chin-up, a man lifts his body 0.40 \(\mathrm{m} .\) (a) How much work must the man do per kilogram of body mass? (b) The muscles involved in doing a chin-up can generate about 70 \(\mathrm{J}\) of work per kilogram of muscle mass. If the man can just barely do a \(0.40-\mathrm{m}\) chin-up, what percentage of his body's mass do these muscles constitute? (For comparison, the total percentage of muscle in a typical \(70-\mathrm{kg}\) man with 14\(\%\) body fat is about 43\(\%\) . (c) Repeat part (b) for the man's young son, who has arms half as long as his father's but whose muscles can also generate 70 \(\mathrm{J}\) of work per kilogram of muscle mass. (d) Adults and children have about the same percentage of muscle in their bodies. Explain why children can commonly do chin-ups more easily than their fathers.

CP A 20.0 -kg crate sits at rest at the bottom of a 15.0 -m-long ramp that is inclined at \(34.0^{\circ}\) above the horizontal. A constant horizontal force of 290 \(\mathrm{N}\) is applied to the crate to push it up the ramp. While the crate is moving, the ramp exerts a constant frictional force on it that has magnitude 65.0 \(\mathrm{N}\) . (a) What is the total work done on the crate during its motion from the bottom to the top of the ramp? (b) How much time does it take the crate to travel to the top of the ramp?

A 5.00 -kg package slides 1.50 \(\mathrm{m}\) down a long ramp that is inclined at \(24.0^{\circ}\) below the horizontal. The coefficient of kinetic friction between the package and the ramp is \(\mu_{k}=0.310 .\) Calculate (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. If the package has a speed of 2.20 \(\mathrm{m} / \mathrm{s}\) at the top of the ramp, what is its speed after sliding 1.50 \(\mathrm{m}\) down the ramp?

Bl0 Whiplash Injuries. When a car is hit from behind, its passengers undergo sudden forward acceleration, which can cause a severe neck injury known as whiplash. During normal acceleration, the neck muscles play a large role in accelerating the head so that the bones are not injured. But during a very sudden acceleration, the muscles do not react immediately because they are flexible, so most of the accelerating force is provided by the neck bones. Experimental tests have shown that these bones will fracture if they absorb more than 8.0 \(\mathrm{J}\) of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 \(\mathrm{ms},\) what is the greatest speed this car and its driver can reach without breaking neck bones if the driver's head has a mass of 5.0 \(\mathrm{kg}\) (which is about right for a 70 -kg person)? Express your answer in \(\mathrm{m} / \mathrm{s}\) and in mph. (b) What is the acceleration of the passengers during the collision in part (a), and how large a force is acting to accelerate their heads? Express the acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} \mathrm{s}\) .

CALC A net force along the \(x\) -axis that has \(x\) -component \(F_{x}=-12.0 \mathrm{N}+\left(0.300 \mathrm{N} / \mathrm{m}^{2}\right) x^{2}\) is applied to a 5.00 \(\mathrm{-kg}\) object that is initially at the origin and moving in the \(-x\) -direction with a speed of 6.00 \(\mathrm{m} / \mathrm{s} .\) What is the speed of the object when it reaches the point \(x=5.00 \mathrm{m} ?\)

CALC An object is attracted toward the origin with a force given by \(F_{x}=-k / x^{2}\) . (Gravitational and electrical forces have this distance dependence.) (a) Calculate the work done by the force \(F_{x}\) when the object moves in the \(x\) -direction from \(x_{1}\) to \(x_{2}\) . If \(x_{2}>x_{1},\) is the work done by \(F_{x}\) positive or negative? (b) The only other force acting on the object is a force that you exert with your hand to move the object slowly from \(x_{1}\) to \(x_{2} .\) How much work do you do? If \(x_{2}>x_{1},\) is the work you do positive or negative? (c) Explain the similarities and differences between your answers to parts (a) and (b).

CALC The gravitational pull of the earth on an object is inversely proportional to the square of the distance of the object from the center of the earth. At the earth's surface this force is equal to the object's normal weight \(m g,\) where \(g=9.8 \mathrm{m} / \mathrm{s}^{2},\) and at large distances, the force is zero. If a \(20,000-\mathrm{kg}\) asteroid falls to earth from a very great distance away, what will be its minimum speed as it strikes the earth's surface, and how much kinetic energy will it impart to our planet? You can ignore the effects of the earth's atmosphere.

CALC Varying Coefficient of Friction. A box is sliding with a speed of 4.50 \(\mathrm{m} / \mathrm{s}\) on a horizontal surface when, at point \(P\) it encounters a rough section. On the rough section, the coefficient of friction is not constant, but starts at 0.100 at \(P\) and increases linearly with distance past \(P\) , reaching a value of 0.600 at 12.5 \(\mathrm{m}\) past point \(P .\) (a) Use the work-energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid if the friction coefficient didn't increase but instead had the constant value of 0.100\(?\)

CALC Consider a spring that does not obey Hooke's law very faithfully. One end of the spring is fixed. To keep the spring stretched or compressed an amount \(x,\) a force along the \(x\) -axis with \(x\) -component \(F_{x}=k x-b x^{2}+c x^{3}\) must be applied to the free end. Here \(k=100 \mathrm{N} / \mathrm{m}, b=700 \mathrm{N} / \mathrm{m}^{2},\) and \(c=12,000 \mathrm{N} / \mathrm{m}^{3} .\) Note that \(x>0\) when the spring is stretched and \(x<0\) when it is compressed. (a) How much work must be done to stretch this spring by 0.050 m from its unstretched length? (b) How much work must be done to compress this spring by 0.050 m from its unstretched length? (c) Is it easier to stretch or compress this spring? Explain why in terms of the dependence of \(F_{x}\) on \(x\) . (Many real springs behave qualitatively in the same way.)

CPA small block with a mass of 0.0900 \(\mathrm{kg}\) is attached to a cord passing through a hole in a frictionless, horizontal surface (Fig. P6.75). The block is originally revolving at a distance of 0.40 \(\mathrm{m}\) from the hole with a speed of 0.70 \(\mathrm{m} / \mathrm{s} .\) The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.10 \(\mathrm{m} .\) At this new distance, the speed of the block is observed to be 2.80 \(\mathrm{m} / \mathrm{s}\) . (a) What is the tension in the cord in the original situation when the block has speed \(v=0.70 \mathrm{m} / \mathrm{s} ?\) (b) What is the tension in the cord in the final situation when the block has speed \(v=2.80 \mathrm{m} / \mathrm{s} ?\) (c) How much work was done by the person who pulled on the cord?

CALC Proton Bombardment. A proton with mass \(1.67 \times 10^{-27} \mathrm{kg}\) is propelled at an initial speed of \(3.00 \times 10^{5} \mathrm{m} / \mathrm{s}\) directly toward a uranium nucleus 5.00 \(\mathrm{m}\) away. The proton is repelled by the uranium nucleus with a force of magnitude \(F=\alpha / x^{2},\) where \(x\) is the separation between the two objects and \(\alpha=2.12 \times 10^{-26} \mathrm{N} \cdot \mathrm{m}^{2} .\) Assume that the uranium nucleus remains at rest. (a) What is the speed of the proton when it is \(8.00 \times 10^{-10} \mathrm{m}\) from the uranium nucleus? (b) As the proton approaches the uranium nucleus, the repulsive force slows down the proton until it comes momentarily to rest, after which the proton moves away from the uranium nucleus. How close to the uranium nucleus does the proton get? (c) What is the speed of the proton when it is again 5.00 \(\mathrm{m}\) away from the uranium nucleus?

CALC A block of ice with mass 4.00 \(\mathrm{kg}\) is initially at rest on a frictionless, horizontal surface. A worker then applies a horizontal force \(\vec{\boldsymbol{F}}\) to it. As a result, the block moves along the \(x\) -axis such that its position as a function of time is given by 3 . \(x(t)=\alpha t^{2}+\beta t^{3},\) where \(\alpha=0.200 \mathrm{m} / \mathrm{s}^{2}\) and \(\beta=0.0200 \mathrm{m} / \mathrm{s}^{3}\). (a) Calculate the velocity of the object when \(t=4.00\) s. (b) Calculate the magnitude of \(F\) when \(t=4.00\) s. (c) Calculate the work done by the force \(\vec{F}\) during the first 4.00 s of the motion.

You and your bicycle have combined mass 80.0 \(\mathrm{kg} .\) When you reach the bridge, you are traveling along the road at 5.00 \(\mathrm{m} / \mathrm{s}(\) Fig. \(\mathrm{P} 6.78)\) . At the top of the bridge, you have climbed a vertical distance of 5.20 \(\mathrm{m}\) and have slowed to 1.50 \(\mathrm{m} / \mathrm{s} .\) You can ignore work done by friction and any inefficiency in the bike or your legs. (a) What is the total work done on you and your bicycle when you go from the base to the top of the bridge? (b) How much work have you done with the force you apply to the pedals?

You are asked to design spring bumpers for the walls of a parking garage. A freely rolling \(1200-\mathrm{kg}\) car moving at 0.65 \(\mathrm{m} / \mathrm{s}\) is to compress the spring no more than 0.090 \(\mathrm{m}\) before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

The spring of a spring gun has force constant \(k=400 \mathrm{N} / \mathrm{m}\) and negligible mass. The spring is compressed \(6.00 \mathrm{cm},\) and a ball with mass 0.0300 \(\mathrm{kg}\) is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 \(\mathrm{cm}\) long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so the barrel is horizontal. (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 \(\mathrm{N}\) acts on the ball as it moves along the barrel. (c) For the situation in part (b), at what position along the barrel does the ball have the greatest speed, and what is that speed? (In this case, the maximum speed does not occur at the end of the barrel.)

A 2.50 -kg textbook is forced against a horizontal spring of negligible mass and force constant \(250 \mathrm{N} / \mathrm{m},\) compressing the spring a distance of 0.250 \(\mathrm{m} .\) When released, the textbook slides on a horizontal tabletop with coefficient of kinetic friction \(\mu_{\mathrm{k}}=0.30 .\) Use the work-energy theorem to find how far the textbook moves from its initial position before coming to rest.

Pushing a Cat. Your cat "Ms." (mass 7.00 kg) is trying to make it to the top of a frictionless ramp 2.00 \(\mathrm{m}\) long and inclined upward at \(30.0^{\circ}\) above the horizontal. Since the poor cat can't get any traction on the ramp, you push her up the entire length of the ramp by exerting a constant \(100-\mathrm{N}\) force parallel to the ramp. If Ms. takes a running start so that she is moving at 2.40 \(\mathrm{m} / \mathrm{s}\) at the bottom of the ramp, what is her speed when she reaches the top of the incline? Use the work-energy theorem.

A student proposes a design for an automobile crash barrier in which a 1700 -kg sport utility vehicle moving at 20.0 \(\mathrm{m} / \mathrm{s}\) crashes into a spring of negligible mass that slows it to a stop. So that the passengers are not injured, the acceleration of the vehicle as it slows can be no greater than 5.00\(g .\) (a) Find the required spring constant \(k,\) and find the distance the spring will compress in slowing the vehicle to a stop. In your calculation, disregard any deformation or crumpling of the vehicle and the friction between the vehicle and the ground. (b) What disadvantages are there to this design?

A physics professor is pushed up a ramp inclined upward at \(30.0^{\circ}\) above the horizontal al as he sits in his desk chair that slides on frictionless rollers. The combined mass of the professor and chair is 85.0 \(\mathrm{kg} .\) He is pushed 2.50 \(\mathrm{m}\) along the incline by a group of students who together exert a constant horizontal force of 600 \(\mathrm{N} .\) The professor's speed at the bottom of the ramp is 2.00 \(\mathrm{m} / \mathrm{s} .\) Use the work-energy theorem to find his speed at the top of the ramp.

A 5.00 -kg block is moving at \(v_{0}=6.00 \mathrm{m} / \mathrm{s}\) along a frictionless, horizontal surface toward a spring with force constant \(k=500 \mathrm{N} / \mathrm{m}\) that is attached to a wall (Fig. P6.85). The spring has negligible mass. \begin{equation} \begin{array}{l}{\text { (a) Find the maximum distance the spring will be compressed. }} \\ {\text { (b) If the spring is to compress by no more than } 0.150 \mathrm{m}, \text { what }} \\\ {\text { should be the maximum value of } v_{0} ?}\end{array} \end{equation}

On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 \(\mathrm{m} / \mathrm{s}\) encounters a rough patch that reduces her speed to 1.65 \(\mathrm{m} / \mathrm{s}\) due to a friction force that is 25\(\%\) of her weight. Use the work-energy theorem to find the length of this rough patch.

Rescue. Your friend (mass 65.0 \(\mathrm{kg} )\) is standing on the ice in the middle of a frozen pond. There is very litle friction between her feet and the ice, so she is unable to walk. Fortunately, a light rope is tied around her waist and you stand on the bank holding the other end. You pull on the rope for 3.00 s and accelerate your friend from rest to a speed of 6.00 \(\mathrm{m} / \mathrm{s}\) whileyou remain at rest. What is the average power supplied by the force you applied?

A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 \(\mathrm{m}\) deep and eject it with a speed of 18.0 \(\mathrm{m} / \mathrm{s}\) (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

BIO All birds, independent of their size, must maintain a power output of \(10-25\) watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (Patagona gigas) has mass 70 \(\mathrm{g}\) and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70 -kg athlete can maintain a power output of 1.4 \(\mathrm{kW}\) for no more than a few seconds; the steady power output of a typical athlete is only 500 \(\mathrm{W}\) or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

A physics student spends part of her day walking between classes or for recreation, during which time she expends energy at an average rate of 280 \(\mathrm{W}\) . The remainder of the day she is sitting in class, studying, or resting; during these activities, she expends energy at an average rate of 100 \(\mathrm{W}\) . If she expends a total of \(1.1 \times 10^{7} \mathrm{J}\) of energy in a 24 -hour day, how much of the day did she spend walking?

The Grand Coulee Dam is 1270 \(\mathrm{m}\) long and 170 \(\mathrm{m}\) high. The electrical power output from generators at its base is approximately 2000 \(\mathrm{MW}\) . How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 92\(\%\) of the work done on the water by gravity is converted to electrical energy? (Each cubic meter of water has a mass of 1000 \(\mathrm{kg.})\)

BIO Power of the Human Heart. The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 \(\mathrm{L}\) of blood. Assume that the work done by the heart is equal to the work required to lift this amount of blood a height equal to that of the average American woman \((1.63 \mathrm{m}) .\) The density (mass per unit volume) of blood is \(1.05 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}\) . (a) How much work does the heart do in a day? (b) What is the heart's power output in watts?

Six diesel units in series can provide 13.4 \(\mathrm{MW}\) of power to the lead car of a freight train. The diesel units have total mass \(1.10 \times 10^{6} \mathrm{kg}\) . The average car in the train has mass \(8.2 \times 10^{4} \mathrm{kg}\) and requires a horizontal pull of 2.8 \(\mathrm{kN}\) to move at a constant 27 \(\mathrm{m} / \mathrm{s}\) on level tracks. (a) How many cars can be in the train under these conditions? (b) This would leave no power for accelerating or climbing hills. Show that the extra force needed to accelerate the train is about the same for a \(0.10-\mathrm{m} / \mathrm{s}^{2}\) acceleration or a 1.0\(\%\) slope (slope angle \(\alpha=\arctan 0.010 )\) . (c) With the 1.0\(\%\) slope, show that an extra 2.9 \(\mathrm{MW}\) of power is needed to maintain the \(27-\mathrm{m} / \mathrm{s}\) speed of the diesel units. (d) With 2.9 \(\mathrm{MW}\) less power available, how many cars can the six diesel units pull up a 1.0\(\%\) slope at a constant 27 \(\mathrm{m} / \mathrm{s} ?\)

It takes a force of 53 \(\mathrm{kN}\) on the lead car of a 16 -car passenger train with mass \(9.1 \times 10^{5} \mathrm{kg}\) to pull it at a constant 45 \(\mathrm{m} / \mathrm{s}\) \((101 \mathrm{mi} / \mathrm{h})\) on level tracks. (a) What power must the locomotive provide to the lead car? (b) How much more power to the lead car than calculated in part (a) would be needed to give the train an acceleration of 1.5 \(\mathrm{m} / \mathrm{s}^{2}\) , at the instant that the train has a speed of 45 \(\mathrm{m} / \mathrm{s}\) on level tracks? (c) How much more power to the lead car than that calculated in part (a) would be needed to move the train up a 1.5\(\%\) grade (slope angle \(\alpha=\arctan 0.015 )\) at a constant 45 \(\mathrm{m} / \mathrm{s} ?\)

CALC An object has several forces acting on it. One of these forces is \(\vec{\boldsymbol{F}}=a x y \hat{\boldsymbol{r}},\) a force in the \(x\) -direction whose magni- tude depends on the position of the object, with \(\alpha=2.50 \mathrm{N} / \mathrm{m}^{2}\) . Calculate the work done on the object by this force for the following displacements of the object: (a) The object starts at the point \(x=0\) , \(y=3.00 \mathrm{m}\) and moves parallel to the \(x\) -axis to the point \(x=2.00 \mathrm{m}, y=3.00 \mathrm{m} .\) (b) The object starts at the point \(x=2.00 \mathrm{m}, \quad y=0\) and moves in the \(y\) -direction to the the point \(x=2.00 \mathrm{m}, y=3.00 \mathrm{m} .\) (c) The object starts at the origin and moves on the line \(y=1.5 x\) to the point \(x=2.00 \mathrm{m},\) \(y=3.00 \mathrm{m} .\)

Cycling. For a touring bicyclist the drag coefficient \(C\left(f_{\text { air }}=\frac{1}{2} C A \rho v^{2}\right)\) is \(1.00,\) the frontal area \(A\) is \(0.463 \mathrm{m}^{2},\) and the coefficient of rolling friction is \(0.0045 .\) The rider has mass 50.0 \(\mathrm{kg}\) , and her bike has mass 12.0 \(\mathrm{kg}\) (a) To maintain a speed of 12.0 \(\mathrm{m} / \mathrm{s}\) (about 27 \(\mathrm{mi} / \mathrm{h} )\) on a level road, what must the rider's power output to the rear wheel be? (b) For racing, the same rider uses a different bike with coefficient of rolling friction 0.0030 and mass 9.00 kg. She also crouches down, reducing her drag coefficient to 0.88 and reducing her frontal area to 0.366 \(\mathrm{m}^{2} .\) What must her power output to the rear wheel be then to maintain a speed of 12.0 \(\mathrm{m} / \mathrm{s} ?\) (c) For the situation in part (b), what power output is required to maintain a speed of 6.0 \(\mathrm{m} / \mathrm{s} ?\) Note the great drop in power requirement when the speed is only halved. (For more on aerodynamic speed limitations for a wide variety of human-powered vehicles, "see "The Aerodynamics of Human- Powered Land Vehicles," Scientific American, December \(1983 . )\)

Automotive Power I. A truck engine transmits 28.0 \(\mathrm{kW}(37.5 \mathrm{hp})\) to the driving wheels when the truck is traveling at a constant velocity of magnitude 60.0 \(\mathrm{km} / \mathrm{h}\) (37.3 \(\mathrm{mi} / \mathrm{h} )\) on a level road. (a) What is the resisting force acting on the truck? (b) Assume that 65\(\%\) of the resisting force is due to rolling friction and the remainder is due to air resistance. If the force of rolling friction is independent of speed, and the force of air resistance is proportional to the square of the speed, what power will drive the truck at 30.0 \(\mathrm{km} / \mathrm{h} ?\) At 120.0 \(\mathrm{km} / \mathrm{h} ?\) Give your answers in kilowatts and in horsepower.

Automotive Power II. (a) If 8.00 hp are required to drive a \(1800-\) -kg automobile at 60.0 \(\mathrm{km} / \mathrm{h}\) on a level road, what is the total retarding force due to friction, air resistance, and so on? (b) What power is necessary to drive the car at 60.0 \(\mathrm{km} / \mathrm{h}\) up a 10.0\(\%\) grade (a hill rising 10.0 \(\mathrm{m}\) vertically in 100.0 \(\mathrm{m}\) horizon- tally)? (c) What power is necessary to drive the car at 60.0 \(\mathrm{km} / \mathrm{h}\) down a 1.00\(\%\) grade? (d) Down what percent grade would the car coast at 60.0 \(\mathrm{km} / \mathrm{h}\) ?

CALC On a winter day in Maine, a warehouse worker is shoving boxes up a rough plank inclined at an angle \(\alpha\) above the horizontal. The plank is partially covered with ice, with more ice near the bottom of the plank than near the top, so that the coefficient of friction increases with the distance \(x\) along the plank: \(\mu=A x,\) where \(A\) is a positive constant and the bottom of the plank is at \(x=0 .\) (For this plank the coefficients of kinetic and static friction are equal: \(\mu_{\mathrm{k}}=\mu_{\mathrm{s}}=\mu .\) The worker shoves a box up the plank so that it leaves the bottom of the plank mov- ing at speed \(v_{0}\) . Show that when the box first comes to rest, it will remain at rest if $$v_{0}^{2} \geq \frac{3 g \sin ^{2} \alpha}{A \cos \alpha}$$

CALC A Spring with Mass. We usually ignore the kinetic energy of the moving coils of a spring, but let's try to get a reasonable approximation to this. Consider a spring of mass \(M,\) equilibrium length \(L_{0},\) and spring constant \(k .\) The work done to stretch or compress the spring by a distance \(L\) is \(\frac{1}{2} k X^{2}\) , where \(X=L-L-L_{0}\) . Consider a spring, as described above, that has one end fixed and the other end moving with speed \(v .\) Assume that the speed of points along the length of the spring varies linearly with distance \(l\) from the fixed end. Assume also that the mass \(M\) of the spring is distributed uniformly along the length of the spring. (a) Calculate the kinetic energy of the spring in terms of the \(M\) and \(v .\) (Hint: Divide the spring into pieces of length \(d l ;\) find the speed of each pivide in terms of \(l, v,\) and \(L ;\) find the mass of each piece in terms of \(d l, M,\) and \(L ;\) and integrate from 0 to \(L .\) The result is \(n o t \frac{1}{2} M v^{2},\) since not all of the spring moves with the same speed.) In a spring gun, a spring of mass 0.243 \(\mathrm{kg}\) and force constant 3200 \(\mathrm{N} / \mathrm{m}\) is compressed 2.50 \(\mathrm{cm}\) from its unstretched length. When the trigger is pulled, the spring pushes horizon- tally on a 0.053 -kg ball. The work done by friction is negligible. Calculate the ball's speed when the spring reaches its uncom- pressed length (b) ignoring the mass of the spring and (c) includ- ing, using the results of part (a), the mass of the spring. (d) In part (c), what is the final kinetic energy of the ball and of the spring?

CALC An airplane in flight is subject to an air resistance force proportional to the square of its speed \(v .\) But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward (Fig. P6.104). The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to \(v^{2},\) so that the total air resistance force can be expressed by \(F_{\text { air }}=\alpha v^{2}+\beta / v^{2},\) where \(\alpha\) and \(\beta\) are positive constants that depend on the shape and size of the airplane and the density of the air. For a Cessna \(150,\) a small single-engine airplane, \(\alpha=0.30 \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2}\) and \(\beta=3.5 \times 10^{5} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{s}^{2} .\) In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed (in \(\mathrm{km} / \mathrm{h} )\) at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. (b) Calculate the speed (in \(\mathrm{km} / \mathrm{h} )\) for which the airplane will have the maximum endurance (that is, remain in the air the longest time).