Chapter 6

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

On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 m/s encounters a rough patch that reduces her speed to 1.65 m/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.

All birds, independent of their size, must maintain a power output of 10$-$25 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird ($Patagonagigas$) has mass 70 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 kW for no more than a few seconds; the $steady$ power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 m deep and eject it with a speed of 18.0 m/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?

The Grand Coulee Dam is 1270 m long and 170 m high. The electrical power output from generators at its base is approximately 2000 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 kg.)

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 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 W. If she expends a total of 1.1 $\times$ 10${}^7$ J of energy in a 24-hour day, how much of the day did she spend walking?

An object has several forces acting on it. One of these forces is $\overrightarrow{F}=axy\hat{ı}$, a force in the $x$-direction whose magnitude 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 ($x=0$, $y=3.00$ m) and moves parallel to the x-axis to the point ($x=2.00$ m, $y=3.00$ m). (b) The object starts at the point ($x=2.00$ m, $y=0$) and moves in the $y$-direction to the point ($x=2.00$ m, $y=3.00$ m). (c) The object starts at the origin and moves on the line $y=1.5x$ to the point ($x=2.00$ m, $y=3.00$ m).

The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 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 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?

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 force constant $k$. The work done to stretch or compress the spring by a distance $L$ is \(\frac{1}{2}\) $k{X}^2$, where $X=L-L_0$. 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 $M$ and $v$. ($Hint$: Divide the spring into pieces of length $dl$; find the speed of each piece in terms of $l$, $v$, and $L$; find the mass of each piece in terms of $dl$, $M$, and $L$; and integrate from $0$ to $L$. The result is $not$ \(\frac{1}{2}\) $M{v}^2$, since not all of the spring moves with the same speed.) In a spring gun, a spring of mass 0.243 kg and force constant 3200 N/m is compressed 2.50 cm from its unstretched length. When the trigger is pulled, the spring pushes horizontally on a 0.053-kg ball. The work done by friction is negligible. Calculate the ball's speed when the spring reaches its uncompressed length (b) ignoring the mass of the spring and (c) including, 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?

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 ($\mathbf{\text{Fig. P6.94}}$). The upward force is the lift force that keeps the airplane aloft, and the backward force is called $induceddrag$. At flying speeds, induced drag is inversely proportional to $v^2$, so the total air resistance force can be expressed by $F_{a}ir=\alpha v^2+\beta /v{}^2$, 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 km/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 km/h) for which the airplane will have the maximum $endurance$(that is, remain in the air the longest time).