Chapter 8

The Lunar Module could make a safe landing if its vertical velocity at impact is \(3.0 \mathrm{~m} / \mathrm{s}\) or less. Suppose that you want to determine the greatest height \(h\) at which the pilot could shut off the engine if the velocity of the lander relative to the surface is (a) zero; (b) \(2.0 \mathrm{~m} / \mathrm{s}\) downward; (c) \(2.0 \mathrm{~m} / \mathrm{s}\) upward. Use conservation of energy to determine \(h\) in each case. The acceleration due to gravity at the surface of the Moon is \(1.62 \mathrm{~m} / \mathrm{s}^{2}\)

Proper design of automobile braking systems must account for heat buildup under heavy braking. Calculate the thermal energy dissipated from brakes in a \(1500-\mathrm{kg}\) car that descends a \(17^{\circ}\) hill. The car begins braking when its speed is \(95 \mathrm{~km} / \mathrm{h}\) and slows to a speed of \(35 \mathrm{~km} / \mathrm{h}\) in a distance of \(0.30 \mathrm{~km}\) measured along the road.

Some electric power companies use water to store energy. Water is pumped by reversible turbine pumps from a low reservoir to a high reservoir. To store the energy produced in 1.0 hour by a 180 -MW electric power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is \(380 \mathrm{~m}\) above the lower one, and we can neglect the small change in depths of each. Water has a mass of \(1.00 \times 10^{3} \mathrm{~kg}\) for every \(1.0 \mathrm{~m}^{3}\)

Suppose the gravitational potential energy of an object of mass \(m\) at a distance \(r\) from the center of the Earth is given by $$ U(r)=-\frac{G M m}{r} e^{-\alpha r} $$ where \(\alpha\) is a positive constant and \(e\) is the exponential function. (Newton's law of universal gravitation has \(\alpha=0\) ). \((a)\) What would be the force on the object as a function of \(r ?(b)\) What would be the object's escape velocity in terms of the Earth's radius \(R_{\mathrm{E}} ?\)

( \(a\) ) If the human body could convert a candy bar directly into work, how high could a \(76-\mathrm{kg}\) man climb a ladder if he were fueled by one bar \((=1100 \mathrm{~kJ}) ?(b)\) If the man then jumped off the ladder, what will be his speed when he reaches the bottom?

Electric energy units are often expressed in the form of "kilowatt-hours." (a) Show that one kilowatt-hour (kWh) is equal to \(3.6 \times 10^{6} \mathrm{~J} .\) (b) If a typical family of four uses electric energy at an average rate of \(580 \mathrm{~W}\), how many \(\mathrm{kWh}\) would their electric bill show for one month, and (c) how many joules would this be? ( \(d\) ) At a cost of \(\$ 0.12\) per \(\mathrm{kWh}\), what would their monthly bill be in dollars? Does the monthly bill depend on the rate at which they use the electric energy?

Chris jumps off a bridge with a bungee cord (a heavy stretchable cord) tied around his ankle, Fig. \(47 .\) He falls for 15 before the bungee cord begins to stretch. Chris's mass is 75 \(\mathrm{kg}\) and we assume the cord obeys Hooke's law, \(F=-k x,\) with \(k=50 \mathrm{N} / \mathrm{m}\) . If we neglect air resistance, estimate how far below the bridge Chris's foot will be before coming to a stop. Ignore the mass of the cord (not realistic, however) and treat Chris as a particle.

In a common test for cardiac function (the "stress test"), the patient walks on an inclined treadmill (Fig. \(8-48\) ). Estimate the power required from a \(75-\mathrm{kg}\) patient when the treadmill is sloping at an angle of \(12^{\circ}\) and the velocity is \(3.3 \mathrm{~km} / \mathrm{h}\). (How does this power compare to the power rating of a lightbulb?

(a) If a volcano spews a \(450-\mathrm{kg}\) rock vertically upward a distance of \(320 \mathrm{m},\) what was its velocity when it left the volcano? \((b)\) If the volcano spews 1000 rocks of this size every minute, estimate its power output.

A film of Jesse Owens's famous long jump (Fig. 49) in the 1936 Olympics shows that his center of mass rose 1.1 \(\mathrm{m}\) from launch point to the top of the arc. What minimum speed did he need at launch if he was traveling at 6.5 \(\mathrm{m} / \mathrm{s}\) at the top of the arc?

An elevator cable breaks when a \(920-\mathrm{kg}\) elevator is \(24 \mathrm{~m}\) above a huge spring \(\left(k=2.2 \times 10^{5} \mathrm{~N} / \mathrm{m}\right)\) at the bottom of the shaft. Calculate ( \(a\) ) the work done by gravity on the elevator before it hits the spring, \((b)\) the speed of the elevator just before striking the spring, and (c) the amount the spring compresses (note that work is done by both the spring and gravity in this part).

A particle moves where its potential energy is given by \(U(r)=U_{0}\left[\left(2 / r^{2}\right)-(1 / r)\right] .(a)\) Plot \(U(r)\) versus \(r .\) Where does the curve cross the \(U(r)=0\) axis? At what value of \(r\) does the minimum value of \(U(r)\) occur? (b) Suppose that the particle has an energy of \(E=-0.050 U_{0} .\) Sketch in the approximate turning points of the motion of the particle on your diagram. What is the maximum kinetic energy of the particle, and for what value of \(r\) does this occur?

A particle of mass \(m\) moves under the influence of a potential energy $$ U(x)=\frac{a}{x}+b x $$ where \(a\) and \(b\) are positive constants and the particle is restricted to the region \(x>0\). Find a point of equilibrium for the particle and demonstrate that it is stable.

(III) The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or \(F(r)=F_{0}\left[2(\sigma / r)^{13}-(\sigma / r)^{7}\right],\) where \(r\) is the separation between the two atoms, and \(\sigma\) and \(F_{0}\) are constant. For an oxygen molecule (which is diatomic) \(F_{0}=9.60 \times 10^{-11} \mathrm{~N}\) and \(\sigma=3.50 \times 10^{-11} \mathrm{~m} .\) (a) Integrate the equation for \(F(r)\) to determine the potential energy \(U(r)\) of the oxygen molecule. (b) Find the equilibrium distance \(r_{0}\) between the two atoms ( \(c\) ) Graph \(F(r)\) and \(U(r)\) between \(0.9 r_{0}\) and \(2.5 r_{0}\)