91Ó°ÊÓ

The number of kilocalories per day required by a person resting under standard conditions is called the basal metabolic rate (BMR). (a) To generate 1 kcal, Jermaine's body needs approximately 0.010 mol of oxygen. If Jermaine's net intake of oxygen through breathing is 0.015 mol/min while he is resting, what is his BMR in kcal/day? (b) If Jermaine fasts for 24 h, how many pounds of fat does he lose? Assume that only fat is consumed. Each gram of fat consumed generates \(9.3 \mathrm{kcal}\)

Jane is running from the ivory hunters in the jungle. Cheetah throws a 7.0 -m-long vine toward her. Jane leaps onto the vine with a speed of $4.0 \mathrm{m} / \mathrm{s} .$ When she catches the vine, it makes an angle of \(20^{\circ}\) with respect to the vertical. (a) When Jane is at her lowest point, she has moved downward a distance \(h\) from the height where she originally caught the vine. Show that \(h\) is given by \(h=L-L \cos 20^{\circ},\) where \(L\) is the length of the vine. (b) How fast is Jane moving when she is at the lowest point in her swing? (c) How high can Jane swing above the lowest point in her swing?

The escape speed from Earth is \(11.2 \mathrm{km} / \mathrm{s},\) but that is only the minimum speed needed to escape Earth's gravitational pull; it does not give the object enough energy to leave the solar system. What is the minimum speed for an object near the Earth's surface so that the object escapes both the Earth's and the Sun's gravitational pulls? Ignore drag due to the atmosphere and the gravitational forces due to the Moon and the other planets. Also ignore the rotation and the orbital motion of the Earth.

A skier starts from rest at the top of a frictionless slope of ice in the shape of a hemispherical dome with radius \(R\) and slides down the slope. At a certain height \(h,\) the normal force becomes zero and the skier leaves the surface of the ice. What is \(h\) in terms of \(R ?\)

Two springs with spring constants \(k_{1}\) and \(k_{2}\) are connected in series. (a) What is the effective spring constant of the combination? (b) If a hanging object attached to the combination is displaced by \(4.0 \mathrm{cm}\) from the relaxed position, what is the potential energy stored in the spring for \(k_{1}=5.0 \mathrm{N} / \mathrm{cm}\) and $k_{2}=3.0 \mathrm{N} / \mathrm{cm} ?$ [See Problem \(83(\mathrm{a}) .]\)

Two springs with spring constants \(k_{1}\) and \(k_{2}\) are connected in parallel. (a) What is the effective spring constant of the combination? (b) If a hanging object attached to the combination is displaced by \(2.0 \mathrm{cm}\) from the relaxed position, what is the potential energy stored in the spring for \(k_{1}=5.0 \mathrm{N} / \mathrm{cm}\) and $k_{2}=3.0 \mathrm{N} / \mathrm{cm} ?$ [See Problem \(83(\mathrm{b}) .]\)

Human feet and legs store elastic energy when walking or running. They are not nearly as efficient at doing so as kangaroo legs, but the effect is significant nonetheless. If not for the storage of elastic energy, a \(70-\mathrm{kg}\) man running at \(4 \mathrm{m} / \mathrm{s}\) would lose about \(100 \mathrm{J}\) of mechanical energy each time he sets down a foot. Some of this energy is stored as elastic energy in the Achilles tendon and in the arch of the foot; the elastic energy is then converted back into the kinetic and gravitational potential energy of the leg, reducing the expenditure of metabolic energy. If the maximum tension in the Achilles tendon when the foot is set down is \(4.7 \mathrm{kN}\) and the tendon's spring constant is $350 \mathrm{kN} / \mathrm{m},$ calculate how far the tendon stretches and how much elastic energy is stored in it.

A wind turbine converts some of the kinetic energy of the wind into electric energy. Suppose that the blades of a small wind turbine have length $L=4.0 \mathrm{m}$ (a) When a \(10 \mathrm{m} / \mathrm{s}(22 \mathrm{mi} / \mathrm{h})\) wind blows head-on, what volume of air (in \(\mathrm{m}^{3}\) ) passes through the circular area swept out by the blades in \(1.0 \mathrm{s} ?\) (b) What is the mass of this much air? Each cubic meter of air has a mass of 1.2 \(\mathrm{kg}\). (c) What is the translational kinetic energy of this mass of air? (d) If the turbine can convert \(40 \%\) of this kinetic energy into electric energy, what is its electric power output? (e) What happens to the power output if the wind speed decreases to \(\frac{1}{2}\) of its initial value? What can you conclude about electric power production by wind turbines?

Use dimensional analysis to show that the electric power output of a wind turbine is proportional to the cube of the wind speed. The relevant quantities on which the power can depend are the length \(L\) of the rotor blades, the density \(\rho\) of air (SI units \(\mathrm{kg} / \mathrm{m}^{3}\) ), and the wind speed \(v\).