Chapter 6

An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 s, and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 kg.

A ski tow operates on a 15.0\(^\circ\) slope of length 300 m. The rope moves at 12.0 km/h and provides power for 50 riders at one time, with an average mass per rider of 70.0 kg. Estimate the power required to operate the tow.

You are applying a constant horizontal force \(\overrightarrow{F} = (-8.00\mathrm{N})\hat{\imath} + (3.00\mathrm{N})\hat{\jmath}\) to a crate that is sliding on a factory floor. At the instant that the velocity of the crate is \(\overrightarrow{\upsilon} = (3.20\mathrm{m/s})\hat{\imath} + (2.20 \mathrm{m/s})\hat{\jmath}\), what is the instantaneous power supplied by this force?

While hovering, a typical flying insect applies an average force equal to twice its weight during each downward stroke. Take the mass of the insect to be 10 g, and assume the wings move an average downward distance of 1.0 cm during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.

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 \(x = 6.9\) m as you apply a force with \(x\)-component \(F_x = - [20.0 \, \mathrm{N} + (3.0 \, \mathrm{N/m})x]\). How much work does the force you apply do on the cow during this displacement?

A luggage handler pulls a 20.0-kg suitcase up a ramp inclined at 32.0\(^\circ\) above the horizontal by a force \(\overrightarrow{F}\) of magnitude 160 N that acts parallel to the ramp. The coefficient of kinetic friction between the ramp and the incline is \(\mu_k = 0.300\). If the suitcase travels 3.80 m along the ramp, calculate (a) the work done on the suitcase by \(\overrightarrow{F}\); (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 m along the ramp?

While doing a chin-up, a man lifts his body 0.40 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 J of work per kilogram of muscle mass. If the man can just barely do a 0.40-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-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 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.

A 5.00-kg package slides 2.80 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. (e) If the package has a speed of 2.20 m/s at the top of the ramp, what is its speed after it has slid 2.80 m down the ramp?

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; most of the accelerating force is provided by the neck bones. Experiments have shown that these bones will fracture if they absorb more than 8.0 J of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 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 kg (which is about right for a 70-kg person)? Express your answer in m/s and in mi/h. (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 m/s\(^2\) and in \(g\)'s.

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