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Chapter 11: Problem 37

A \(31.4-\mathrm{kg}\) wheel with radius \(1.21 \mathrm{~m}\) is rotating at 283 rev/min. It must be brought to a stop in \(14.8 \mathrm{~s}\). Find the required average power. Assume the wheel to be a thin hoop.

Short Answer

Expert verified
The required average power to stop the wheel is 1125 Watts.

Step by step solution

01

Calculate the initial angular velocity

Firstly, we have to convert the rotational speed from rev/min to rad/sec. 1 rev = \(2π\) radian and 1 minute = 60 seconds, so the initial angular velocity ω = \(283 rev/min \times \frac{2π rad}{1 rev} \times \frac{1 min}{60 sec} = 29.6 rad/sec\)
02

Calculate the Moment of Inertia

The moment of inertia I for a hoop of mass m and radius r is \(I = m * r^2\). So, \(I = 31.4 kg * (1.1 m)^2 = 38.4 kg.m^2).
03

Calculate the initial Rotational Kinetic Energy

A body with moment of inertia I and angular velocity ω has the rotational kinetic energy KE = \(0.5 * I * ω^2\). So the initial rotational kinetic energy KE = \(0.5 * 38.4 kg.m^2 * (29.6 rad/sec)^2 = 16625 J\)
04

Calculate the required Power

The power required P to do work W in a time t is \(P = \frac{W}{t}\). In stopping the wheel, we do work equal to its initial rotational kinetic energy. So the required power P = \( \frac{16625 J}{14.8 s} = 1125 W\).

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Most popular questions from this chapter

Calculate the kinetic energies of the following objects moving at the given speeds: ( \(a\) ) a \(110-\mathrm{kg}\) football linebacker running at \(8.1 \mathrm{~m} / \mathrm{s} ;\) (b) a \(4.2-\mathrm{g}\) bullet at \(950 \mathrm{~m} / \mathrm{s} ;\) ( \(c\) ) the aircraft carrier Nimitz, 91,400 tons at \(32.0\) knots.

Suppose that your car averages \(30 \mathrm{mi} / \mathrm{gal}\) of gasoline. \((a)\) How far could you travel on \(1 \mathrm{~kW} \cdot \mathrm{h}\) of energy consumed? \((b)\) If you are driving at \(55 \mathrm{mi} / \mathrm{h}\), at what rate are you expending energy? The heat of combustion of gasoline is \(140 \mathrm{MJ} / \mathrm{gal}\).

Delivery trucks that operate by making use of energy stored in a rotating flywheel have been used in Europe. The trucks are charged by using an electric motor to get the flywheel up to its top speed of \(624 \mathrm{rad} / \mathrm{s}\). One such flywheel is a solid, homogeneous cylinder with a mass of \(512 \mathrm{~kg}\) and a radius of \(97.6 \mathrm{~cm} .\) (a) What is the kinetic energy of the flywheel after charging? ( \(b\) ) If the truck operates with an average power requirement of \(8.13 \mathrm{~kW}\), for how many minutes can it operate between chargings?

A swimmer moves through the water at a speed of \(0.22 \mathrm{~m} / \mathrm{s}\). The drag force opposing this motion is \(110 \mathrm{~N}\). How much power is developed by the swimmer?

A force acts on a \(2.80-\mathrm{kg}\) particle in such a way that the position of the particle as a function of time is given by \(x=\) \((3.0 \mathrm{~m} / \mathrm{s}) t-\left(4.0 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}+\left(1.0 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3} .(a)\) Find the work done by the force during the first \(4.0 \mathrm{~s}\). (b) At what instantaneous rate is the force doing work on the particle at the in\(\operatorname{stan} t=3.0 \mathrm{~s} ?\)
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