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Chapter 9: Problem 37

A pulley having a rotational inertia of \(1.14 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) and a radius of \(9.88 \mathrm{~cm}\) is acted on by a force, applied tangentially at its rim, that varies in time as \(F=A t+B t^{2}\), where \(A=0.496 \mathrm{~N} / \mathrm{s}\) and \(B=0.305 \mathrm{~N} / \mathrm{s}^{2} .\) If the pulley was initially at rest, find its angular speed after \(3.60 \mathrm{~s}\).

Short Answer

Expert verified
The final answer will be in rad/s after calculating the definite integral in the last step.

Step by step solution

01

Calculation of Net Torque

The torque on the pulley can be calculated by multiplying the force by the radius. However, given that the force varies with time - \(F = At + Bt^2\), we need to express the torque as a function of time as well. We convert the radius from centimeter to meter for the correct SI unit. So, the net torque, \(\tau\), is: \(\tau = rF = r(At + Bt^2)\), where \(r = 9.88/100 = 0.0988m\).
02

Calculation of Angular Acceleration

The angular acceleration, represented by \(\alpha\), can be calculated by dividing the net torque by the rotational inertia. So, \(\alpha = \frac{(At + Bt^2)r}{I}\), where \(I = 1.14 \times 10^{-3} kgm^2\).
03

Calculate Angular Velocity

The angular speed, represented by \(\omega\), can be calculated by multiplying the angular acceleration by the time, given that the initial angular speed is zero. Here, the time is given as \(t = 3.60s\). So, \(\omega = \int_0^3.60 \alpha dt = \int_0^3.60 \frac{(At + Bt^2)r}{I} dt \).

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Most popular questions from this chapter

(a) Show that a solid cylinder of mass \(M\) and radius \(R\) is equivalent to a thin hoop of mass \(M\) and radius \(R / \sqrt{2}\), for rotation about a central axis. ( \(b\) ) The radial distance from a given axis at which the mass of a body could be concentrated without altering the rotational inertia of the body about that axis is called the radius of gyration. Let \(k\) represent the radius of gyration and show that $$ k=\sqrt{I / M} $$ This gives the radius of the "equivalent hoop" in the general case.

Each of three helicopter rotor blades shown in Fig. 9-42 is \(5.20 \mathrm{~m}\) long and has a mass of \(240 \mathrm{~kg}\). The rotor is rotating at 350 rev/min. What is the rotational inertia of the rotor assembly about the axis of rotation? (Each blade can be considered a thin rod.)

What minimum force \(F\) applied horizontally at the axle of the wheel in Fig. \(9-49\) is necessary to raise the wheel over an obstacle of height \(h\) ? Take \(r\) as the radius of the wheel and \(W\) as its weight.

An automobile traveling \(78.3 \mathrm{~km} / \mathrm{h}\) has tires of \(77.0-\mathrm{cm}\) diameter. (a) What is the angular speed of the tires about the axle? (b) If the car is brought to a stop uniformly in \(28.6\) turns of the tires (no skidding), what is the angular acceleration of the wheels? (c) How far does the car advance during this braking period?

A 160 -lb person is walking across a level bridge and stops three-fourths of the way from one end. The bridge is uniform and weighs 600 lb. What are the values of the vertical forces exerted on each end of the bridge by its supports?
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