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Chapter 9: Problem 34

A thin spherical shell has a radius of \(1.88 \mathrm{~m}\). An applied torque of \(960 \mathrm{~N} \cdot \mathrm{m}\) imparts an angular acceleration equal to \(6.23 \mathrm{rad} / \mathrm{s}^{2}\) about an axis through the center of the shell. Calculate \((a)\) the rotational inertia of the shell about the axis of rotation and \((b)\) the mass of the shell.

Short Answer

Expert verified
The rotational inertia of the shell is \( I = \tau / \alpha \) and the mass of the shell is \( m = I / (2/3 * r^2) \).

Step by step solution

01

Find the Rotational Inertia

Using the formula for rotational inertia \( I = \tau / \alpha \), insert the provided values: \( I = 960 \mathrm{~N} \cdot \mathrm{m} / 6.23 \mathrm{rad} / \mathrm{s}^{2}\). Calculate the solution to find \( I \).
02

Calculate the Mass

Rearrange the formula \( I = 2/3 m * r^2 \) to solve for \( m \), giving \( m = I / (2/3 * r^2) \). Insert the known values of \( I \) and \( r \) to calculate the mass.

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Most popular questions from this chapter

Let \(\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{b}}=4 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}} .\) Let \(\overrightarrow{\mathbf{c}}=\) \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}\). ( \(a\) ) Find \(\mathbf{c}\), expressed in unit vector notation. ( \(b\) ) Find the angle between \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\).

Two identical blocks, each of mass \(M\), are connected by a light string over a frictionless pulley of radius \(R\) and rotational inertia \(I\) (Fig. 9-55). The string does not slip on the pulley, and it is not known whether or not there is friction between the plane and the sliding block. When this system is released, it is found that the pulley turns through an angle \(\theta\) in time \(t\) and the acceleration of the blocks is constant. (a) What is the angular acceleration of the pulley? ( \(b\) ) What is the acceleration of the two blocks? ( \(c\) ) What are the tensions in the upper and lower sections of the string? All answers are to be expressed in terms of \(M, I, R, \theta, g\), and \(t\).

Vectors \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) lie in the \(x y\) plane. The angle between \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) is \(\phi\), which is less than \(90^{\circ}\). Let \(\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})\). Find the magnitude of \(\overrightarrow{\mathbf{c}}\) and the angle between \(\overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\).

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 parked automobile of mass \(1360 \mathrm{~kg}\) has a wheel base (distance between front and rear axles) of \(305 \mathrm{~cm}\). Its center of gravity is located \(178 \mathrm{~cm}\) behind the front axle. Determine \((a)\) the upward force exerted by the level ground on each of the front wheels (assumed the same) and \((b)\) the upward force exerted by the level ground on each of the rear wheels (assumed the same).
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