/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 A basketball of mass \(610 \math... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 8

A basketball of mass \(610 \mathrm{~g}\) and circumference \(76 \mathrm{~cm}\) is rolling without slipping across a gymnasium floor. Treating the ball as a hollow sphere, what fraction of its total kinetic energy is associated with its rotational motion? a) 0.14 b) 0.19 c) 0.29 d) 0.40 e) 0.67

Short Answer

Expert verified
Answer: The fraction of the total kinetic energy associated with the rotational motion of the basketball is approximately 0.14.

Step by step solution

01

Calculate the moment of inertia of a hollow sphere

The formula for the moment of inertia I of a hollow sphere with mass m and radius R is given by: I = \frac{2}{3}mR^2 We know the mass m = 610 g and the circumference C = 76 cm. We can find the radius R using the relation: R = \frac{C}{2\pi}
02

Find radius of the basketball

Using the circumference, calculate the radius of the basketball: R = \frac{76}{2\pi} = \frac{76}{6.28} \approx 12.1 \mathrm{~cm} Now, we can calculate the moment of inertia I.
03

Calculate the moment of inertia

Substitute the values for mass and radius in the formula for the moment of inertia: I = \frac{2}{3}(610)(12.1)^2 \approx 14926 \mathrm{~g~ cm^2}
04

Determine the relationship between linear and rotational kinetic energies

The total kinetic energy is the sum of the linear kinetic energy (K_l) and the rotational kinetic energy (K_r). The linear kinetic energy is given by: K_l = \frac{1}{2}mv^2 The rotational kinetic energy is given by: K_r = \frac{1}{2}I蠅^2 Since the basketball is rolling without slipping, the relationship between linear velocity v and angular velocity 蠅 is: v = R蠅 Now, we can express the rotational kinetic energy in terms of linear velocity as follows: K_r = \frac{1}{2}I\frac{v^2}{R^2}
05

Calculate the fraction of the total kinetic energy due to rotation

Find the fraction of the total kinetic energy associated with rotational motion: Fraction = \frac{K_r}{K_r + K_l} = \frac{\frac{1}{2}I\frac{v^2}{R^2}}{\frac{1}{2}I\frac{v^2}{R^2} + \frac{1}{2}mv^2} Upon simplification, we get: Fraction = \frac{I}{I + mR^2} Now, substitute the values for the moment of inertia I, mass m, and radius R into the equation: Fraction = \frac{14926}{14926 + 610(12.1)^2} = \frac{14926}{14926+91142} = \frac{14926}{106068} \approx 0.14 The fraction of the total kinetic energy associated with the rotational motion is a) 0.14.

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Most popular questions from this chapter

-10.62 The Earth has an angular speed of \(7.272 \cdot 10^{-5} \mathrm{rad} / \mathrm{s}\) in its rotation. Find the new angular speed if an asteroid \(\left(m=1.00 \cdot 10^{22} \mathrm{~kg}\right)\) hits the Earth while traveling at a speed of \(1.40 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\) (assume the asteroid is a point mass compared to the radius of the Earth) in each of the following cases: a) The asteroid hits the Earth dead center. b) The asteroid hits the Earth nearly tangentially in the direction of Earth's rotation. c) The asteroid hits the Earth nearly tangentially in the direction opposite to Earth's rotation.

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A couple is a set of two forces of equal magnitude and opposite directions, whose lines of action are parallel but not identical. Prove that the net torque of a couple of forces is independent of the pivot point about which the torque is calculated and of the points along their lines of action where the two forces are applied.

A round body of mass \(M\), radius \(R,\) and moment of inertia \(I\) about its center of mass is struck a sharp horizontal blow along a line at height \(h\) above its center (with \(0 \leq h \leq R,\) of course). The body rolls away without slipping immediately after being struck. Calculate the ratio \(I /\left(M R^{2}\right)\) for this body.

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