/*! 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 53 A ball of mass \(m\) and radius ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 53

A ball of mass \(m\) and radius \(r\) is released in viscous liquid. The value of its terminal velocity is proportional to (A) \((1 / r)\) only (B) \(\mathrm{m} / \mathrm{r}\) (C) \((m / r)^{1 / 2}\) (D) \(m\) only

Short Answer

Expert verified
The correct relationship between the terminal velocity of a ball falling through a viscous liquid, its mass, and radius, is given by the terminal velocity equation: \(v = \frac{mg}{6\pi\eta r}\). The terminal velocity is directly proportional to the mass of the ball and inversely proportional to its radius. Thus, the correct option is (B) \(m / r\).

Step by step solution

01

Identify forces acting on the ball

In this problem, we have two main forces acting on the ball: the gravitational force pulling it downwards and the viscous drag force opposing its motion. The gravitational force is given by: \(F_g = mg\), where \(m\) is the mass of the ball and \(g\) is the acceleration due to gravity. The viscous drag force can be expressed as: \(F_d = 6\pi\eta rv\), where \(\eta\) is the coefficient of viscosity of the liquid, \(r\) is the radius of the ball, and \(v\) is the velocity of the ball.
02

Balance the forces

At terminal velocity, the gravitational force equals the viscous drag force, so we can set the two equations equal to each other: \(mg = 6\pi\eta rv\)
03

Solve for terminal velocity

Now, we need to solve the equation for the terminal velocity, \(v\): \(v = \frac{mg}{6\pi\eta r}\) Notice that the relationship between \(v\) and \(m\) is a direct proportionality, and the relationship between \(v\) and \(r\) is an inverse proportionality. Now we can compare the given options to our result: (A) \(1 / r\): Terminal velocity is inversely proportional to the radius, but this does not include mass. (B) \(m / r\): Terminal velocity is directly proportional to mass and inversely proportional to the radius. (C) \((m / r)^{1 / 2}\): This is not applicable as we have linear relationships established. (D) 饾憵: Terminal velocity is directly proportional to mass, but this does not include the inverse proportionality to the radius.
04

Choose the correct option

Comparing the given options with the terminal velocity equation we derived, the correct answer is: (B) \(m / r\) Terminal velocity is directly proportional to the mass of the ball and inversely proportional to its radius.

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