Q.75

Expert-verified
Found in: Page 387

### Physics for Scientists and Engineers: A Strategic Approach with Modern Physics

Book edition 4th
Author(s) Randall D. Knight
Pages 1240 pages
ISBN 9780133942651

# In addition to the buoyant force, an object moving in a liquid experiences a linear drag force , direction opposite the motion), where is a constant. For a sphere of radius , the drag constant can be shown to be , where is the viscosity of the liquid. Consider a sphere of radiusand density that is released from rest at the surface of a liquid with density .a. Find an expression in terms of , and the densities for the sphere's terminal speed as it falls through the liquid.b. Solve Newton's second law to find an expression for , the sphere's vertical velocity as a function of time as it falls. Pay careful attention to signs!c. Water at has viscosity Pas. Aluminum has density . If a -mm-diameter aluminum pellet is dropped into water, what is its terminal speed, and how long does it take to reach of its terminal speed?

a.Sphere's terminal speedis.

b.Sphere's vertical velocityis .

c.The time taken for reaching of terminal speed is .

See the step by step solution

## Step 1: Calculation of Sphere's terminal speed (part a)

(a)

The volume of the fluid displaced is the same as the volume of the sphere. The acceleration of the sphere will be zero when the sphere is falling with the terminal speed and the sum of the buoyant force and drag force will be equal to the gravitational force.

The force expression on the sphere is:

Where, is the buoyant force, is the drag force and is the gravitational force.

## Step 2: Calculation of Sphere's vertical velocity (part b)

(b)

The volume of the fluid displaced is the same as the volume of the sphere. The acceleration of the sphere will not be zero when the sphere is not falling with the terminal speed and the sum of the buoyant force and drag force will be equal to the sum of gravitational force and force due to acceleration.

The force expression on the sphere is:

Where, is the buoyant force, is the drag force, is the gravitational force, is the mass of the sphere and is the acceleration due to gravity.

Thus,

The sign of is negative because the sphere is going down. Integrating for the time,

Here, for and for .

## Step 3: Calculation of Time taken (part c)

(c)

The terminal velocity is the velocity of a body is the maximum speed it can attain in the real physical conditions. The acceleration of the body is zero when the body is moving with terminal velocity.

The expression for the terminal velocity is:

Substitute for , for ,for, for and for in the equation.

Solve for the time taken to achieve terminal speed is: