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Chapter 9: Q.72 (page 230)

A uniform solid bar with mass m and length L rotates with angular velocity v about an axle at one end of the bar. What is the bar鈥檚 kinetic energy?

Short Answer

Expert verified

The total kinetic energy of the bar is $\frac{1}{6} (m 蝇^{2} L^{2}) .$

Step by step solution

01

Content Introduction

The formula to find kinetic energy for the small element of bar is

$d K = \frac{1}{2} v^{2} d m$

here, $d K$ is the kinetic energy of the small element of bar, $v$ is linear velocity, $d m$ is mass element of the same.

02

Content Explanation

Integrate the kinetic energy of each small element of the bar to find the total kinetic energy,

$K = {鈭�}_{0}^{L} d K K = {鈭�}_{0}^{L} \frac{1}{2} v^{2} d m K = {鈭�}_{0}^{L} \frac{1}{2} {(蝇 x)}^{2} d m$

The formula to find the small mass is

$d m = 位 d x$

Substitute

$K = {鈭�}_{0}^{L} \frac{1}{2} {(蝇 x)}^{2} d m K = \frac{1}{2} 蝇^{2} 位 {鈭�}_{0}^{L} {(x)}^{2} d x K = \frac{1}{6} 蝇^{2} 位 L^{3} K = \frac{1}{6} m 蝇^{2} L^{2}$

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Most popular questions from this chapter

A horizontal spring with spring constant $250 N / m$ is compressed by $12 c m$ and then used to launch a $250 g$ box across the floor. The coefficient of kinetic friction between the box and the floor is $0.23$ . What is the box鈥檚 launch speed?

A $45$ g bug is hovering in the air. A gust of wind exerts a force role="math" localid="1648061852575" $\overset{鈫�}{F} = (4.0 \hat{i} - 6.0 \hat{j}) 脳 10^{- 2}$ N on the bug.

a. How much work is done by the wind as the bug undergoes displacement $鈭� \overset{鈫�}{r} = (2.0 \hat{i} - 2.0 \hat{j})$ m?

b. What is the bug鈥檚 speed at the end of this displacement? Assume that the speed is due entirely to the wind.

A $50$ kg sprinter, starting from rest, runs $50$ m in $7.0$ s at constant acceleration.

a. What is the magnitude of the horizontal force acting on the sprinter?

b. What is the sprinter鈥檚 power output at $2.0$ s, $4.0$ s, and $6.0$ s?

Evaluate the dot product $\overset{鈫�}{A} 路 \overset{鈫�}{B}$ if

a). role="math" localid="1647759176820" $\overset{鈫�}{A} = 3 \hat{i} + 4 \hat{j}$ and $\overset{鈫�}{B} = 2 \hat{i} - 6 \hat{j}$

b). $\overset{鈫�}{A} = 3 \hat{i} - 2 \hat{j}$ and $\overset{鈫�}{B} = 6 \hat{i} + 4 \hat{j}$

How much energy is consumed by

(a) a $1.2$ kW hair dryer used for $10$ min and

(b) a $10$ W night light left on for $24$ h?

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