• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q93P

Expert-verified Found in: Page 293 ### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718 # A wheel of radius ${\mathbf{0}}{\mathbf{.}}{\mathbf{20}}{\mathbf{}}{\mathbit{m}}{\mathbf{}}$ is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is $\mathbf{0}\mathbf{.}\mathbf{050}\mathbit{k}\mathbit{g}\mathbf{.}{\mathbit{m}}^{\mathbf{2}}$ . A mass less cord wrapped around the wheel is attached to a ${\mathbf{2}}{\mathbf{.}}{\mathbf{0}}{\mathbf{}}{\mathbit{k}}{\mathbit{g}}$ block that slides on a horizontal frictionless surface. If a horizontal force of magnitude ${\mathbit{P}}{\mathbf{=}}{\mathbf{}}{\mathbf{3}}{\mathbf{.}}{\mathbf{0}}{\mathbit{N}}$ is applied to the block as shown in Fig.${\mathbf{10}}{\mathbf{-}}{\mathbf{56}}$ , what is the magnitude of the angular acceleration of the wheel? Assume the cord does not slip on the wheel. The magnitude of the angular acceleration of the wheel is $4.62{\text{rad/s}}^{\text{2}}\text{.}$

See the step by step solution

## Step 1: Given

1. Mass of block, $M=2.0\text{kg}$
2. Radius of wheel, $R=0.20\text{m}$
3. Rotational inertia of wheel, $I=0.050{\text{kgm}}^{2}$
4. Magnitude of force, $P=3.0\text{N}$

## Step 2: Determining the concept

Due to applied force$\stackrel{\mathbf{\to }}{\mathbf{P}}$ , the block will accelerate. So, using Newton’s second law, write a net force equation for the block. Similarly, use Newton’s second law for rotating the wheel. Then by using the relation between angular acceleration and linear acceleration, find the magnitude of the angular acceleration of the wheel.

Formulae are as follow:

$P-T=ma\phantom{\rule{0ex}{0ex}}-TR=I\alpha \phantom{\rule{0ex}{0ex}}{a}_{t}=R\alpha \phantom{\rule{0ex}{0ex}}$

where, T, P are forces, m is mass, R is radius, I is moment of inertia, a is acceleration and is angular acceleration.

## Step 3: Determining the magnitude of the angular acceleration of the wheel

Taking rightward motion to be positive for the block and clockwise motion to be negative for the wheel.

Applying Newton’s second law to the block gives,

$P-T=ma ---\left(1\right)$

Similarly, applying Newton’s second law to the wheel gives,

$-TR=I\alpha$

Now,

${a}_{t}=R\alpha$

As tangential acceleration ${a}_{t}$ is opposite to that of block’s acceleration ,

${a}_{t}=-a\phantom{\rule{0ex}{0ex}}-a=R\alpha \phantom{\rule{0ex}{0ex}}\alpha =-\frac{a}{R}$

Therefore,

$-TR=-I\frac{a}{R}\phantom{\rule{0ex}{0ex}}T=\frac{Ia}{{R}^{2}} ---\left(2\right)$

Use equation (2) into (1),

$P-\frac{Ia}{{R}^{2}}=ma\phantom{\rule{0ex}{0ex}}P-\frac{Ia}{{R}^{2}}=-m\alpha \phantom{\rule{0ex}{0ex}}P=-\left(m\alpha R+\frac{I\alpha R}{{R}^{2}}\right)\phantom{\rule{0ex}{0ex}}P=-\alpha R\left(m+\frac{I}{{R}^{2}}\right)\phantom{\rule{0ex}{0ex}}\alpha =-\frac{P}{R\left(m+\frac{I}{{R}^{2}}\right)}\phantom{\rule{0ex}{0ex}}\alpha =-\frac{3.0\text{N}}{\left(0.20\text{m}\right)\left[\left(2.0\text{kg}\right)+\frac{\left(0.0050\text{kg}.{\text{m}}^{2}}{{\left(0.20\text{m}\right)}^{2}}\right]}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\alpha =-4.62rad}{{s}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Hence, the magnitude of the angular acceleration of the wheel is $4.62\text{rad}/{\text{s}}^{2}$

Therefore, the magnitude of the angular acceleration of the wheel can be found using Newton’s second law for the block and for the rotating wheel by using the relation between angular acceleration and linear acceleration.

## Recommended explanations on Physics Textbooks

94% of StudySmarter users get better grades. 