Chapter 8.2, Problem 26PP

To determine

The force that should be exerted on the strap if the same angular velocity were desired.

Answer to Problem 26PP

  $9\text{N}$

Explanation of Solution

Given:

The angular acceleration of the solid wheel is $\alpha =3.25{\text{rad/s}}^{\text{2}}$ .

Initial force applied on the wheel, $F_i=4.5\text{N}$

When all the mass of the wheel gets concentrated on the rim, then the moment of Inertia of the wheel is $I=m{r}^2$ .

Formula used:

The torque can be obtained by:

  $\tau =Fr\sin \theta$

Where, F is the applied force, r is the distance of the point of application of force from the axis of rotation, and $\theta$ is the angle between F and r .

And, angular acceleration can be written as,

  $\alpha =\frac{{\tau }_{net}}{I}$

Where, ${\tau }_{net}$ is the net torque on a rotating object, and $I$ is the moment of inertia of the object.

Calculation:

When the mass of wheel is evenly distributed across its surface, then the moment of inertia of the wheel is:

  $I_i=\frac{m{r}^2}{2}$

Where, m is the mass of the wheel and r is the radius of the wheel.

Then, the angular acceleration of the wheel for the initial case can be written as,

  $\begin{array}{c}{\alpha }_i=\frac{{\tau }_i}{I_i}\\ =\frac{F_{i}r\sin 90°}{\left(\frac{m{r}^2}{2}\right)}\end{array}$

Now, when all the mass of the wheel is concentrated on its rim, then its moment of inertia will be $I=m{r}^2$ . The angular acceleration for this case can be written as,

  $\begin{array}{c}{\alpha }_2=\frac{{\tau }_2}{I_2}\\ =\frac{F_{2}r\sin 90°}{m{r}^2}\end{array}$

Now, the same angular velocity and thereby the same angular acceleration is desired. Thus, ${\alpha }_i={\alpha }_2$

Then,

  $\begin{array}{c}\frac{F_{i}r}{\left(\frac{m{r}^2}{2}\right)}=\frac{F_{2}r}{m{r}^2}\\ F_2=2F_i\end{array}$

Using the given values, the new force required is,

  $\begin{array}{c}{F}_2=2F_i\\ =2\left(4.5\text{N}\right)\\ =9\text{N}\end{array}$

Conclusion:

Thus, the force that should be exerted on the strap if the same angular velocity were desired is $9\text{N}$ .