Chapter 8, Problem 87A

(a)

To determine

The angular acceleration of the door.

Answer to Problem 87A

The angular acceleration of the wooden door is $\alpha =\left(\frac{9.80}{l}\right){\text{rad/s}}^2$ .

Explanation of Solution

Given:

The mass of the wooden door is denoted by $m$ .

The length of the wooden door is denoted by $l$ .

Consider the acceleration due to gravity is $g=9.80{\text{m/s}}^2$ .

Formula used:

The expression for the angular acceleration of the object is,

  $\alpha =\frac{a}{r}$

Here, $a$ is the linear acceleration and $r$ is the radius of rotation.

Calculation:

Consider the wooden door of length $l$ is held at both the ends by person A and D. Ifperson D leaves his hand, the door will rotate such that person A acts as the axis of rotation.

The acceleration with which the wooden door falls is same as the acceleration due to gravity $\left(g\right)$ .

The radius of axis of rotation is same as the length of the wooden door.

The angular acceleration of the wooden door is,

  $\begin{array}{l}\alpha =\frac{a}{r}\\ \alpha =\frac{g}{l}\\ \alpha =\left(\frac{9.80{\text{m/s}}^2}{l\text{m}}\right)\\ \alpha =\left(\frac{9.80}{l}\right){\text{rad/s}}^2\end{array}$

Conclusion:

Therefore, the angular acceleration of the wooden door is $\alpha =\left(\frac{9.80}{l}\right){\text{rad/s}}^2$ .

(b)

To determine

To identify:Whether the acceleration of the door is constant or not.

Answer to Problem 87A

Yes, the acceleration of the door is constant.

Explanation of Solution

Given:

The mass of the wooden door is denoted by $m$ .

The length of the wooden door is denoted by $l$ .

Consider the acceleration due to gravity is $g=9.80{\text{m/s}}^2$ .

Formula used:

The expression for the angular acceleration of the object is,

  $\alpha =\frac{a}{r}$

Here, $a$ is the linear acceleration and $r$ is the radius of rotation.

Calculation:

From part (a), the angular acceleration of the wooden door is,

  $\begin{array}{l}\alpha =\frac{g}{l}\\ \alpha =\left(\frac{9.80}{l}\right){\text{rad/s}}^2\end{array}$

It is observed that the value of angular acceleration of the door depends upon the acceleration due to gravity of Earth $\left(g\right)$ and length of the wooden-door $\left(l\right)$ .

The value of acceleration due to gravity of Earth $\left(g\right)$ is a constant.

The value of acceleration length of the wooden-door $\left(l\right)$ is a constant.

Hence, the angular acceleration of the door is a constant.

Conclusion:

Thus, the angular acceleration of the door is a constant.