Question

A uniform disk of radius 0.453 m and unknown mass is constrained to rotate about a perpendicular axis through its center. A ring with the same mass as the disk is attached around the disk's rim. A tangential force of 0.221 N applied at the rim causes an angular acceleration of 0.111 rad/s2. Find the mass of the disk.

Answer 1

Answer:

The mass of the disk is 0.7179 kg.

Explanation:

Given that,

Radius = 0.453 m

Tangential force = 0.221 N

Angular acceleration = 0.111 rad/s²

We need to calculate the moment of inertia of both the disk and ring

Using formula of moment of inertia

For disk,

$I=\dfrac{1}{2}mr^2$

For ring,

$I=mr^2$

The net moment of inertia

$I=\dfrac{1}{2}mr^2+mr^2$

$I=1.5mr^2$

We need to calculate the torque

Using formula of  torque

$\tau=I\alpha$

$Fd=1.5mr^2\times\alpha$

Put the value into the formula

$0.221\times0.111=1.5\times m\times(0.453)^2\times0.111$

$m=\dfrac{0.221\times0.111}{1.5\times(0.453)^2\times0.111}$

$m=0.7179\ kg$

Hence,  The mass of the disk is 0.7179 kg.

Answer 2

Moment of inertia is a quantitative measure of a body's rotational inertia . Mass is an important component in moment of inertia formulas. The mass of the disk will be 0.7179 Kg.

What is the moment of inertia?

Moment of inertia is a quantitative measure of a body's rotational inertia the body's resistance to having its speed of rotation about an axis is changed by the application of torque.

The given data in the problem is;

r is the radius = 0.453 m

F is the tangential force = 0.221 N

α is the angular acceleration = 0.111 rad/s²

The moment of inertia of the disk

$\rm I = \frac{1}{2}mr^2$

The moment of inertia of the ring

$\rm I = mr^2$

The total moment of inertia will be;

$\rm I = \frac{1}{2}mr^2+\rm mr^2 \\\\ \rm I = 1.5 mr^2$

The torque is found by;

$\rm \tau =i\alpha \\\\ Fd= 1.5 mr^2 \times \alpha \\\\ 0.221 \times 0.111 = 1.5 m(0.453)^2 \times 0.1111 \\\\ \rm m= \frac{0.221 \times 0.111}{1.5 \times (0.453)^2 \times 0.111} \\\\ \rm m= 0.7179\ Kg$

Hence the mass of the disk will be 0.7179 Kg.

To learn more about the moment of inertia refer to the link;

https://brainly.com/question/6953943