Question

A uniform solid disk of mass m=3.00Kg and radius r=0.200m rotates about a fixed axis perpendicular to its face with angular frequency 6.00rad/s. Calculate the magnitude of the angular momentum of the disk when the axis of rotation (a) passes through its centre of mass and (b) passes through a point midway between the centre and the rim.

Short answer

(a)Calculate the magnitude of the angular momentum of the disk when the axis of rotation passes through its center of mass is$\text{L=}0.36kg\cdot m^2/s$.

(b)Calculate the magnitude of the angular momentum of the disk when the axis of rotation passes through a point midway between the center and the rim is $\text{L=}0.54kg\cdot m^2/s$.

Step-by-step solution

Concept

The angular momentum is the amount of rotation of the body, which is the product of its inertia time and angular velocity.

(a) Angular momentum when passes through the center of mass

Mass of solid disk M=3.00 Kg

Radius R=0.20m

The angular velocity, $\omega =6rad/s$

When the axis of rotation passes through its center of mass.

Moment of inertia is,

$\begin{array}{c}{I}_{cm}=\frac{1}{2}M{R}^2\\ =\frac{1}{2}\times 3k\text{Invalid <msup> element}g\times \\ =0.06kg\cdot m^2\end{array}$

Angular momentum is define as below.

$\begin{array}{c}L=I\omega \\ =0.06kg\cdot m^2\times 6rad/s\\ =0.36kg.m^2/s\end{array}$

angular momentum when passes through a point midway

When axis of rotation passes through midway between the center and rim

$I=I+M{D}^2$

Where, D is distance of point from origin.

Substitute known values in the above equation, you get

$\begin{array}{c}I=0.06kg\cdot m^2+M{\left(\frac{R}{2}\right)}^2\\ =0.06kg\cdot m^2+3{\left(\frac{0.2m}{2}\right)}^2\\ =0.09kg\cdot m^2\end{array}$

Define the angular Momentum as follow.

$L=I\omega$

Substitute known values in the above equation.

$\begin{array}{c}L=0.09kg\cdot m^2\times 6rad/s\\ =0.54kg\cdot m^2/s\end{array}$

Result

(a) Calculate the magnitude of the angular momentum of the disk when the axis of rotation passes through its center of mass is $\text{L=}0.36kg\cdot m^2/s$.

(b) Calculate the magnitude of the angular momentum of the disk when the axis of rotation passes through a point midway between the center and the rim is$\text{L=}0.54kg\cdot m^2/s$ .