How to Calculate the Angle of Shifting for a Rotating Disc with Removed Portion?

[d_vsuresh]

Homework Statement

From a uniform circular disc of radius 2 centimeter( its centre of mass is at 'O') , a circular portion of radius 1 cm is removed such that shift in centre of mass is maximum.The disc is now rotated about about 'O' perpendicular to the plane through an angle then the magnitude of displacement of new centre of mass is 1/square root of 3 centimeter. Then the angle of shifting is how much ?

Answer given as 120 degree.

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi d_vsuresh!

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[tomcue]

To calculate the angle of shifting for a rotating disc with a removed portion, we can use the formula for the displacement of the center of mass:

Δx = xcm * sin(θ)

Where Δx is the displacement of the center of mass, xcm is the distance from the center of mass to the axis of rotation, and θ is the angle of rotation.

In this problem, we are given that the magnitude of the displacement of the new center of mass is 1/sqrt(3) cm. We also know that the radius of the original disc is 2 cm and the radius of the removed portion is 1 cm. Since the center of mass is shifted by the maximum amount, we can assume that the center of mass of the removed portion is at the edge of the disc, which means that xcm = 1 cm.

Substituting these values into the formula, we get:

1/sqrt(3) = 1 * sin(θ)

Solving for θ, we get θ = 120 degrees.

Therefore, the angle of shifting for the rotating disc with a removed portion is 120 degrees. This makes sense since the removed portion is exactly one-third of the original disc, so rotating it by 120 degrees would result in the maximum displacement of the center of mass.