Question

Initial angular velocity of a circular disc of mass \(\mathrm{M}\) is \(\omega_{1}\) Then two spheres of mass \(\mathrm{m}\) are attached gently two diametrically opposite points on the edge of the disc what is the final angular velocity of the disc? \(\\{\mathrm{A}\\}[(\mathrm{M}+\mathrm{m}) / \mathrm{M}] \omega_{1}\) \(\\{\mathrm{B}\\}[(\mathrm{M}+4 \mathrm{~m}) / \mathrm{M}] \omega_{1}\) \(\\{\mathrm{C}\\}[\mathrm{M} /(\mathrm{M}+4 \mathrm{~m})] \omega_{1}\) \\{D\\} \([\mathrm{M} /(\mathrm{M}+2 \mathrm{~m})] \omega_{1}\)

Short answer

The short answer is: \(\\{C\\} [\mathrm{M} /(\mathrm{M}+4 \mathrm{~m})] \omega_{1}\)

Step-by-step solution

Moment of Inertia of the Circular Disc

To find the moment of inertia of the circular disc, use the following formula: \[I_{disc} = \frac{1}{2}MR^2\] Where I is the moment of inertia, M is the mass of the disc, and R is the radius of the disc.

Moment of Inertia of the Attached Spheres

Since the two spheres are attached at the edge of the disc, their moment of inertia can be calculated using the following formula: \[I_{sphere} = mR^2\] Where I is the moment of inertia, m is the mass of a sphere, and R is the radius of the disc. Because there are two spheres, the total moment of inertia of the spheres is: \[I_{spheres} = 2mR^2\]

Calculate the Initial Angular Momentum

Using the moment of inertia of the circular disc, we can calculate the initial angular momentum, L1: \[L_{1} = I_{disc} \omega_{1} = \frac{1}{2}MR^2 \omega_{1}\]

Calculate the Final Angular Momentum

The final angular momentum, L2, will be the sum of the initial angular momentum of the disc and the angular momentum of the attached spheres: \[L_{2} = (I_{disc} + I_{spheres}) \omega_{2} = (\frac{1}{2}MR^2 + 2mR^2) \omega_{2}\]

Apply the Law of Conservation of Angular Momentum

Since L1 = L2, we have: \[\frac{1}{2}MR^2 \omega_{1} = (\frac{1}{2}MR^2 + 2mR^2) \omega_{2}\] Now, we can solve for the final angular velocity, ω2: \[\omega_{2} = \frac{\frac{1}{2}MR^2 \omega_{1}}{\frac{1}{2}MR^2 + 2mR^2}\]

Simplify the Expression for ω2

We can simplify the expression for ω2: \[\omega_{2} = \frac{MR^2 \omega_{1}}{MR^2 + 4mR^2} = \frac{M \omega_{1}}{M+4m}\] Comparing this result with the multiple-choice options, the correct answer is: \(\\{C\\} [\mathrm{M} /(\mathrm{M}+4 \mathrm{~m})] \omega_{1}\)