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

Key concepts

Moment of Inertia

Moment of inertia is a fundamental concept in rotational dynamics. It is essentially the rotational equivalent of mass in linear motion. It describes how difficult it is to change the rotation of an object around an axis.
For a simple rotating object like a disc, the moment of inertia depends not only on the mass but also on how that mass is distributed relative to the axis of rotation. The formula used to compute the moment of inertia for a disc about its central axis is \(I_{disc} = \frac{1}{2}MR^2\). Here, \(M\) is the mass of the disc and \(R\) is its radius.
When additional masses, such as the two spheres in this exercise, are added to the disc, they increase the total moment of inertia because their mass is distributed further from the axis. This can be calculated for each sphere and then combined to get the total moment of inertia due to the added masses. For spheres attached at the edge, the moment of inertia is \(I_{sphere} = mR^2\), where \(m\) is the mass of a sphere, and again \(R\) is the radius of the disc. Thus, the total for two spheres would be \(I_{spheres} = 2mR^2\).
Understanding moment of inertia helps us predict how an object will behave when it is set into rotation or when the rotational speed is changed, as seen in the case of the disc and spheres.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle or rotation along a circular path. A key characteristic of circular motion is that the object remains at a constant distance from the center of the circle, undergoing what is called centripetal acceleration.
In the context of this problem, we're examining how additional masses, when attached to a rotating disc, influence its motion. While the disc initially rotates with angular velocity \(\omega_1\), upon attaching the masses, the distribution of mass changes and alters its moment of inertia, which in turn affects the angular velocity.
Angular velocity, denoted as \(\omega\), is a measure of how quickly an object rotates around a circular path. Just as with linear velocity, it reflects the speed and direction of rotation but is measured in radians per second. The relationship between angular velocity and moment of inertia is pivotal when additional objects are added to a rotating system, as they can cause the system to conserve angular momentum, altering \(\omega\) if no external forces act upon it.
Circular motion and the conservation principles are integral to understanding how changes in mass distribution impact rotational systems like the disc in the exercise.

Rotational Dynamics

Rotational dynamics deals with the motion of objects that rotate. It encompasses concepts like torque, angular velocity, angular acceleration, and involves the laws of motion as applied to rotating bodies.
A key principle within rotational dynamics is the conservation of angular momentum, which states that if no external torque is acting on a system, the total angular momentum remains constant. Angular momentum \(L\) can be calculated as the product of moment of inertia \(I\) and angular velocity \(\omega\), expressed as \(L = I\omega\).
In our example, adding masses to the spinning disc produces no net external torque, so the system's angular momentum before and after adding the masses should be equal. Initially, the angular momentum \(L_1\) is determined by the disc alone: \(L_1 = I_{disc} \omega_1\). After adding the spheres, the new total moment of inertia \(I_{total}\) includes both the disc and spheres, and the new angular velocity \(\omega_2\) can be found through \(L_2 = I_{total} \omega_2\). Using conservation, \(L_1 = L_2\), allowing for solving for \(\omega_2\), clarifies how the system's dynamics adjust with added mass.
Rotational dynamics explain real-world phenomena like why spinning objects slow down when additional mass is added away from the center, similar to the disc in the problem.