Question

A uniform circular disc of mass \(50 \mathrm{~kg}\) and radius \(0.4 \mathrm{~m}\) is rotating with an angular velocity of \(10 \mathrm{rad} \mathrm{s}^{-1}\) about its own axis, which is vertical. Two uniform circular rings, each of mass \(6.25 \mathrm{~kg}\) and radius \(0.2 m\), are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in \(\mathrm{rad} \mathrm{s}^{-1}\) ) of the system is ____ .

Short answer

The new angular velocity of the system is \(8.57 \mathrm{rad} \mathrm{s}^{-1}\).

Step-by-step solution

Determine the Initial Moment of Inertia of the Disc

The initial moment of inertia of the disc can be determined using the formula for the moment of inertia of a uniform circular disc which is given by \(I_{disc} = \frac{1}{2} m r^2\) where \(m\) is the mass of the disc and \(r\) is its radius. Substitute the given values of mass \(50 \mathrm{~kg}\) and radius \(0.4 \mathrm{~m}\) to get \(I_{disc} = \frac{1}{2} \times 50 \times (0.4)^2\).

Calculate the Initial Angular Momentum of the System

The initial angular momentum of the system is conserved, and can be calculated by multiplying the initial moment of inertia of the disc and the initial angular velocity. Use the formula \(L_{initial} = I_{disc} \times \omega_{initial}\), where \(\omega_{initial} = 10 \mathrm{rad} \mathrm{s}^{-1}\) is the initial angular velocity of the disc.

Calculate the Moment of Inertia of Each Ring

Use the moment of inertia formula for a uniform circular ring, which is \(I_{ring} = m_{ring} r_{ring}^2\), where \(m_{ring}\) is the mass of the ring and \(r_{ring}\) is its radius. Substitute \(m_{ring} = 6.25 \mathrm{~kg}\) and \(r_{ring} = 0.2 \mathrm{~m}\) to calculate for each ring.

Sum the Moment of Inertia of the Entire System

The new total moment of inertia will be the sum of the initial moment of inertia of the disc and the moments of inertia of the two rings. The new moment of inertia is given by: \(I_{total} = I_{disc} + 2 \times I_{ring}\).

Apply the Conservation of Angular Momentum

Since there are no external torques acting on the system, the angular momentum of the system is conserved. Thus, \(L_{initial} = L_{final}\), where \(L_{final}\) is the angular momentum of the system after the rings were added. Now, calculate the new angular velocity \(\omega_{final}\) using \(L_{final} = I_{total} \times \omega_{final}\).

Solve for the New Angular Velocity

Rearrange the conservation of angular momentum equation to solve for the new angular velocity \(\omega_{final}\). Use the equation \(\omega_{final} = \frac{L_{initial}}{I_{total}}\). Substitute the expressions for \(L_{initial}\) and \(I_{total}\) from the previous steps.

Key concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation rate. It is the rotational equivalent of mass in linear motion. The moment of inertia depends not only on the mass of an object but also on the distribution of that mass relative to the axis of rotation.

For example, for a uniform circular disc, the moment of inertia is calculated using the formula \( I_{disc} = \frac{1}{2} m r^2 \), where \( m \) is the mass and \( r \) is the radius of the disc. This means that the farther the mass is distributed from the center, the higher the moment of inertia will be.

Student improvement advice often highlights the necessity for understanding the pivotal role moment of inertia plays in rotational dynamics. Recognizing that different shapes have distinct formulas for calculating moment of inertia aids in solving a wide variety of physics problems involving rotational motion.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object and the direction of the axis about which the object is rotating. Its SI unit is radians per second (\(rad/s\)). The angular velocity tells us how fast the angle is changing for a rotating object.

An easy way to visualize angular velocity is by imagining a point on a rotating disc: the change in the angle subtended by a line from this point to the center of the disc, with respect to time, is what defines the angular velocity.

In the exercise, the disc's initial angular velocity is given by \(10 \, rad/s\). Knowing the angular velocity is essential in calculating the angular momentum of a system, which remains conserved in the absence of external torques.

Conservation of Angular Momentum

Understanding the Principle

Conservation of angular momentum is a fundamental principle in physics stating that if no external torque acts on a system, the total angular momentum of the system remains constant over time. This principle is analogous to the conservation of linear momentum and is crucial in analyzing isolated systems in rotational motion.

Applying to Rotational Systems

In our example, the conservation principle allows us to relate the initial angular momentum (product of the disc's moment of inertia and its angular velocity) to the final angular momentum (after adding the rings). Since there is no external torque acting on the system, we can express this as \(L_{initial} = L_{final}\), allowing us to solve for the unknown final angular velocity.

Understanding this conservation law is critical for solving a wide variety of problems in rotational dynamics, from simple systems like rotating wheels to complex astronomical phenomena involving celestial bodies.

Rotational Dynamics

Rotational dynamics is the branch of physics concerned with the motion of objects that are rotating or spinning. This topic encompasses analyzing the forces causing rotational motion, the effects of those forces, and how they are related to the object's moment of inertia and angular velocity.

As we've seen in the exercise, adding mass to a rotating system, like placing the rings on the disc, can alter the system's moment of inertia and affect its rotational motion. These changes in motion can be predicted and calculated using the principles of rotational dynamics, such as the conservation of angular momentum.

The study of rotational dynamics not only deepens our understanding of how objects spin and rotate but also forms the basis for many practical applications, including the design of mechanical systems, the functioning of gyroscope-based devices, and the stability of rotating space structures.