Question

A disk of clay is rotating with angular velocity \(\omega .\) A blob of clay is stuck to the outer rim of the disk, and it has a mass \(\frac{1}{10}\) of that of the disk. If the blob detaches and flies off tangent to the outer rim of the disk, what is the angular velocity of the disk after the blob separates? a) \(\frac{5}{6} \omega\) b) \(\frac{10}{11} \omega\) c) \(\omega\) d) \(\frac{11}{10} \omega\) e) \(\frac{6}{5} \omega\)

Short answer

In this problem, a disk of clay is rotating with an initial angular velocity ω. A blob of clay with mass equal to one-tenth of the disk's mass is stuck to its outer rim, and we need to determine the angular velocity of the disk after the blob separates and flies off tangent to the outer rim. Using conservation of angular momentum, we find that the final angular velocity of the disk remains equal to its initial angular velocity, ω.

Step-by-step solution

Define the problem and identify relevant variables and constants

We want to find the new angular velocity of the disk after the blob separates. Let's denote the initial angular velocity of the disk as ω_initial, the final angular velocity of the disk as ω_final, the mass of the disk (without the blob) as M_disk, and the mass of the blob as M_blob. Given in the problem, the mass of the blob is represented as \(\frac{1}{10}\) the mass of the disk: \(M_{blob} = \frac{1}{10} M_{disk}\).

Conservation of angular momentum

Using conservation of angular momentum (L), we can write the equation: \(L_{initial} = L_{final}.\) Now let's break down the initial and final angular momentum in terms of disk and blob. Initially, both the disk and the blob are rotating together with the same angular velocity ω_initial. Their total combined mass is \(M_{total} = M_{disk} + M_{blob}\). Thus, the initial angular momentum can be written as: \(L_{initial} = I_{initial} \omega_{initial}\), where I_initial is the initial moment of inertia of the system. After the separation, the final angular momentum consists of the angular momentum of the disk and the linear momentum of the blob. Hence, we can write: \(L_{final} = I_{final} \omega_{final} + M_{blob}v_{blob}R\), where I_final is the final moment of inertia of the disk, v_blob is the linear velocity of the blob, and R is the radius of the disk.

Calculate the final angular velocity of the disk

Using the principle of conservation of angular momentum, we have: \(I_{initial} \omega_{initial} = I_{final} \omega_{final} + M_{blob}v_{blob}R.\) Since this is a solid disk, we use the moment of inertia for a solid disk, \(I_{disk} = \frac{1}{2}M_{disk}R^2\). Initially, both the disk and the blob are rotating together. Therefore, the initial moment of inertia and final moment of inertia can be given as: \(I_{initial} = \left( M_{disk} + M_{blob} \right) \frac{1}{2}R^2\) and \(I_{final} = \frac{1}{2}M_{disk}R^2\). Now, we can replace the values in the equation: \(\left( M_{disk} + M_{blob} \right) \frac{1}{2}R^2 \omega_{initial} = \frac{1}{2}M_{disk}R^2 \omega_{final} + M_{blob}v_{blob}R.\) The linear velocity of the blob can be written as \(v_{blob} = R\omega_{initial}\), and the mass of the blob is \(\frac{1}{10} M_{disk}\). Now, substitute these values into the equation: \(\left( M_{disk} + \frac{1}{10} M_{disk} \right) \frac{1}{2}R^2 \omega_{initial} = \frac{1}{2}M_{disk}R^2 \omega_{final} + \frac{1}{10}M_{disk}R\omega_{initial}R.\) Simplify the equation and solve for ω_final: \(\frac{11}{10}M_{disk} \frac{1}{2}R^2 \omega_{initial} = \frac{1}{2}M_{disk}R^2 \omega_{final} + \frac{1}{10}M_{disk}R^2\omega_{initial}\). Divide both sides by \(\frac{1}{2}M_{disk}R^2\): \(\frac{11}{10} \omega_{initial} = \omega_{final} + \frac{1}{10}\omega_{initial}\). Rearranging and solving for ω_final, we get: \(\omega_{final} = \frac{11}{10}\omega_{initial} - \frac{1}{10}\omega_{initial}\). \(\omega_{final} = \frac{10}{10}\omega_{initial}\). Thus, the final angular velocity of the disk is equal to its initial angular velocity, and the answer is: \(\boxed{\text{(c)}\;\omega}\).

Key concepts

Rotational Motion

Rotational motion is a concept that forms the basis of understanding how objects move in a circular path around a pivot point or axis. It's similar to linear motion, where an object travels from one point to another, but instead follows a circular trajectory.

Key parameters of rotational motion include the angle of rotation, measured in radians, and the angular velocity, which tells us how quickly an object rotates. The angular version of Newton's second law also applies, where torque is the equivalent of force, and angular acceleration is derived from changes in angular velocity.

Rotational motion is underpinned by the conservation of angular momentum. This principle states that if no external torque acts on a system, the total angular momentum remains constant over time, even if individual parts of the system change their motion. This was directly applied in the problem where the 'fly-off' of the blob from the rotating disk was expected to alter the angular velocity of the disk but adhered to conservation principles.

Moment of Inertia

The moment of inertia, often represented as 'I', is a measure of an object's resistance to changes in its rotational motion. It's akin to mass in linear dynamics but for rotation. The greater the moment of inertia, the harder it is to change the rotational state of the object.

For different shapes and mass distributions, the moment of inertia has distinct equations. For example, the moment of inertia of a solid disk about its center is given by \( I_{disk} = \frac{1}{2}M_{disk}R^2 \), where \( M_{disk} \) is the mass of the disk and \( R \) its radius.

In the context of the textbook problem, the moment of inertia plays a key role. When the blob, which had its own share of mass—and thus inertia—detaches, the system’s total moment of inertia changes. According to the conservation of angular momentum, this will influence the disk's angular velocity because the moment of inertia and angular velocity are inversely related in a conserved system.

Angular Velocity

Angular velocity is the rate of change of angular displacement and is a vector quantity, which means it has both a magnitude and a direction. It tells us how fast an object rotates or spins about an axis and is measured in radians per second (rad/s).

The angular velocity concept is essential in understanding rotational dynamics. When an ice skater pulls in her arms and reduces her moment of inertia, she spins faster to conserve angular momentum. The same principle is at work in our disk and clay blob problem. Before the blob detaches, both it and the disk are rotating together with a uniform angular velocity. Once the blob is gone, the distribution of mass—and thus the moment of inertia—of the system changes. To conserve angular momentum, the disk's angular velocity must adjust accordingly.

Ultimately, the problem's solution hinges on recognizing that the angular velocity will remain unchanged. This is a counter-intuitive result thanks to the exact proportions of mass leaving the system, balancing the equation perfectly and leaving the final angular velocity equal to the initial velocity, demonstrating a beautiful symmetry to the conservation laws in physics.