Question

A uniform disk with a mass of \(800 \mathrm{g}\) and radius \(17.0 \mathrm{cm}\) is rotating on friction less bearings with an angular speed of \(18.0 \mathrm{Hz}\) when Jill drops a 120 -g clod of clay on a point \(8.00 \mathrm{cm}\) from the center of the disk, where it sticks. What is the new angular speed of the disk?

Short answer

The new angular speed of the disk is approximately 16.6 Hz.

Step-by-step solution

Convert Units

First, convert the mass of the disk and the clay from grams to kilograms, and the radius of the disk and position of the clay from centimeters to meters.\[ m_{disk} = 0.8 \text{ kg}, \quad m_{clay} = 0.12 \text{ kg} \]\[ r = 0.17 \text{ m}, \quad d = 0.08 \text{ m} \]

Calculate Initial Moment of Inertia

The moment of inertia for a disk is given by \( I = \frac{1}{2} m r^2 \).\[ I_{disk} = \frac{1}{2} \times 0.8 \times (0.17)^2 = 0.01156 \text{ kg} \cdot \text{m}^2 \]

Calculate Initial Angular Momentum

Angular momentum \( L = I \times \omega \), where \( \omega = 2\pi \times \text{frequency} \).\[ \omega_{initial} = 2\pi \times 18 = 113.097 \text{ rad/s} \]\[ L_{initial} = 0.01156 \times 113.097 = 1.3078 \text{ kg} \cdot \text{m}^2/\text{s} \]

Calculate the Moment of Inertia with Clay

The new moment of inertia includes the clay as a point mass: \( I = I_{disk} + m_{clay} \times d^2 \).\[ I_{new} = 0.01156 + 0.12 \times (0.08)^2 = 0.01254 \text{ kg} \cdot \text{m}^2 \]

Apply Conservation of Angular Momentum

Since there is no external torque, angular momentum is conserved: \( L_{initial} = L_{new} \).\[ 0.01254 \times \omega_{new} = 1.3078 \]\[ \omega_{new} = \frac{1.3078}{0.01254} \approx 104.28 \text{ rad/s} \]

Find New Angular Speed in Hz

Convert the new angular velocity from rad/s to Hz by dividing \( \omega_{new} \) by \( 2\pi \).\[ \text{Frequency}_{new} = \frac{104.28}{2\pi} \approx 16.6 \text{ Hz} \]

Key concepts

Moment of Inertia

Understanding the moment of inertia is crucial when studying rotational motion. This concept is analogous to mass in linear motion. It tells us how difficult it is to change the rotational speed of an object. In this exercise, we looked at a uniform disk, which is a common object used to explain this idea.
The formula for the moment of inertia (\(I\)) of a disk is:\[ I = \frac{1}{2} m r^2 \]where \(m\) is the mass and \(r\) is the radius. For our disk, the initial moment of inertia was calculated with its given mass and radius. When a clod of clay is added, the new moment of inertia also includes this addition as a point mass. This is crucial for correctly applying the principle of conservation of angular momentum later.

Conservation of Angular Momentum

The principle of conservation of angular momentum plays a key role in solving rotational motion problems. It states that if no external torque acts on a system, the total angular momentum remains constant. This principle allows us to predict the changes in rotational speed when mass distribution in a system is altered.
In our case, the system consists of a rotating disk and a clod of clay. Initially, only the disk contributes to the angular momentum. When the clay sticks to the disk, it becomes part of the system. By considering the initial and final moments of inertia, we find the new angular velocity after accounting for the added mass of the clay. This illustrates how changes within a system, without external influences, maintain angular momentum, leading to a new equilibrium state.

Rotational Motion

Rotational motion involves objects moving around an axis. Unlike linear motion, where objects move along a path, rotational motion is circular. Understanding this difference is important.
Here, the disk spins around its center, which acts as the axis of rotation. The angular speed of the disk is initially high due to its low moment of inertia. When clay is added, the rotation slows because the system adjusts to conserve angular momentum. This example underscores how rotational motion requires understanding both speed and the distribution of mass around the axis. It showcases how interconnected these factors are in determining how quickly or slowly an object rotates.

Unit Conversion

Unit conversion is a basic yet crucial step in solving physics problems. It ensures that all calculations are consistent and that results are interpreted correctly.
In the given exercise, the mass of the disk and clay were initially in grams and had to be converted to kilograms. Likewise, the radius and distance measurements needed conversion from centimeters to meters. These unit conversions ensure that when calculating physical quantities like moment of inertia and angular momentum, they are expressed in the standard units of the International System of Units (SI), avoiding errors from inconsistent units.
Converting the final result from angular velocity (in rad/s) to frequency (in Hz) also illustrates the importance of unit conversion in providing meaningful and interpretable results, ensuring that the final output fits the common understanding and usage in rotational dynamics.