Problem 55 - Fundamentals of Physics

Question

A horizontal vinyl record of mass \(0.10 \mathrm{~kg}\) and radius \(0.10 \mathrm{~m}\) rotates freely about a vertical axis through its center with an angular speed of \(4.7 \mathrm{rad} / \mathrm{s}\) and a rotational inertia of \(5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}\). Putty of mass \(0.020 \mathrm{~kg}\) drops vertically onto the record from above and sticks to the edge of the record. What is the angular speed of the record immediately afterwards?

Step-by-Step Solution

Answer

The angular speed of the record after the putty sticks is approximately 4.5 rad/s.

Step 1: Identify Key Variables

From the problem, the key variables are: \( m = 0.10 \, \text{kg} \) (mass of the record), \( R = 0.10 \, \text{m} \) (radius of the record), \( \omega_i = 4.7 \, \text{rad/s} \) (initial angular speed), \( I_i = 5.0 \times 10^{-4} \, \text{kg} \cdot \text{m}^2 \) (initial rotational inertia), and \( m_p = 0.020 \, \text{kg} \) (mass of putty).

Step 2: Use Conservation of Angular Momentum

Angular momentum is conserved in this scenario because there are no external torques. The formula for angular momentum \( L \) is \( L = I \cdot \omega \). The initial angular momentum \( L_i \) is \( I_i \cdot \omega_i \).

Step 3: Calculate Final Inertia

After the putty sticks to the record, the total moment of inertia \( I_f \) is the sum of the initial moment of inertia and the contribution from the putty at the edge. Calculate using \( I_f = I_i + m_p \cdot R^2 \). Substitute in: \( I_f = 5.0 \times 10^{-4} + 0.020 \times (0.10)^2 \).

Step 4: Substitute to Find Final Angular Speed

Using \( L_i = L_f \) with \( L_f = I_f \cdot \omega_f \) and \( \omega_f \) is the final angular speed, solve for \( \omega_f \). Set \( I_i \cdot \omega_i = I_f \cdot \omega_f \) and calculate for \( \omega_f \).

Step 5: Perform Calculations

Calculate \( I_f = 5.0 \times 10^{-4} + 0.020 \times 0.01 = 5.2 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}\). Then, substitute back into the equation: \( (5.0 \times 10^{-4}) \cdot (4.7) = (5.2 \times 10^{-4}) \cdot \omega_f \). Solve for \( \omega_f \): \( \omega_f = \frac{(5.0 \times 4.7)}{5.2} \).

Step 6: Final Calculation

Perform the final calculation: \( \omega_f \approx 4.5 \mathrm{~rad/s} \).

Key Concepts

Rotational Inertia

Rotational inertia, commonly referred to as the moment of inertia, is an essential concept when dealing with rotating bodies. It's a measure of how much an object resists changes to its rotation. Just like how mass is a measure of an object's resistance to linear motion, rotational inertia plays a similar role in rotational motion.
In a rotating system, rotational inertia depends on two main factors:

• The mass of the object.
• How the mass is distributed relative to the axis of rotation.

For example, in the exercise with the vinyl record, the moment of inertia is given as \(5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}\). This value represents how hard it is to change the angular velocity of the record without any putty on it. Generally, the more mass located further from the axis, the larger the inertia, which means more effort is required to change the spin of the object. Rotational inertia is a foundational concept in understanding how bodies behave when they rotate.

Angular Speed

Angular speed, represented by the symbol \(\omega\), measures how fast an object is rotating. It is the rate of change of the angle through which the object rotates and is measured in radians per second \(\mathrm{rad/s}\).
The exercise provides an initial angular speed \(\omega_i\) of \(4.7 \mathrm{rad/s}\). This is the speed at which the record is spinning before the putty impacts and alters its rotation. Angular speed is crucial for analyzing rotational dynamics because it illustrates how rapid the motion is around an axis.
When the putty lands on the record, the angular speed decreases due to the increased rotational inertia. By the conservation of angular momentum, the speed at which the record spins changes, leading us to calculate a new speed. After the putty sticks to the edge, using the conservation of angular momentum, we determine the final angular speed \(\omega_f\) to be approximately \(4.5 \mathrm{~rad/s}\).

Moment of Inertia

The moment of inertia is a quantity that expresses an object's inclination to resist angular acceleration. It plays a pivotal role in rotational motion, similar to how mass affects linear motion.
For an object like our vinyl record in the exercise, the moment of inertia changes due to the addition of the putty. Initially, its moment of inertia is \(5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}\). When the putty, with its mass of \(0.020 \mathrm{~kg}\), sticks to the record's edge (\(0.10 \mathrm{~m}\) from the center), it adds to the system’s moment of inertia. The new moment of inertia \(I_f\) can be calculated using the relation:

• Initial moment of inertia of the record
• Additional moment of inertia due to the putty: \(m_p \times R^2\)

So, \(I_f = I_i + m_p \cdot R^2 = 5.0 \times 10^{-4} + 0.020 \times 0.01 = 5.2 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}\). This increased moment of inertia influences the final angular speed, as explained through the conservation of angular momentum in our exercise.