Problem 69 - College Physics

Question

Circular disks are used in automobile clutches and transmissions. When a rotating disk couples to a stationary one through frictional force, the energy from the rotating disk can transfer to the stationary one. (a) Is the angular speed of the coupled disks (1) greater than, (2) less than, or (3) the same as the angular speed of the original rotating disk? Why? (b) If a disk rotating at 800 rpm couples to a stationary disk with three times the moment of inertia, what is the angular speed of the combination?

Step-by-Step Solution

Answer

(a) Less; (b) 200 rpm.

Step 1: Concept Understanding

When two disks couple by friction, they form a single system, conserving angular momentum if no external torques act on the system. This means that the total angular momentum before coupling (only rotating disk) equals the total angular momentum after coupling (combined disks).

Step 2: Apply Conservation Law

Let \( I_1 \) be the moment of inertia of the rotating disk and \( \omega_1 \) its initial angular speed. The stationary disk has a moment of inertia \( I_2 = 3I_1 \) and an initial angular speed \( \omega_2 = 0 \). The angular momentum conservation equation is: \[ I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2) \omega_f \] where \( \omega_f \) is the angular speed of the combined system.

Step 3: Simplify the Equation

Substitute the known values into the equation: \[ I_1 \cdot 800 + 3I_1 \cdot 0 = (I_1 + 3I_1) \cdot \omega_f \] This simplifies to: \[ 800I_1 = 4I_1 \cdot \omega_f \]

Step 4: Solve for Final Angular Speed

Divide both sides of the equation by \( 4I_1 \): \[ \omega_f = \frac{800}{4} = 200 \] Thus, the final angular speed \( \omega_f \) of the combination is 200 rpm.

Step 5: Compare Speeds

Compare the initial angular speed and the final angular speed: Initially, \( \omega_1 = 800 \) rpm. Finally, \( \omega_f = 200 \) rpm. Thus, the angular speed of the coupled disks is less than the angular speed of the original rotating disk.

Key Concepts

Angular Speed

Angular speed refers to how fast an object rotates or revolves relative to another point, typically the center of a circle. It is measured in revolutions per minute (rpm) or radians per second (rad/s). When dealing with problems like coupling rotating disks, understanding angular speed is crucial.

In the given exercise, one disk is stationary while the other is initially rotating at 800 rpm. Angular speed after the disks couple is determined by conserving momentum. Since momentum is transferred in the process, the combined disks rotate at a different rate. Here, that rate is lower, at 200 rpm. This happens because the combined rotational inertia of both disks dilutes the initial energy input from the rotating disk. Understanding the change in angular speed helps us determine how energy gets redistributed among components of the system. Thus, recognizing these nuances aids in predicting behavior in mechanical systems like automotive clutches.

Moment of Inertia

Moment of inertia is a key concept in rotational dynamics. It measures an object’s resistance to changes in its rotation. Think of it as rotational mass. Just like mass in linear motion, if the moment of inertia is high, it’s harder to change the object's rotation. In the problem, two circles are disks. The stationary one has a moment of inertia three times that of the rotating one. This imbalance affects how energy is shared between them. The concept here is critical: when objects rotate together, their inertias add up, influencing the system's rotational behavior. Understanding moment of inertia clarifies why the final angular speed is decreased. As the combined inertia becomes greater, the system rotates slower. This concept shows up in many areas, like flywheels and vehicle dynamics, where momentum and rotational control are essential.

Rotational Dynamics

Rotational dynamics involves the study of motions and forces in rotating systems. Applying principles such as torque and angular momentum gives insights into how objects behave when they rotate.

In the exercise, the frictional coupling of the disks is a great example of dynamics in action. Angular momentum, which is conserved in this case, is a product of moment of inertia and angular speed. When the disks couple, they form a new system, and without external torques, the total angular momentum remains constant. This means that even though the speed reduces, the movement of the disks continues in a predictable manner. These principles have practical applications in scenarios involving gears, pulleys, and wheels. They are foundational concepts crucial for anyone keen on understanding the mechanics of rotating systems.