Problem 27 - College Physics

Question

The spin cycles of a washing machine have two angular speeds, 423 rev \(/ \mathrm{min}\) and \(640 \mathrm{rev} / \mathrm{min} .\) The internal diameter of the drum is \(0.470 \mathrm{~m}\). (a) What is the ratio of the maximum radial force on the laundry for the higher angular speed to that for the lower speed? (b) What is the ratio of the maximum tangential speed of the laundry for the higher angular speed to that for the lower speed? (c) Find the laundry's maximum tangential speed and the maximum radial acceleration, in terms of \(g\).

Step-by-Step Solution

Answer

(a) 2.28; (b) 1.51; (c) Maximum tangential speed: 15.74 m/s; Radial acceleration: 107.3g.

Step 1: Convert RPM to Radians per Second

To solve the problem, first convert the given angular speeds from revolutions per minute to radians per second. Use the formula: \( \omega = \frac{2\pi \cdot \, \text{RPM}}{60} \). For the lower speed: \( \omega_1 = \frac{2\pi \cdot 423}{60} = 44.3 \, \text{rad/s} \). For the higher speed: \( \omega_2 = \frac{2\pi \cdot 640}{60} = 67.0 \, \text{rad/s} \).

Step 2: Calculate Maximum Radial Force Ratio

The maximum radial force is proportional to the square of the angular velocity: \( F = m r \omega^2 \). Thus, the ratio of the radial forces is \( \left(\frac{\omega_2}{\omega_1}\right)^2 \). Substituting the angular speeds: \( \left(\frac{67.0}{44.3}\right)^2 = 2.28 \).

Step 3: Calculate Maximum Tangential Speed Ratio

The maximum tangential speed is given by \( v = r \omega \). The ratio of the maximum tangential speeds is simply \( \frac{\omega_2}{\omega_1} \). Therefore, \( \frac{67.0}{44.3} \approx 1.51 \).

Step 4: Find Maximum Tangential Speed and Radial Acceleration

With \( r = \frac{0.470}{2} = 0.235 \text{ m} \), the maximum tangential speed at the higher angular speed is \( v_2 = r \omega_2 = 0.235 \times 67.0 = 15.74 \, \text{m/s} \). For radial acceleration: \( a_r = r \omega^2 \). At higher speed: \( a_{r2} = 0.235 \times 67.0^2 = 1052.5 \text{ m/s}^2 \). In terms of \( g \), \( \frac{1052.5}{9.81} \approx 107.3 \).

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast something spins around an axis. When thinking about objects in circular motion, angular velocity tells us how many revolutions occur per unit of time. It's like the speedometer of rotating objects.

In our example of a washing machine, angular velocity is initially given in revolutions per minute (RPM). To work with it more easily in physics equations, it is often converted to radians per second. This conversion is done using the formula:

• \( \omega = \frac{2\pi \cdot \text{RPM}}{60} \)

This allows us to describe angular velocity in a standard unit that fits well with other measurements like force and acceleration. As seen in the solution steps, converting the angular speeds of 423 RPM and 640 RPM to radians per second results in 44.3 rad/s and 67.0 rad/s, respectively. This change is crucial for calculating radial force and tangential speed.

Radial Force

Radial force, often related to centripetal force, is the force that keeps an object moving in a circular path. It pulls the object toward the center of the rotation. Imagine swinging a ball on a string; the tension in the string is what stops the ball from flying off and is directed inward

• The radial force formula is: \( F = m r \omega^2 \)

where \( m \) is mass, \( r \) is radius, and \( \omega \) is angular velocity. In a washing machine drum, this force acts to keep the laundry against the sides of the drum.

The maximum radial force increases greatly with higher angular velocity, as seen in the problem solution. This is because the force is proportional to the square of angular velocity, so even a small increase in speed results in a much larger increase in force. Thus, the ratio of radial forces at the higher speed to the lower speed ends up being 2.28.

Tangential Speed

Tangential speed can be thought of as the "linear speed" of an object moving along a circular path. Unlike angular velocity, which measures the rate of rotation, tangential speed tells us how fast a point on the circumference of the rotating object is traveling in a straight line.

The formula for finding tangential speed is:

• \( v = r \omega \)

This means tangential speed is directly proportional to both the radius \( r \) of the circular path and the angular velocity \( \omega \).

In the washing machine example, the maximum tangential speed ratio between the higher and lower angular speeds is approximately 1.51, demonstrating that while tangential speed increases with angular velocity, it doesn't increase as sharply as radial force does. The actual maximum tangential speed at the higher angular velocity is calculated as 15.74 m/s, given the drum radius.

Radial Acceleration

Radial acceleration, or centripetal acceleration, refers to the rate of change of tangential velocity as an object travels in a circular path. It acts inward, towards the center of the rotation, similar to radial force.

The formula for radial acceleration is:

• \( a_r = r \omega^2 \)

This shows that radial acceleration depends on both the radius of the circle and the square of the angular velocity.

In our washing machine scenario, higher angular velocity results in significant radial acceleration. The maximum radial acceleration for the higher angular speed is substantial, at 1052.5 m/s². When we express this acceleration in gravitational terms (factor of \( g \), where \( g = 9.81 \) m/s²), it becomes approximately 107.3g. This contextualizes the immense forces experienced in a high-speed spin cycle.