Problem 29 - College Physics

Question

\(\bullet\) The spin cycles of a washing machine have two angular speeds, 423 \(\mathrm{rev} / \mathrm{min}\) 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) Max tangential speed = 15.74 m/s; Max radial acceleration = 107.58g

Step 1: Understanding Angular Speed

The washing machine spins at two speeds, 423 revolutions per minute (rpm) and 640 rpm. We need to find the ratios of various forces and speeds using these given speeds.

Step 2: Find the Angular Speeds in Radians per Second

Convert the angular speeds from revolutions per minute to radians per second. Use the conversion factor: 1 revolution = \(2\pi\) radians and 1 minute = 60 seconds. For 423 rpm: \(\text{Angular Speed 1} = \frac{423 \times 2\pi}{60} = 44.34 \, \text{rad/s} \)For 640 rpm: \(\text{Angular Speed 2} = \frac{640 \times 2\pi}{60} = 67.02 \, \text{rad/s} \)

Step 3: Calculate Radial Force Ratio

The radial force for an object moving in a circle is given by \(F_r = m \cdot \omega^2 \cdot r\), where \(m\) is the mass, \(\omega\) is angular speed, and \(r\) is the radius.Since mass and radius are constant, the ratio of the radial forces is:\( \text{Force Ratio} = \left(\frac{\omega_2}{\omega_1}\right)^2 = \left(\frac{67.02}{44.34}\right)^2 \approx 2.28 \)

Step 4: Calculate Tangential Speed Ratio

Tangential speed \(v\) is given by \(v = \omega \cdot r\). So the ratio of tangential speeds is:\( \text{Tangential Speed Ratio} = \frac{\omega_2}{\omega_1} = \frac{67.02}{44.34} \approx 1.51 \)

Step 5: Calculate Maximum Tangential Speed

Using \(r = \frac{0.470}{2} = 0.235\) m (since the radius is half the diameter), For the higher speed:\(\text{Max Tangential Speed} = \omega_2 \cdot r = 67.02 \cdot 0.235 = 15.74 \, \text{m/s} \)

Step 6: Calculate Maximum Radial Acceleration

Radial acceleration \(a_r\) is given by \(a_r = \omega^2 \cdot r\).For the higher speed:\(a_r = 67.02^2 \cdot 0.235 = 1055.37 \, \text{m/s}^2\)Express in terms of \(g\), where \(g\) is the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)):\(a_r \approx \frac{1055.37}{9.81} \approx 107.58 \, g\)

Key Concepts

Radial Force

Radial force plays a crucial role in understanding how objects behave when they move in a circle. When objects, like laundry in a washing machine drum, spin in a circular path, a force acts perpendicular to their velocity, pulling them towards the center of the circle. This force is known as radial or centripetal force.

The formula to calculate radial force is:

• \( F_r = m \cdot \omega^2 \cdot r \)

where:

• \( F_r \) is the radial force,
• \( m \) is the mass of the object,
• \( \omega \) is the angular speed in radians per second, and
• \( r \) is the radius of the circular path.

For the washing machine problem, the radial force changes with speed. As we increase the angular speed from 423 to 640 revolutions per minute, the centrifugal force increases dramatically. Using the given formula, we found that the maximum radial force for the higher speed is approximately 2.28 times greater than at the lower speed.

This increase is because radial force is proportional to the square of the angular speed. When you double the speed, you quadruple the force exerted inwards.

Tangential Speed

Tangential speed is another important concept when dealing with circular motion. Unlike radial force which acts towards the center, tangential speed refers to how fast the object moves along the circular path. It is the linear speed of any point rotating at radius \( r \).

For an object in circular motion, tangential speed \( v \) is calculated as:

• \( v = \omega \cdot r \)

where:

• \( v \) is the tangential speed,
• \( \omega \) is the angular speed, and
• \( r \) is the radius of the circular path.

In the context of the washing machine exercise, we calculated the tangential speed ratios for speeds of 423 rpm and 640 rpm, resulting in a ratio of approximately 1.51. This means that the tangential speed at the higher speed is about 1.51 times that at the lower speed.

Understanding tangential speed helps in visualizing how quickly a point on a spinning drum is moving. It is especially critical in applications like washing machines, where high tangential speeds help clean clothes by forcing water out more efficiently.

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, is the rate at which an object's velocity changes direction towards the center of its circular path. This happens not because the speed itself changes, but because the direction of the velocity changes continuously during circular motion.

We calculate radial acceleration using the formula:

• \( a_r = \omega^2 \cdot r \)

where:

• \( a_r \) is the radial acceleration,
• \( \omega \) is the angular speed, and
• \( r \) is the radius.

In the solution, for a drum with a diameter of 0.470 m, the maximum radial acceleration during the higher spin cycle (640 rpm) was very high, approximately 1055.37 \( \text{m/s}^2 \). When expressed in terms of gravity ('\( g \) where \( g = 9.81 \text{m/s}^2 \)'), this equated to about 107.58 \( g \).

This high value indicates how powerful circular motions are regarding force and acceleration. It explains why it's essential for machines and rotating systems to be engineered to endure these high forces without suffering damage.