Problem 31 - University Physics with Modern Physics

Question

The spin cycles of a washing machine have two angular speeds, 423 rev/min and 640 rev/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 maxi- mum radial acceleration, in terms of \(g .\)

Step-by-Step Solution

Answer

(a) 2.28, (b) 1.51, (c) 15.74 m/s; 107.42g

Step 1: Convert angular speeds to radians per second

The angular speeds given are in revolutions per minute (rev/min). To convert to radians per second, use the formula: \( \omega = \frac{2\pi n}{60} \), where \( n \) is the angular speed in rev/min. Calculate \( \omega_1 \) for 423 rev/min and \( \omega_2 \) for 640 rev/min. \[ \omega_1 = \frac{2\pi \times 423}{60} = 44.35 \, \text{rad/s} \] \[ \omega_2 = \frac{2\pi \times 640}{60} = 67.02 \, \text{rad/s} \]

Step 2: Calculate radial forces ratio

The radial force is proportional to the square of the angular speed, given by \( F_r = m r \omega^2 \). Since we require the ratio, the mass \(m\) and radius \(r\) are constant and thus cancel out: \[ \text{Force ratio} = \left(\frac{\omega_2}{\omega_1}\right)^2 = \left(\frac{67.02}{44.35}\right)^2 = 2.28 \]

Step 3: Calculate tangential speeds ratio

The tangential speed \( v \) is connected to angular speed by \( v = r \omega \). Take the ratio of these speeds at higher and lower angular speeds: \[ \text{Tangential Speed Ratio} = \frac{r \omega_2}{r \omega_1} = \frac{\omega_2}{\omega_1} = \frac{67.02}{44.35} = 1.51 \]

Step 4: Find the maximum tangential speed

Using the given radius \(r = 0.235\) m (half of the diameter), calculate the maximum tangential speed at \( \omega_2 \): \[ v_\text{max} = r \omega_2 = 0.235 \times 67.02 = 15.74 \, \text{m/s} \]

Step 5: Find the maximum radial acceleration

The radial acceleration \( a_r \) at maximum speed is provided by \( a_r = r \omega^2 \), and is often expressed in terms of \( g \). Calculate for the higher speed: \[ a_r = 0.235 \times (67.02)^2 = 1053.41 \, \text{m/s}^2 \] To express this in terms of \( g = 9.81 \, \text{m/s}^2 \): \[ \text{Radial Acceleration in terms of } g = \frac{1053.41}{9.81} = 107.42g \]

Key Concepts

Radial Force

Radial force is essential when it comes to understanding circular motion. It is the force acting towards the center of the circular path an object follows. This force is critical in maintaining the circular motion, preventing objects from flying off the tangent. In mathematical terms, radial force \( F_r \) is given by the equation:

• \( F_r = m r \omega^2 \)

where:

• \( m \) is the mass of the object.
• \( r \) is the radius of the circular path.
• \( \omega \) is the angular speed in rad/s.

When comparing radial forces at different angular speeds, as in the washing machine example, you can see the relationship is quadratic. By squaring the ratio of angular speeds, you determine how much stronger the radial force becomes with increased speed.

• For example, increasing from 44.35 to 67.02 rad/s resulted in a radial force ratio of 2.28.

This means the radial force at the higher speed is more than double that at the lower speed, highlighting how essential it is to manage these forces effectively to safeguard rotating systems.

Tangential Speed

While radial force focuses on the center of the circle, tangential speed pertains to how fast an object travels along the circle's edge. This concept represents the linear speed of any point on a rotating object, assuming no slippage. Tangential speed \( v \) is calculated by:

• \( v = r \omega \)

Here, \( r \) is the radius, and \( \omega \) is the angular speed. When you change the angular speed, the tangential speed changes proportionally. Thus, the speed increases directly with angular speed.

• This is evident in the washing machine example, where the ratio of tangential speeds at different angular velocities was 1.51.

It indicates the speed along the washing machine drum's surface increased by 51% as the angular speed rose from 423 rev/min to 640 rev/min. Understanding tangential speed is crucial in scenarios where effective timing and speed are critical, such as in washing machines for optimal cleaning efficiency.

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, refers to the acceleration that pulls or pushes an object toward the center of its circular path. It's necessary for keeping an object moving in that path. The radial acceleration \( a_r \) is given by:

• \( a_r = r \omega^2 \)

This formula demonstrates that radial acceleration is directly proportional to the square of the angular speed and the radius.

• For example, in the washing machine's maximum setting, the radial acceleration was calculated to be \( 107.42g \).
• Here, \( g \) is the acceleration due to gravity, which is \( 9.81 \, \text{m/s}^2 \).

This reveals that the force pulling the laundry inward was over 107 times stronger than the force of gravity. Such high levels of radial acceleration are typical in appliances like washing machines, where fast spinning capabilities enhance the cleaning efficiency by exerting larger radial forces.