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Chapter 3: Problem 28

On your first day at work for an appliance manufacturer, you are told to figure out what to do to the period of rotation during a washer spin cycle to triple the centripetal accelerntion. You inpress your boss by answering immediately. What do you tell her?

Short Answer

Expert verified
Reduce the period to \( \frac{1}{\sqrt{3}} \) of the original to triple centripetal acceleration.

Step by step solution

01

Understanding the Problem

The goal is to triple the centripetal acceleration of the washing machine drum during its spin cycle.
02

Recognizing the Relationship

Centripetal acceleration is calculated using the formula \( a_c = \omega^2 r \), where \( a_c \) is centripetal acceleration, \( \omega \) is angular speed, and \( r \) is the radius of the drum.
03

Expressing Angular Speed in Terms of Period

The angular speed \( \omega \) can be expressed in terms of the period \( T \) using \( \omega = \frac{2\pi}{T} \).
04

Substituting Angular Speed Formula

Substitute \( \omega = \frac{2\pi}{T} \) into centripetal acceleration formula to get \( a_c = \left(\frac{2\pi}{T}\right)^2 r = \frac{4\pi^2 r}{T^2} \).
05

Setting Up the Equation for Triple Acceleration

To triple the acceleration, set \( 3a_c = 3\frac{4\pi^2 r}{T^2} = \frac{4\pi^2 r}{T_{new}^2} \) and solve for \( T_{new} \).
06

Solving for New Period

Solving \( 3\frac{4\pi^2 r}{T^2} = \frac{4\pi^2 r}{T_{new}^2} \), remove common factors to find \( T_{new}^2 = \frac{T^2}{3} \). Therefore, \( T_{new} = \frac{T}{\sqrt{3}} \).
07

Conclusion

To triple the centripetal acceleration, the period of rotation must be reduced to \( \frac{1}{\sqrt{3}} \) of its original value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Speed
Angular speed, denoted by \( \omega \), is a measure of how quickly an object rotates around a central point. It is commonly expressed in radians per second. In practical applications like a washing machine, angular speed helps determine how fast the drum spins during a wash cycle. You can calculate angular speed using the equation \( \omega = \frac{2\pi}{T} \), where \( T \) represents the period of rotation. Thus, understanding angular speed is crucial to adjusting the dynamics of rotating systems like a washer.
  • Faster angular speed means quicker rotation.
  • Slower angular speed results in gentler spins.
  • Angular speed influences the centripetal acceleration exerted on clothes in the washer.
Grasping this concept allows for precise adjustments to washer settings, impacting the washing performance, energy efficiency, and fabric care. Knowing how to translate period into angular speed is essential for practical applications, such as optimizing a washing machine's spin cycle.
Period of Rotation
The period of rotation, represented by \( T \), is the time it takes for a rotating object, like a washing machine drum, to complete one full spin. It is usually measured in seconds. This metric is essential for determining the speed of rotation indirectly.
  • Shorter period means faster drum spinning and hence greater angular speed.
  • Longer period indicates a more leisurely rotation, which can be gentler on fabrics.
  • In physical systems, the period is tightly linked with angular speed \( \omega = \frac{2\pi}{T} \).
By understanding the period of rotation, we can make informed adjustments in washing cycle settings to achieve desired outcomes. In the case of a washer, reducing the period of rotation is a method to increase centripetal acceleration, thereby improving spin efficiency. By decreasing \( T \) by a factor of \( \sqrt{3} \), we can obtain three times the original centripetal acceleration, enhancing water extraction during the spin cycle.
Washing Machine Dynamics
Washing machine dynamics involve the interplay of mechanical forces that determine how efficiently clothes are washed and dried. Central to these dynamics is the concept of centripetal acceleration, which keeps clothes stuck to the drum's wall during a spin cycle. This force can be controlled by tweaking the angular speed and period of rotation.
  • Centripetal force depends on the drum's rotation speed.
  • Acceleration increases with faster spins but requires careful balance to protect fabrics.
  • By controlling period \( T \) and angular speed \( \omega \), optimal washing and drying efficiency can be achieved.
This understanding of dynamics not only enhances washing efficiency but also contributes to effective maintenance and operation of the appliance. Adjustments to the washing machine parameters are often made by modifying settings like speed or cycle length. In this way, enhanced performance can be established, meeting both manufacturing and consumer efficiency standards while maintaining fabric care.

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Most popular questions from this chapter

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a borizontal path 90.0 \(\mathrm{m}\) above the ground and with a speed of \(64.0 \mathrm{m} / \mathrm{s}(143 \mathrm{mi} / \mathrm{h}),\) at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

A rininoceros is at the origin of coordinates at time \(t_{1}=0 .\) For the time interval from \(t_{1}=0\) to \(t_{2}=12.0 \mathrm{s}\) , the rhino's average velocity has \(x-\) component \(-3.8 \mathrm{m} / \mathrm{s}\) and \(y\) -component 4.9 \(\mathrm{m} / \mathrm{s}\) . At time \(t_{2}=12.0 \mathrm{s},(\mathrm{a})\) what are the \(x\) - and \(y\) -coordinates of the rhino? (b) How far is the rhino from the origin?

A physics book slides off a horizontal tabletop with a speed of 1.10 \(\mathrm{m} / \mathrm{s}\) . It strikes the floor in 0.350 \(\mathrm{s}\) . Ignore air resistance. Find (a) the height of the tabletop above the floor; (b) the horizontal distance from the edge of the table to the point where the book strikes the floor, (c) the horizontal and vertical components of the book's velocity, and the magnitude and direction of its velocity, just before the book reaches the floor. (d) Draw \(x-t, y-t, v_{x}-t,\) and \(v_{y}-t\) graphs for the motion.

The earth has a radius of 6380 \(\mathrm{km}\) its axis in 24 \(\mathrm{h}\) . (a) What is the radial acceleration of an object at the earth's equator? Give your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and as a fraction of g. (b) If \(a_{n d}\) at the equator is greater than \(g\) , objects would fly off the earth's surface and into space. (We will see the reason for this in Chapter \(5 . .\) What would the period of the earth's rotation have to be for this to occur?

A 400.0 -m-wide river flows from west to east at 30.0 \(\mathrm{m} / \mathrm{min}\) . Your boat moves at 100.0 \(\mathrm{m} / \mathrm{min}\) relative to the water no matter which direction you point it. To cross this river, you start from a dock at point \(A\) on the south bank. There is a boat landing directly opposite at point \(B\) on the north bank, and also one at point \(C, 75.0 \mathrm{m}\) downstream from \(B(\text { Fig. } 3.53)\) . (a) Where on the north shore will you land if you point your boat perpendicular to the water current, and what distance will you have traveled? (b) If you initially aim your boat directly toward point \(C\) and do not change that bearing relative to the shore, where on the north shore will you land? (c) To reach point \(C :(i)\) at what bearing must you aim your boat, (ii) how long will it take to cross the river, (iii) what distance do you travel, and (iv) and what is the speed of your boat as measured by an observer standing on the river bank?
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