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Chapter 7: Problem 4

A potter's wheel moves uniformly from rest to an angular speed of \(1.00 \mathrm{rev} / \mathrm{s}\) in \(30.0 \mathrm{~s}\). (a) Find its angular acceleration in radians per second per second. (b) Would doubling the angular acceleration during the given period have doubled final angular speed?

Short Answer

Expert verified
The angular acceleration of the potter's wheel is \(\frac{\pi}{15} \, \text{rad/s}^2\). Yes, doubling the angular acceleration would have doubled the final angular speed.

Step by step solution

01

Conversion of Units

The angular speed given is in revolution per second, and we must convert this to radians per second since the unit of angular acceleration is radians per second squared. Note that one revolution is equal to \(2\pi\) radians. So, \(1.00 \, \text{rev/s} = 1.00 \times 2\pi \, \text{rad/s} = 2\pi \, \text{rad/s}\).
02

Calculate Angular Acceleration

Now, use the formula for angular acceleration, \(\alpha = \frac{\Delta \omega}{\Delta t}\), where \(\Delta \omega\) is the change in angular speed and \(\Delta t\) is the change in time. Using the values obtained from above and the given time duration, \(\alpha = \frac{2\pi \, \text{rad/s}}{30.0 \, \text{s}} = \frac{\pi}{15} \, \text{rad/s}^2\).
03

Understanding Acceleration-Speed Relationship

For the question part (b), let's use the formula for final angular speed \(蠅_f = 蠅_i + 伪t\). Doubling the angular acceleration while the time remains the same, the final angular speed would have been doubled. Thus, the answer is yes.

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Key Concepts

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

Angular Speed
Angular speed describes how fast an object rotates or revolves around a certain point. It is a measure of the angle covered per unit of time. Angular speed is crucial in many fields like physics and engineering because it helps in analyzing rotational motion with respect to time.
Often, angular speed is given in revolutions per second or revolutions per minute. However, in scientific calculations, it is typically converted into radians per second for consistency. This makes further calculations, like finding angular acceleration, much more manageable. Remember, one revolution equals an angle of \(2\pi\) radians.
Radians per Second
Radians per second is a standard unit for measuring angular speed in the International System of Units (SI). It represents the angular displacement through an angle in radians over time.
For example, if a wheel completes one full rotation (one revolution) every second, the angular speed in radians per second would be \(2\pi\), because a full circle is \(2\pi\) radians. This unit allows for simplified calculations of other related metrics, such as angular acceleration.
Using radians ensures that all the elements of the calculation are in the same unit system, preventing conversion errors and making equations simpler to work with.
Conversion of Units
Converting units is a fundamental skill in physics, particularly for rotational motion calculations. To find the angular acceleration in this exercise, you need to convert angular speed from revolutions per second to radians per second.
Understanding that there are \(2\pi\) radians in one revolution is key. For instance, if a potter's wheel has an angular speed of 1 revolution per second, it should be converted to \(1 \times 2\pi = 2\pi\) radians per second for calculation purposes. This conversion is essential to derive correct values for calculations involving angular properties.
Final Angular Speed
Final angular speed is the speed at which an object rotates after a certain period, having accelerated at a consistent rate. It can be calculated using the initial angular speed, the time duration, and the angular acceleration.
  • Formula: \(\omega_f = \omega_i + \alpha t\)
  • \(\omega_i\) is the initial angular speed
  • \(\alpha\) is the angular acceleration
  • \(t\) is the time
In the given exercise, the wheel starts from rest, meaning the initial angular speed \(\omega_i\) is zero. Therefore, the final angular speed largely depends on the angular acceleration and the time period. If you double the angular acceleration, keeping the time constant, the final angular speed would also double, illustrating the direct relationship between these variables.

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Most popular questions from this chapter

A piece of mud is initially at point \(A\) on the rim of a bicycle wheel of radius \(R\) rotating clockwise about a horizontal axis at a constant angular speed \(\omega\) (Fig. P7.8). The mud dislodges from point \(A\) when the wheel diameter through \(A\) is horizontal. The mud then rises vertically and returns to point \(A\). (a) Find a symbolic expression in terms of \(R, \omega\), and \(g\) for the total time the mud is in the air and returns to point \(A\). (b) If the wheel makes one complete revolution in the time it takes the mud to return to point \(A\), find an expression for the angular speed of the bicycle wheel \(\omega\) in terms of \(\pi, g\), and \(R\).

In a popular amusement park ride, a rotating cylin- der of radius \(3.00 \mathrm{~m}\) is set in rotation at an angular speed of \(5.00 \mathrm{rad} / \mathrm{s}\), as in Figure P7.75. The floor then drops away, leav- ing the riders suspended against the wall in a verti- cal position. What mini- mum coefficient of friction between a rider's clothing and the wall is needed to keep the rider from slipping?

Neutron stars are extremely dense objects that are formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is \(10.0 \mathrm{~km}\). Determine the greatest possible angular speed the neutron star can have so that the matter at its surface on the equator is just held in orbit by the gravitational force.

\(\mathrm{}\) A \(40.0-\mathrm{kg}\) child swings in a swing supported by two chains, each \(3.00 \mathrm{~m}\) long. The tension in each chain at the lowest point is \(350 \mathrm{~N}\). Find (a) the child's speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

A car moves at speed \(v\) across a bridge made in the shape of a circular arc of radius \(r\). (a) Find an expression for the normal force acting on the car when it is at the top of the arc. (b) At what minimum speed will the normal force become zero (causing the occupants of the car to seem weightless) if \(r=30.0 \mathrm{~m}\) ?
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