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Chapter 4: Problem 29

A tire \(0.500 \mathrm{m}\) in radius rotates at a constant rate of 200 rev/min. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge).

Short Answer

Expert verified
The speed of the stone is approximately \(10.47 \: m/s\) and the acceleration is approximately \(209.44 \: m/s^2\).

Step by step solution

01

Convert Rotational Speed to Radians Per Second

The rotational speed is given in revolutions per minute. To use this speed in calculations involving the circumference of a circle or the radius (since the stone is on the circumference of the tire), it needs to be converted into radians per second. Use the formula: \[\omega=\frac{(200 \: rev/min) * 2\pi}{60 s/min}= \frac{20\pi}{3} \: rad/s\]
02

Calculate the Speed of the Stone

The speed of the stone (assuming it is stuck to the tread) will be the same as the speed of the point on the outer edge of the tire. The linear speed (v) can be calculated using the formula: \(v=r*\omega\). Substituting the given values, \[v=(0.500 \: m)*\frac{20\pi}{3} \: rad/s = \frac{10\pi}{3} \: m/s\]
03

Calculate the Acceleration of the Stone

The linear acceleration (a) of the stone can be calculated using the formula: \(a=r*\omega^{2}\) . Substituting the given values, \[a=(0.500 \: m)*(\frac{20\pi}{3} \: rad/s)^{2}=\frac{200\pi^{2}}{3} \: m/s^2\]

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Key Concepts

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

Understanding Angular Velocity
Angular velocity is a term used in rotational motion to describe how fast an object spins around a central point. In the context of the exercise, the tire is rotating, and we need to know how quickly it turns. Angular velocity is usually expressed in radians per second. This unit might seem odd at first, but it鈥檚 essential for expressing how an object rotates. Radians are a type of angle measurement that relate to the radius of a circle. To convert the rotational speed from revolutions per minute to radians per second, you use the formula:
  • Convert revolutions to radians by multiplying with \(2\pi\), since a complete revolution is \(2\pi\) radians.
  • Convert minutes to seconds, knowing there are 60 seconds in a minute.
So 200 revolutions per minute becomes \(\frac{20\pi}{3}\) radians per second.
Exploring Linear Speed
Linear speed, often just called speed, tells us how fast an object is moving through space. It's slightly different from angular velocity because it focuses on straight-line movement, rather than rotation. For the stone in the tire tread, its linear speed is simply how fast it would be moving if you were watching it fly past in its circular track.To determine linear speed from angular velocity, you multiply the angular velocity by the radius of the path. The formula is \(v = r\omega\), where:
  • \(v\) is the linear speed
  • \(r\) is the radius of the circle
  • \(\omega\) is the angular velocity
From our example, a radius of 0.5 meters and an angular velocity of \(\frac{20\pi}{3}\) rad/s gives a linear speed of \(\frac{10\pi}{3}\) m/s. This means the stone is zipping around the tire track at this impressive speed!
Demystifying Tangential Acceleration
Tangential acceleration is all about how quickly something鈥檚 speed is changing as it spins. Even if the overall rotation speed doesn't change, a point on the tire is constantly changing its direction, making this concept crucial.Tangential acceleration is calculated using the formula \(a = r\omega^2\), where:
  • \(a\) is the tangential acceleration
  • \(r\) is the radius of the circle
  • \(\omega\) is the angular velocity
In our problem, with a radius of 0.5 meters and an angular velocity of \(\frac{20\pi}{3}\) radians per second, the acceleration becomes \(\frac{200\pi^2}{3}\) m/s虏. This value indicates the rate of change of speed for the stone, attached to the tire's edge, as it follows the circular path.

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Most popular questions from this chapter

The water in a river flows uniformly at a constant speed of \(2.50 \mathrm{m} / \mathrm{s}\) between parallel banks \(80.0 \mathrm{m}\) apart. You are to deliver a package directly across the river, but you can swim only at \(1.50 \mathrm{m} / \mathrm{s} .\) (a) If you choose to minimize the time you spend in the water, in what direction should you head? (b) How far downstream will you be carried? (c) What If? If you choose to minimize the distance downstream that the river carries you, in what direction should you head? (d) How far downstream will you be carried?

An archer shoots an arrow with a velocity of \(45.0 \mathrm{m} / \mathrm{s}\) at an angle of \(50.0^{\circ}\) with the horizontal. An assistant standing on the level ground 150 m downrange from the launch point throws an apple straight up with the minimum initial speed necessary to meet the path of the arrow. (a) What is the initial speed of the apple? (b) At what time after the arrow launch should the apple be thrown so that the arrow hits the apple?

A fireworks rocket explodes at height \(h,\) the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed \(v .\) Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal.

A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of \(8.00 \mathrm{m} / \mathrm{s}\) at an angle of \(20.0^{\circ}\) below the horizontal. It strikes the ground \(3.00 \mathrm{s}\) later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point \(10.0 \mathrm{m}\) below the level of launching?

A motorist drives south at \(20.0 \mathrm{m} / \mathrm{s}\) for \(3.00 \mathrm{min}\) then turns west and travels at \(25.0 \mathrm{m} / \mathrm{s}\) for \(2.00 \mathrm{min},\) and finally travels northwest at \(30.0 \mathrm{m} / \mathrm{s}\) for 1.00 min. For this 6.00 -min trip, find (a) the total vector displacement, (b) the average speed, and (c) the average velocity. Let the positive \(x\) axis point east.
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