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Chapter 10: Problem 9

A drum rotates around its central axis at an angular velocity of \(12.60 \mathrm{rad} / \mathrm{s} .\) If the drum then slows at a constant rate of \(4.20 \mathrm{rad} / \mathrm{s}^{2}\) (a) how much time does it take and (b) through what angle does it rotate in coming to rest?

Short Answer

Expert verified
(a) 3 seconds, (b) 18.9 radians.

Step by step solution

01

Understanding the problem

To solve this problem, we need to determine two things: the time it takes for the drum to stop (coming to rest) and the total angle through which the drum rotates while stopping. We know the initial angular velocity and the constant deceleration.
02

Use the angular motion formula for time

To find the time, we can use the formula \( \omega_f = \omega_i + \alpha t \), where \( \omega_f \) is the final angular velocity, \( \omega_i \) is the initial angular velocity, and \( \alpha \) is the angular acceleration. In this case, \( \omega_f = 0 \ \text{rad/sec} \), \( \omega_i = 12.60 \ \text{rad/sec} \), and \( \alpha = -4.20 \ \text{rad/sec}^2 \). Rearranging gives us \( t = \frac{\omega_f - \omega_i}{\alpha} \).
03

Calculate the time for the drum to stop

Plug the values into the formula: \( t = \frac{0 - 12.60}{-4.20} = \frac{-12.60}{-4.20} = 3 \ \text{seconds} \).
04

Use the angular motion formula for angle

To find the angle through which the drum rotates, we use the equation \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \). We already calculated the time \( t = 3 \ \text{seconds} \).
05

Calculate the angle through which the drum rotates

Substitute the known values into the equation: \( \theta = 12.60 \times 3 + \frac{1}{2} (-4.20) \times 3^2 \). Calculate each part: \( \theta = 37.8 - 18.9 = 18.9 \ \text{radians} \).

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Key Concepts

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

Angular Velocity
Angular velocity is a measure of how fast something rotates around a specific axis. It tells us the rate at which an object changes its position in terms of rotation per unit time. Imagine it as the speed of rotation.
- **Expressed in radians per second (rad/s):** This is the standard unit for angular velocity. It gives us how many radians (a full circle is 2蟺 radians) the object goes through in one second.
- **Initial angular velocity:** In the exercise, the drum starts with an angular velocity of 12.60 rad/s. This means if it continued spinning at this rate, it would cover 12.60 radians every second.
- **Final angular velocity:** When a problem asks about coming to rest, the final angular velocity ( 蠅_f ) is 0 rad/s, meaning the object has stopped rotating.
Angular Acceleration
Once we know how fast an object is rotating, we may also want to know how quickly that speed is changing. This is where angular acceleration comes into play. It describes how the angular velocity changes over time. If the object is slowing down, this value will be negative, indicating deceleration.
- **Units:** Angular acceleration is measured in radians per second squared (rad/s^2). This tells us how much the angular velocity changes per second. - **In the exercise:** The drum decelerates at a constant rate of -4.20 rad/s^2. This means each second, the angular velocity decreases by 4.20 rad/s until it reaches zero.
- **Using it in formulas:** For instance, to find out how long it will take to stop completely, you can use the formula:\[\omega_f = \omega_i + \alpha t\]where solving for tgives us the time elapsed until the drum comes to a rest.
Angular Displacement
Understanding angular displacement is crucial to find out how far a rotating object has turned during its motion. It is similar to distance, but in terms of angular motion.
- **Definition:** Angular displacement is the angle (in radians) through which an object moves on a circular path.
- **Equation in use:** In the solution, we use the formula:\[\theta = \omega_i t + \frac{1}{2} \alpha t^2\]where \theta represents the angular displacement.
- **Exercise specifics:** The drum starts with an angular velocity of 12.60 rad/s and slows down due to an angular acceleration of -4.20 rad/s^2. Using the calculated time of 3 seconds, we find the total angle the drum rotates until it stops is 18.9 radians.
- **Visual Understanding:** Imagine the drum making just short of 3 full turns since 1 full revolution is about 6.28 radians (which is equivalent to 2蟺 radians). This aids in visualizing how much the drum rotates as it comes to rest.

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Most popular questions from this chapter

A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period \(T\) of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of \(T=0.033 \mathrm{~s}\) that is increasing at the rate of \(1.26 \times 10^{-5} \mathrm{~s} / \mathrm{y}\). (a) What is the pulsar's angular acceleration \(\alpha ?\) (b) If \(\alpha\) is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year \(1054 .\) Assuming constant \(\alpha,\) find the initial \(T\)

A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length \(55.0 \mathrm{~m}\) At the instant it makes an angle of \(35.0^{\circ}\) with the vertical as it falls, what are (a) the radial acceleration of the top, and (b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.) (c) At what angle \(\theta\) is the tangential acceleration equal to \(g ?\)

A uniform cylinder of radius \(10 \mathrm{~cm}\) and mass \(20 \mathrm{~kg}\) is mounted so as to rotate freely about a horizontal axis that is parallel to and \(5.0 \mathrm{~cm}\) from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the angular speed of the cylinder as it passes through its lowest position?

A record turntable rotating at \(33 \frac{1}{3}\) rev \(/\) min slows down and stops in \(30 \mathrm{~s}\) after the motor is turned off. (a) Find its (constant) angular acceleration in revolutions per minute-squared. (b) How many revolutions does it make in this time?

Two uniform solid cylinders, each rotating about its central (longitudinal) axis at \(235 \mathrm{rad} / \mathrm{s},\) have the same mass of \(1.25 \mathrm{~kg}\) but differ in radius. What is the rotational kinetic energy of (a) the smaller cylinder, of radius \(0.25 \mathrm{~m},\) and \((\mathrm{b})\) the larger cylinder, of radius \(0.75 \mathrm{~m} ?\)
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