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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
It takes 3 seconds to stop, and the drum rotates through 18.90 radians.

Step by step solution

01

Identify the Given Variables

In the problem, the initial angular velocity \(\omega_i\) is \(12.60\, \text{rad/s}\), and the angular deceleration \(\alpha\) is \(4.20\, \text{rad/s}^2\). The final angular velocity \(\omega_f\) is \(0\, \text{rad/s}\) since the drum comes to rest.
02

Apply Angular Motion Equation for Time

To find the time it takes for the drum to come to rest, use the formula \( \omega_f = \omega_i + \alpha t \). Solving for \(t\), we rearrange the equation to get:\[ t = \frac{\omega_f - \omega_i}{\alpha} \]Substituting the known values:\[ t = \frac{0 - 12.60}{-4.20} = 3 \text{ seconds} \]
03

Calculate the Angle Rotated until the Drum Stops

For the angle \(\theta\) the drum rotates, use the formula:\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]Using \(t = 3\,\text{s}\) from earlier, the angle is:\[ \theta = 12.60 \times 3 + \frac{1}{2} (-4.20) \times (3)^2 \]Calculating step-by-step:- \(12.60 \times 3 = 37.80 \)- \(-4.20 \times 9 = -37.80\)- Total \(\theta = 37.80 - 18.90 = 18.90\, \text{rad}\)

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Key Concepts

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

Angular Velocity
Angular velocity describes how quickly an object rotates or spins around a central axis. It is measured in radians per second (rad/s), providing insight into the speed of rotation. Imagine looking at the hands of a clock. The faster they move, the higher their angular velocity.
  • The initial angular velocity, denoted as \(\omega_i\), is the rate at which the object begins spinning.
  • In our exercise, the drum starts with an angular velocity of \(12.60\; \text{rad/s}\).
  • Angular velocity can decrease over time due to forces like friction or applied brakes.
Understanding angular velocity is crucial in predicting how quickly a rotating object will reach a desired state, such as coming to a standstill.
Angular Deceleration
Angular deceleration occurs when the rate of rotation of an object decreases over time. It is the negative acceleration that opposes the motion of the object. In mathematical terms, it is represented as \( \alpha \), the angular acceleration, and has units of \(\text{rad/s}^2\).
  • For the drum problem, the angular deceleration is \(-4.20\; \text{rad/s}^2\).
  • This indicates a steady reduction in the drum鈥檚 speed.
  • A constant angular deceleration means the change in rotational speed is uniform across each second.
By understanding angular deceleration, one can calculate how long it will take for a rotating object to come to a halt. This is critical in areas such as vehicle braking systems and machinery operations.
Rotational Kinematics
Rotational kinematics deals with the motion of objects that rotate around an axis. Just like linear kinematics focuses on straight-line motion, rotational kinematics uses similar principles for circular motion. It involves angular position, velocity, and acceleration.
One key equation used in rotational kinematics is \[ \omega_f = \omega_i + \alpha t \]This equation helps determine how long it takes for a rotating object to stop or reach a certain angular velocity.
  • Applying this in the exercise, we find the time for the drum to come to rest is 3 seconds.
  • The same principles used here apply to any spinning systems like wheels, gears, or even planets.
Mastering rotational kinematics is essential for predicting rotational behavior and designing mechanisms that incorporate rotational motion.
Angular Displacement
Angular displacement refers to the total angle through which an object rotates or spins from its starting position. It is usually measured in radians and is a fundamental concept in understanding motion in circular paths.
In our drum example, the angular displacement can be calculated using the formula: \[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
  • From this, we find the drum rotates through \(18.90\; \text{rad}\).
  • Angular displacement gives a comprehensive picture of how far an object turns during a specific time or event.
  • Applications include determining the total rotation of engines, turbines, or wheels within a given period.
Understanding angular displacement helps in analyzing and controlling systems requiring precise angular position management.

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Most popular questions from this chapter

A wheel, starting from rest, rotates with a constant angular acceleration of \(2.00 \mathrm{rad} / \mathrm{s}^{2}\). During a certain \(3.00 \mathrm{~s}\) interval, it turns through \(90.0\) rad. (a) What is the angular velocity of the wheel at the start of the \(3.00 \mathrm{~s}\) interval? (b) How long has the wheel been turning before the start of the \(3.00\) s interval?

A vinyl record is played by rotating the record so that an approximately circular groove in the vinyl slides under a stylus. Bumps in the groove run into the stylus, causing it to oscillate. The equipment converts those oscillations to electrical signals and then to sound. Suppose that a record turns at the rate of \(33 \frac{1}{3}\) rev/min, the groove being played is at a radius of \(10.0 \mathrm{~cm}\), and the bumps in the groove are uniformly separated by \(1.75 \mathrm{~mm}\). At what rate (hits per second) do the bumps hit the stylus?

A high-wire walker always attempts to keep his center of mass over the wire (or rope). He normally carries a long, heavy pole to help: If he leans, say, to his right (his com moves to the right) and is in danger of rotating around the wire, he moves the pole to his left (its com moves to the left) to slow the rotation and allow himself time to adjust his balance. Assume that the walker has a mass of \(70.0 \mathrm{~kg}\) and a rotational inertia of \(15.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about the wire. What is the magnitude of his angular acceleration about the wire if his com is \(5.0 \mathrm{~cm}\) to the right of the wire and (a) he carries no pole and (b) the \(14.0 \mathrm{~kg}\) pole he carries has its \(\operatorname{com} 10 \mathrm{~cm}\) to the left of the wire?

An object rotates about a fixed axis, and the angular position of a reference line on the object is given by \(\theta=0.40 e^{2 t}\), where \(\theta\) is in radians and \(t\) is in seconds. Consider a point on the object that is \(4.0\) \(\mathrm{cm}\) from the axis of rotation. At \(t=0\), what are the magnitudes of the point's (a) tangential component of acceleration and (b) radial component of acceleration?

Trucks can be run on energy stored in a rotating flywheel, with an electric motor getting the flywheel up to its top speed of \(200 \pi \mathrm{rad} / \mathrm{s}\). One such flywheel is a solid, uniform cylinder with a mass of \(500 \mathrm{~kg}\) and a radius of \(1.0 \mathrm{~m} .\) (a) What is the kinetic energy of the flywheel after charging? (b) If the truck uses an average power of \(8.0 \mathrm{~kW}\), for how many minutes can it operate between chargings?
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