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Chapter 10: Problem 80

A disk rotates at constant angular acceleration, from angular position \(\theta_{1}=10.0\) rad to angular position \(\theta_{2}=70.0\) rad in \(6.00 \mathrm{~s}\). Its angular velocity at \(\theta_{2}\) is \(15.0 \mathrm{rad} / \mathrm{s}\). (a) What was its angular velocity at \(\theta_{1} ?\) (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph \(\theta\) versus time \(t\) and angular speed \(\omega\) versus \(t\) for the disk, from the beginning of the motion \((\) let \(t=0\) then \()\)

Short Answer

Expert verified
(a) 11 rad/s, (b) 0.67 rad/s虏, (c) -42.5 rad, (d) See graph description in step 6.

Step by step solution

01

Identify known values

We know that the initial angular position is \(\theta_1 = 10.0\) rad, the final angular position \(\theta_2 = 70.0\) rad, the time interval \(t = 6.00\) s, and the final angular velocity is \(\omega_2 = 15.0\, \text{rad/s}\).
02

Use the rotational kinematics equation to solve for initial angular velocity

We use the equation \(\theta_2 = \theta_1 + \omega_1 t + \frac{1}{2}\alpha t^2\). We can also express the final angular velocity as \(\omega_2 = \omega_1 + \alpha t\). Solving these two equations together will allow us to find the unknowns, \(\omega_1\) and \(\alpha\). First express \(\alpha\) from the second equation as \(\alpha = \frac{\omega_2 - \omega_1}{t}\). Substitute \(\alpha\) in the first equation to isolate \(\omega_1\).
03

Solve for \(\omega_1\) (initial angular velocity)

Substituting \(\alpha\) into the first equation gives:\[70 = 10 + 6 \omega_1 + \frac{1}{2} \left( \frac{15 - \omega_1}{6} \right) 6^2\]Simplify and solve for \(\omega_1\):\[60 = 6\omega_1 + 27 - 3\omega_1\]\[60 = 3\omega_1 + 27\]\[33 = 3\omega_1\]\[\omega_1 = 11\, \text{rad/s}\]
04

Solve for \(\alpha\) (angular acceleration)

Using \(\alpha = \frac{\omega_2 - \omega_1}{t}\), we substitute the known values:\[\alpha = \frac{15 - 11}{6} = \frac{4}{6} = 0.67\text{ rad/s}^2\].
05

Calculate the initial rest position

The disk was initially at rest when \(\omega_1 = 0\). Using the equation \(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\), and letting \(\omega_0 = 0\), we find \(\theta = \frac{\omega_2^2 - 0^2}{2\alpha} + \theta_0\). Solving \(\theta_0\):\[\theta_0 = 70 - \frac{15^2}{2 \times 0.67}\]\[\theta_0 = 70 - 167.91 \approx 70 - 112.5\]\[\theta_0 = -42.5\text{ rad}\]
06

Graph \(\theta\) versus time \(t\) and \(\omega\) versus \(t\)

Using the equations obtained, plot \(\theta = 10 + 11t + \frac{1}{2}(0.67)t^2\) for \(t\) from \(0\) to \(6\) s. For \(\omega\), use \(\omega = 11 + 0.67t\). The graph of \(\theta\) is a parabolic curve starting at \(10\) rad when \(t = 0\). The graph of \(\omega\) is linear, starting at \(11\) rad/s and ending at \(15\) rad/s at \(t = 6\) s.

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Key Concepts

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

Angular Velocity
Angular velocity refers to how fast an object rotates or spins around a central point or axis. It is specifically concerned with the rate of change of angular position (typically measured in radians) over time. In rotational kinematics, understanding angular velocity is pivotal because it allows us to predict how the rotational motion of an object will evolve over time.
  • Angular velocity is typically denoted by the symbol \( \omega \).
  • The unit of angular velocity is radians per second (rad/s).
In the given exercise, we were provided with the final angular velocity (15 rad/s) and required to find the initial angular velocity \( \omega_1 \). By using the rotational kinematic equations, we calculated it to be 11 rad/s. This crucial step is often applied when you need to derive unknown motion characteristics from known quantities.
Angular Acceleration
Angular acceleration is the rate of change of angular velocity. When an object's rotation speed changes, either speeding up or slowing down, the angular acceleration comes into play. Understanding angular acceleration helps to describe how quickly a rotating object is picking up or shedding its speed around an axis.
  • It is typically represented by the symbol \( \alpha \).
  • The unit of angular acceleration is radians per second squared (rad/s虏).
In our problem, the angular acceleration was initially unknown. Using provided data and the equations of motion for constant angular acceleration, we solved for \( \alpha \) and found it to be 0.67 rad/s虏. Grasping this concept assists in solving various real-world problems where rotational dynamics are involved.
Initial Angular Position
The initial angular position of a rotating object is the angle at which it starts its motion. It is crucial in determining the entire trajectory of the rotational movement.
  • Denoted as \( \theta_0 \), this is the starting point for angle measurements.
In our case, the disk's initial position was at 10 rad. However, part of the task was to backtrack the disk's journey to ascertain at what angular position it was initially at rest, before receiving any angular velocity. Calculating rest position helps in understanding how far back any given movement extends, offering insight into the energy or effort needed to start the rotation.
Graphing Rotational Motion
Graphing rotational motion involves plotting variables such as angular position and angular velocity over time. These graphs offer visual representation and deeper insights into how rotational variables evolve through the different stages of motion.
  • The angle-time graph usually depicts a parabolic curve for constant acceleration, illustrating the change in position over time.
  • The velocity-time graph generally shows a linear trend, indicating a constant rate of change of angular velocity over time.
In the given solution, we derived equations for both the angle vs. time and velocity vs. time, allowing visual representation of the disk's behavior during the 6-second interval. Graphs are invaluable tools in physics for predicting future motion and validating theoretical calculations.

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Most popular questions from this chapter

Calculate the rotational inertia of a wheel that has a kinetic energy of \(24400 \mathrm{~J}\) when rotating at 602 rev \(/ \mathrm{min}\).

A good baseball pitcher can throw a baseball toward home plate at \(85 \mathrm{mi} / \mathrm{h}\) with a spin of \(1800 \mathrm{rev} / \mathrm{min} .\) How many revolutions does the baseball make on its way to home plate? For simplicity, assume that the \(60 \mathrm{ft}\) path is a straight line.

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 \(0.75 \mathrm{~m} ?\)

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?

A uniform helicopter rotor blade is \(7.80 \mathrm{~m}\) long, has a mass of \(110 \mathrm{~kg},\) and is attached to the rotor axle by a single bolt. (a) What is the magnitude of the force on the bolt from the axle when the rotor is turning at 320 rev/min? (Hint: For this calculation the blade can be considered to be a point mass at its center of mass. Why?) (b) Calculate the torque that must be applied to the rotor to bring it to full speed from rest in \(6.70 \mathrm{~s}\). Ignore air resistance. (The blade cannot be considered to be a point mass for this calculation. Why not? Assume the mass distribution of a uniform thin rod.) (c) How much work does the torque do on the blade in order for the blade to reach a speed of 320 rev/min?
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