Chapter 8: Problem 48
What torque is required to give a disk of mass \(6.1 \mathrm{~kg}\) and radius \(0.58 \mathrm{~m}\) an angular acceleration of \(17 \mathrm{rad} / \mathrm{s}^{2}\) ?
Short Answer
Expert verified
The required torque is approximately 17.442 N·m.
Step by step solution
01
Understanding Torque
Torque \(\tau\) is the measure of the force that can cause an object to rotate about an axis. Torque is calculated via the formula \(\tau = I \cdot \alpha\), where \(I\) is the moment of inertia and \(\alpha\) is the angular acceleration.
02
Calculate Moment of Inertia for a Disk
For a disk rotating about its center, the moment of inertia \(I\) is given by the formula \(I = \frac{1}{2} m r^2\), where \(m\) is the mass and \(r\) is the radius of the disk. Substituting the given values, we have:\[ I = \frac{1}{2} \times 6.1 \, \text{kg} \times (0.58 \, \text{m})^2 \]
03
Perform Calculations
Calculate the moment of inertia:\[ I = \frac{1}{2} \times 6.1 \times 0.3364 \]\[ I \approx 1.026 \text{ kg} \cdot \text{m}^2 \]
04
Calculate the Torque
Use the formula for torque, \(\tau = I \cdot \alpha\), where \(\alpha = 17 \, \text{rad/s}^2\) is the angular acceleration:\[ \tau = 1.026 \cdot 17 \]\[ \tau \approx 17.442 \text{ N} \cdot \text{m} \]
05
Conclusion
The torque required to give the disk the specified angular acceleration is approximately \(17.442 \, \text{N} \cdot \text{m}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Moment of Inertia
The moment of inertia is a crucial concept in rotational dynamics that represents how difficult it is to change the rotation of an object. It is the rotational analog of mass in linear motion. For different shapes, the moment of inertia is calculated differently. For a disk rotating about its central axis, the formula is given by:
- \[I = \frac{1}{2} m r^2\]
- \(m\) is the mass of the disk.
- \(r\) is its radius.
- The computation goes like this: \[I = \frac{1}{2} \times 6.1 \times (0.58)^2\]\[I = \frac{1}{2} \times 6.1 \times 0.3364\]\[I \approx 1.026 \text{ kg} \cdot \text{m}^2\]
Angular Acceleration
Angular acceleration, denoted by the symbol \(\alpha\), describes how quickly the rotational speed of an object is changing. Often measured in radians per second squared (\(\text{rad/s}^2\)), it plays a similar role in rotational dynamics as linear acceleration does in linear motion.To cause an object to rotate more quickly or to slow it down, a torque must be applied. Angular acceleration is a key factor in determining the amount of torque required using the formula:
- \[\tau = I \cdot \alpha\]
- \(\tau\) is the torque.
- \(I\) is the moment of inertia.
- \(\alpha\) is the angular acceleration.
Rotational Dynamics
Rotational dynamics is the branch of physics that deals with the effects of forces on the motion of rotating objects. This field is analogous to linear dynamics but involves rotation about a fixed axis. The main components of rotational dynamics include:
- Torque (\(\tau\)): The rotational equivalent of force.
- Moment of Inertia (\(I\)): The measure of how mass is distributed relative to the axis of rotation, affecting an object's resistance to changes in rotation.
- Angular Acceleration (\(\alpha\)): Describes changes in rotational speed.