Question

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

The required torque is approximately 17.442 N·m.

Step-by-step solution

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.

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:\[ I = \frac{1}{2} \times 6.1 \, \text{kg} \times (0.58 \, \text{m})^2 \]

Perform Calculations

Calculate the moment of inertia:\[ I = \frac{1}{2} \times 6.1 \times 0.3364 \]\[ I \approx 1.026 \text{ kg} \cdot \text{m}^2 \]

Calculate the Torque

Use the formula for torque, \(\tau = I \cdot \alpha\), where \(\alpha = 17 \, \text{rad/s}^2\) is the angular acceleration:\[ \tau = 1.026 \cdot 17 \]\[ \tau \approx 17.442 \text{ N} \cdot \text{m} \]

Conclusion

The torque required to give the disk the specified angular acceleration is approximately \(17.442 \, \text{N} \cdot \text{m}\).

Key concepts

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:

• \[I = \frac{1}{2} m r^2\]

where:

• \(m\) is the mass of the disk.
• \(r\) is its radius.

To determine the moment of inertia for a specific disk, simply substitute the mass and the radius into the formula. As seen in the example, substituting a mass of 6.1 kg and a radius of 0.58 m:

• The computation goes like this: \[I = \frac{1}{2} \times 6.1 \times (0.58)^2\]\[I = \frac{1}{2} \times 6.1 \times 0.3364\]\[I \approx 1.026 \text{ kg} \cdot \text{m}^2\]

This calculated value of the moment of inertia indicates how much resistance the disk offers to a change in its rotational motion.

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:

• \[\tau = I \cdot \alpha\]

where:

• \(\tau\) is the torque.
• \(I\) is the moment of inertia.
• \(\alpha\) is the angular acceleration.

In the given exercise, the angular acceleration is provided as \(17 \text{ rad/s}^2\). This information is essential for calculating the torque required to achieve the specified change in rotational speed. By substituting this value into the torque formula along with the moment of inertia, one can accurately determine the torque needed.

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.

Using the fundamental relationship \(\tau = I \cdot \alpha\), we can understand how these elements interact. For example:A larger moment of inertia means more torque is needed for the same angular acceleration.Conversely, a larger angular acceleration can be achieved with less torque if the moment of inertia is smaller.In practical applications like machinery and vehicle engineering, understanding these relationships is vital to designing systems that efficiently manage rotational forces. By mastering the principles of rotational dynamics, we can predict, control, and optimize the motion of rotating systems.