Question

Calculate 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 17.43 Nm.

Step-by-step solution

Understand the Problem

We need to find the torque required to give a disk a specific angular acceleration. We are given the mass of the disk (6.1 kg), its radius (0.58 m), and the angular acceleration (17 rad/s²).

Calculate the Moment of Inertia

The moment of inertia for a solid disk rotating about its central axis is given by the formula: \( I = \frac{1}{2} m r^2 \). Substitute the mass \( m = 6.1 \, \text{kg} \) and the radius \( r = 0.58 \, \text{m} \) into the formula: \[ I = \frac{1}{2} \times 6.1 \, \text{kg} \times (0.58 \, \text{m})^2 = 1.02546 \, \text{kg} \, \text{m}^2 \].

Use Torque Formula

Torque \( \tau \) is related to moment of inertia \( I \) and angular acceleration \( \alpha \) by the formula: \( \tau = I \alpha \). Substitute \( I = 1.02546 \, \text{kg} \, \text{m}^2 \) and \( \alpha = 17 \, \text{rad/s}^2 \): \[ \tau = 1.02546 \, \text{kg} \, \text{m}^2 \times 17 \, \text{rad/s}^2 = 17.43282 \, \text{Nm} \].

State the Final Answer

The torque required to give the disk an angular acceleration of 17 rad/s² is approximately 17.43 Nm.

Key concepts

Moment of Inertia

The moment of inertia is a core principle in rotational dynamics, akin to mass in linear motion. It measures an object's resistance to changes in its rotation. For different shapes, the moment of inertia is calculated using various formulas. For a solid disk, it is expressed as:

• \( I = \frac{1}{2} m r^2 \)

where \( m \) is the mass and \( r \) is the radius.
In our case, substituting the given values:

• Mass, \( m = 6.1 \, \text{kg} \)
• Radius, \( r = 0.58 \, \text{m} \)

Using these, the moment of inertia \( I \) becomes \( 1.02546 \, \text{kg} \, \text{m}^2 \).
The higher the moment of inertia, the more torque is required to alter the angular velocity.

Angular Acceleration

Angular acceleration is the rate at which an object's angular velocity changes with time. It's a vital parameter in rotational motion, reflecting how quicky the object speeds up or slows down as it spins.
Represented by \( \alpha \), angular acceleration is measured in radians per second squared (rad/s²). In the context of the exercise, the disk's angular acceleration is given as 17 rad/s².
Knowing the angular acceleration helps us determine the torque needed to achieve such change, connecting directly to both the moment of inertia and the applied force.

Solid Disk

A solid disk is a common shape in physics problems dealing with rotational dynamics. It's a simple, perfectly round, flat object which rotates about its own central axis.
Understanding its properties is essential:

• For a solid disk, the mass is evenly distributed.
• This uniform distribution affects its moment of inertia, calculated through \( I = \frac{1}{2} m r^2 \).
• When analyzing forces and rotations, a solid disk's simplistic nature allows consistent application of physics formulas based on its rotational attributes.

When solving exercises, recognizing the distribution helps in correctly applying fundamental concepts like the moment of inertia.

Newton's Second Law for Rotation

Newton's Second Law in rotational form is essential for calculating torque in rotational dynamics. It’s expressed through the equation:

• \( \tau = I \alpha \)

where \( \tau \) is torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration.
Torques cause changes in angular motion, similar to how forces cause changes in linear motion.

• The law highlights the relationship between applied torque and resulting angular acceleration.
• For the solid disk in the exercise, substituting the given moment of inertia (1.02546 kg m²) and angular acceleration (17 rad/s²) into the formula provides the torque required — 17.43 Nm.

Remember, the greater the moment of inertia, the more torque is needed for the same angular change.