Question

Explain How is torque related to angular acceleration?

Short answer

Torque is directly proportional to angular acceleration (\( \tau = I \cdot \alpha \)).

Step-by-step solution

Understanding Torque

Torque (\( \tau \)) is a measure of the rotational force applied to an object. It is calculated using the formula \( \tau = r \cdot F \cdot \sin(\theta) \), where \( r \) is the distance from the pivot point, \( F \) is the force applied, and \( \theta \) is the angle between the force and the lever arm.

Understanding Angular Acceleration

Angular acceleration (\( \alpha \)) is the rate at which the angular velocity of an object changes. It is measured in radians per second squared (\( \text{rad/s}^2 \)). It describes how quickly an object is speeding up or slowing down in its rotational motion.

Applying Newton's Second Law for Rotation

Just as Newton's Second Law for linear motion states \( F = m \cdot a \), it can be applied to rotational motion as \( \tau = I \cdot \alpha \). Here, \( I \) is the moment of inertia, a measure of an object's resistance to change in angular motion.

Relating Torque to Angular Acceleration

From Newton's Second Law for rotation (\( \tau = I \cdot \alpha \)), we can see that torque and angular acceleration are directly proportional to each other. This means more torque will result in more angular acceleration if the moment of inertia remains constant.

Key concepts

Newton's Second Law for Rotation

Newton's Second Law is a fundamental principle in physics that explains how the motion of objects changes when forces are applied. In terms of rotational motion, this law is adapted to relate the torque applied to an object to its angular acceleration. The formula used is \( \tau = I \cdot \alpha \), where \( \tau \) (tau) represents torque, \( I \) stands for moment of inertia, and \( \alpha \) symbolizes angular acceleration. This relationship is very much like the linear form \( F = m \cdot a \), where force \( F \) leads to acceleration \( a \), but in rotational scenarios.

• **Torque** is the rotational equivalent of force.
• **Angular acceleration** represents how quickly an object speeds up or slows down its rotation.
• **Moment of inertia** is a measure of an object's resistance to rotational acceleration.

From this, we see that if the moment of inertia \( I \) stays constant, an increase in torque \( \tau \) will directly increase angular acceleration \( \alpha \). This direct proportionality is fundamental to understanding how objects spin and rotate under different forces.

Moment of Inertia

The moment of inertia is a critical concept when analyzing rotational motion. It is often compared to mass in linear motion, as both quantities measure an object's resistance to change in motion. However, while mass is a single value, the moment of inertia involves the distribution of mass around a pivot point.

Mathematically, it is expressed as \( I = \sum m_i r_i^2 \), summing over each mass element \( m_i \) at a distance \( r_i \) from the rotational axis. This means objects with more mass distributed further from the axis have a higher moment of inertia and are harder to spin.

• A figure skater pulling in their arms to spin faster is a real-life example of reducing the moment of inertia.
• Different shapes and axis configurations can drastically alter an object's moment of inertia.
• The moment of inertia helps determine how much torque is needed to achieve a desired angular acceleration.

Understanding the moment of inertia allows us to predict how different forces will affect the rotational motion of objects and design systems that control rotational behavior.

Rotational Motion

Rotational motion occurs when an object spins around an axis. This can involve any object, from a spinning top to the rotating Earth itself. Unlike linear motion that involves straight paths, rotational motion involves movement around a pivot or center.

In rotational motion, several key quantities define how objects move:

• **Angular displacement** \( \theta \), measured in radians, represents the angle through which an object rotates.
• **Angular velocity** \( \omega \), measured in radians per second, is how quickly the angle changes over time.
• **Angular acceleration** \( \alpha \), measures the change in angular velocity over time.

These quantities relate closely to their linear counterparts: displacement, velocity, and acceleration.

The study of rotational motion also involves understanding the effects of forces and torques on how objects spin. Much like how linear motion can be described by Newton's laws, rotational motion hinges upon these principles to predict and analyze behavior.