Torque & Rotational Dynamics

Introduction to Torque

Torque is the rotational equivalent of force, describing how a force causes an object to rotate about an axis. It is a fundamental concept in rotational dynamics, analogous to how force causes linear acceleration in translational motion.

• Torque (τ): The measure of the tendency of a force to rotate an object about an axis.

• Formula: where r is the lever arm (distance from axis), F is the force, and θ is the angle between r and F.

• Units: Newton-meter (N·m)

• Direction: Determined by the right-hand rule (counterclockwise is positive, clockwise is negative).

Force vs. Torque: Analogy Table

Moment of Inertia (I)

The moment of inertia quantifies an object's resistance to changes in rotational motion, depending on mass distribution relative to the axis of rotation.

• Formula (point mass):

• Formula (rigid body):

• Units: kg·m²

Rotational Dynamics: Newton's Second Law for Rotation

Just as force causes linear acceleration, torque causes angular acceleration:

• Equation:

• α (alpha): Angular acceleration (rad/s²)

Calculating Torque: Examples & Applications

• Example 1: Solid Disc A solid disc of mass M and radius R is rotated by a force at its edge. The torque is (if force is perpendicular).

• Example 2: Fishing Pole If a force is applied at an angle, use to find the torque about the axis of rotation.

• Example 3: Wrench To maximize torque, apply the force perpendicular to the lever arm at the farthest possible point from the axis.

Torque Due to Weight

The weight of an object always acts at its center of mass. For uniform objects, the center of mass is at the geometric center.

• Torque from weight:

Net Torque and the Sign of Torque

The net torque is the sum of all individual torques acting on an object. The sign depends on the direction of rotation:

• Counterclockwise (CCW): Positive (+)

• Clockwise (CW): Negative (−)

• Net Torque:

Torque on Discs and Pulleys

Problems involving discs and pulleys are common in rotational dynamics. The torque depends on the radius at which the force is applied, not the total radius of the disc.

• Example: Two masses connected by a string over a pulley produce a torque on the pulley:

How to Solve: Torque vs. Conservation of Energy

Some rotational motion problems can be solved using either torque and motion equations or conservation of energy. Choose the method based on the information given and what is being asked.

• Use for problems asking for angular acceleration or torque.

• Use conservation of energy for problems asking for speed or energy.

Practice Problems & Applications

• Piano Bar: Calculating moment of inertia and torque for irregular shapes.

• Seesaw: Finding net torque from multiple weights at different positions.

• Barbell: Calculating torque produced by holding a weight at a distance from the axis (shoulder).

• Wrench: Determining the force needed to achieve a required torque.

• Disc and Square: Calculating net torque from multiple forces at various angles.

Summary Table: Key Equations

Additional info: These notes include both conceptual explanations and worked/practice problems, making them suitable for exam preparation in a college-level physics course on rotational dynamics.