Rotational Motion

Torque

Torque is a fundamental concept in rotational dynamics, describing the tendency of a force to rotate an object about an axis, pivot, or fulcrum. It is the rotational analogue of force in linear motion and is essential for understanding how objects begin to spin or change their rotational motion.

• Definition: Torque (denoted by ) is the product of the applied force and the lever arm (the perpendicular distance from the axis of rotation to the line of action of the force).

• Formula: or where is the moment arm (perpendicular distance from the axis to the force's line of action).

• Units: Newton-meter (N·m)

• Dimensions:

• Notation:

Techniques for Calculating Torque

There are two main techniques to determine the magnitude of torque:

1. Force Decomposition:

   • Draw a line from the axis of rotation to the point of force application (radial line).

   • Decompose the force into two components: one parallel () and one perpendicular () to the radial line.

   • Only the perpendicular component produces torque:

2. Lever Arm (Moment Arm) Method:

   • Draw the line of action of the force.

   • Measure the perpendicular distance from this line to the pivot (axis).

   • Torque is given by: In many cases,

Worked Examples

• Example 1: Force applied at the end of a bar of length , perpendicular to the bar.

• Example 2: Same force and bar, but at to the bar.

• Example 3:

• Example 4: (force acts along the bar) (no torque produced)

Properties of Torque

• Reference Point: Torque is always calculated about a specific point (pivot, hinge, or axis).

• Sign Convention: By convention, counterclockwise torques are positive, and clockwise torques are negative.

• Net Torque: The net torque is the sum of all individual torques acting on an object:

Lever Classes

Levers are classified based on the relative positions of the force, load, and fulcrum:

Conceptual Questions & Applications

• Ranking Torques: The magnitude of torque depends on both the force and its distance from the pivot, as well as the angle of application. For rods of equal length, the largest torque is produced by the largest force applied furthest from the pivot and at the most effective angle (perpendicular).

• Zero Net Torque: To achieve zero net torque, a third force can be applied such that the sum of all torques about the pivot is zero.

• Factors Affecting Torque: Torque depends on the magnitude of the force, the angle of application, and the distance from the axis. It does not depend on the object's angular velocity.

• Opening a Door: The most effective force is applied perpendicular to the door and as far from the hinge as possible, maximizing the lever arm.

• Effect of Net Torque: A net torque causes a change in the angular velocity of an object (rotational acceleration), analogous to how a net force causes linear acceleration.

Summary Table: Factors Affecting Torque

Key Equations

Example Application

Opening a Door: To maximize the torque and open a door easily, apply the force perpendicular to the door and as far from the hinge as possible.

Additional info: These notes cover the core principles of torque in rotational motion, including calculation methods, physical interpretation, and practical applications such as levers and doors. The content is suitable for college-level introductory physics.