Rotational Motion, Torque, and Static Equilibrium – Physics 201 Study Notes

Rotational Motion and Torque

Introduction to Rotational Motion

Rotational motion involves the movement of objects around a fixed axis. Unlike linear motion, rotational motion is characterized by angular quantities such as angular displacement, angular velocity, and angular acceleration.

• Angular displacement (θ): The angle through which an object rotates, measured in radians.

• Angular velocity (ω): The rate of change of angular displacement, measured in radians per second (rad/s).

• Angular acceleration (α): The rate of change of angular velocity, measured in radians per second squared (rad/s2).

Example: A rotating disk or rod attached to a hinge, as in the exercises shown in the slides.

Torque

Torque is the rotational equivalent of force. It measures the tendency of a force to rotate an object about an axis, pivot, or fulcrum.

• Definition: Torque (τ) is given by the product of the force (F) and the lever arm (r), which is the perpendicular distance from the axis of rotation to the line of action of the force.

Formula:

$\tau = r F \sin\theta$

• Where θ is the angle between the force vector and the lever arm.

• The moment arm or lever arm is the perpendicular distance from the axis to the line of action of the force.

Right-Hand Rule: Used to determine the direction of the torque vector. Curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the torque vector.

Moment of Inertia

The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation.

• Formula for a point mass: $I = m r^2$

• Formula for a solid disk: $I = \frac{1}{2} M R^2$

• Formula for a uniform rod (about center): $I = \frac{1}{12} M L^2$

Example: A disk of mass 4.0 kg and radius 0.55 m subject to a force at its rim. The moment of inertia is used to calculate angular acceleration.

Newton's Second Law for Rotation

Newton's second law for rotational motion relates the net torque acting on an object to its angular acceleration:

$\sum \tau = I \alpha$

• Where $\sum \tau$ is the net torque, $I$ is the moment of inertia, and $\alpha$ is the angular acceleration.

Example: Calculating the angular acceleration of a rod or disk when a force is applied or when a supporting string is cut.

Static Equilibrium

Conditions for Static Equilibrium

An object is in static equilibrium if it is at rest and remains at rest. For extended objects, two conditions must be satisfied:

• Translational equilibrium: The net force on the object is zero.

• Rotational equilibrium: The net torque on the object is zero.

Mathematical conditions:

$\sum \vec{F} = 0$

$\sum \tau = 0$

• All vector components must be considered.

• These conditions ensure no linear or angular acceleration.

Choosing the Pivot Point

When analyzing static equilibrium, you can choose any point as the pivot for calculating torques. However, selecting a point where unknown forces act can simplify calculations, as the torque due to those forces will be zero.

• Natural axis of rotation: The axis about which the object would rotate if not in equilibrium.

Example: A hammer balanced on two fingers, or a rod supported at one end and by a string at the other.

Applications: Human Body and Everyday Objects

Many real-world systems can be analyzed using static equilibrium, including the human body and objects like crates or vehicles.

• Human foot as a lever: The foot can be modeled as a rigid bar rotating about the ankle. Forces from the Achilles tendon and the ground act to maintain equilibrium.

• Example: Calculating the force exerted by the floor on a person standing on tiptoe, or the force required by the Achilles tendon.

Stability and Balance

Center of Mass and Base of Support

The center of mass (or center of gravity) is the point at which the mass of an object is considered to be concentrated for the purposes of analyzing motion and stability.

• An object is stable if its center of mass remains over its base of support.

• If the center of mass moves outside the base, the object will tip over due to the torque caused by gravity.

Example: A person standing on tiptoe must adjust their center of mass to remain balanced. If the toes are against a wall, it is impossible to lean forward enough to maintain balance, and the person will fall backward.

Stability in Vehicles and Structures

Stability is also crucial in the design of vehicles and buildings. A lower center of gravity and a wider base of support increase stability.

• Critical angle: The angle at which the center of gravity is directly over the pivot point; beyond this, the object will tip.

Summary Table: Key Quantities in Rotational Motion

Quantity	Symbol	SI Unit	Formula
Torque	$\tau$	Newton-meter (N·m)	$\tau = r F \sin\theta$
Moment of Inertia (disk)	$I$	kg·m2	$I = \frac{1}{2} M R^2$
Angular Acceleration	$\alpha$	rad/s2	$\alpha = \frac{\tau}{I}$
Static Equilibrium (force)			$\sum \vec{F} = 0$
Static Equilibrium (torque)			$\sum \tau = 0$

Additional info:

• Some context and examples were inferred from standard physics curriculum and the visible content of the slides.

• Specific textbook references (e.g., Ch. 6, 7.4, 7.5, 8.1–8.3) suggest these topics align with standard introductory physics courses covering mechanics, rotation, and equilibrium.