module 5 lecture 9: Rotational Motion, Torque, and Gravitation

Work, Energy, and Simple Machines

Work and Energy

Work and energy are fundamental concepts in physics, describing how forces cause motion and how energy is transferred or transformed.

• Work: The product of force and the distance moved in the direction of the force. $\text{Work} = \text{Force} \times \text{distance}$

• Gravitational Potential Energy: The energy an object possesses due to its position in a gravitational field. $U = mgh$ where $m$ is mass, $g$ is acceleration due to gravity, and $h$ is height above a reference point.

• Kinetic Energy (KE): The energy of motion. $KE = \frac{1}{2}mv^2$

• Work-Energy Theorem: The net work done on an object is equal to its change in kinetic energy. $\text{Work} = \Delta KE$

• Law of Conservation of Energy: Energy cannot be created or destroyed, only transformed from one form to another.

Example: Lifting a box increases its gravitational potential energy; dropping it converts this energy to kinetic energy.

Simple Machines

Simple machines make work easier by changing the magnitude or direction of a force. Examples include levers, pulleys, and inclined planes.

• Mechanical Advantage: The ratio of output force to input force in a machine.

Circular and Rotational Motion

Circular Motion

When an object turns about an internal axis, it undergoes circular motion or rotation. Circular motion is characterized by two types of speed:

• Tangential (Linear) Speed ($v$): The speed of a point on the circumference, tangent to the circle.

• Rotational (Angular) Speed ($\omega$): The number of rotations per unit time.

All points on a rotating object have the same angular speed, but points farther from the axis have greater tangential speed.

• Relation: $v = r\omega$ where $r$ is the radial distance from the axis.

Example: On a spinning merry-go-round, a child sitting farther from the center moves faster (in terms of linear speed) than one near the center, though both complete a rotation in the same time.

Rotational Inertia (Moment of Inertia)

Rotational inertia is the property of an object to resist changes in its rotational motion. It depends on the mass of the object and how that mass is distributed relative to the axis of rotation.

• Symbol: $I$

• Greater rotational inertia means it is harder to change the rotational state (start, stop, or change speed).

• Depends on:

  • Mass of the object

  • Distribution of mass (distance from axis)

  • Axis about which it rotates

Example: A tightrope walker uses a long pole to increase rotational inertia and maintain balance.

Comparison: A hoop and a disk of the same mass and radius released from an incline: the disk (lower rotational inertia) reaches the bottom first.

Torque and Rotational Dynamics

Torque

Torque is a measure of how much a force acting on an object causes that object to rotate. It is the rotational analogue of force.

• Formula: $\tau = rF$ (when force is perpendicular to the lever arm)

• Lever Arm: The shortest distance from the axis of rotation to the line along which the force acts.

• Unit: Newton-meter (N·m)

Example: Opening a door is easier when pushing at the edge farthest from the hinges (longer lever arm).

Seesaw and Rotational Equilibrium

A seesaw balances when the torques produced by the weights on either side of the pivot are equal in magnitude but opposite in direction.

• Rotational Equilibrium: Occurs when the sum of clockwise torques equals the sum of counterclockwise torques.

Center of Mass and Center of Gravity

Center of Mass

The center of mass is the point in a system or object where the mass can be considered to be concentrated for the purpose of analyzing translational motion.

• For a system of particles: $\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$

Example: A thrown baseball bat wobbles about its center of mass, which follows a smooth parabolic trajectory.

Center of Gravity

The center of gravity is the point where the total gravitational force (weight) can be considered to act. If gravity is uniform, the center of gravity coincides with the center of mass.

• Stability: An object is stable if a vertical line from its center of gravity falls within its base.

Centripetal Force and Circular Motion

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, always directed toward the center of the circle.

• Formula: $F_c = \frac{mv^2}{r}$ where $m$ is mass, $v$ is tangential speed, and $r$ is radius.

Example: A stone tied to a string and whirled in a circle experiences a centripetal force provided by the tension in the string.

Angular Momentum and Its Conservation

Angular Momentum

Angular momentum is the rotational analogue of linear momentum. It is a measure of the quantity of rotation of an object and is conserved in the absence of external torques.

• Formula: $L = I\omega$ where $I$ is rotational inertia and $\omega$ is angular velocity.

• Law of Conservation of Angular Momentum: If no external net torque acts on a system, its angular momentum remains constant. $I_i\omega_i = I_f\omega_f$

Example: A figure skater spins faster by pulling in her arms, reducing $I$ and increasing $\omega$ to conserve $L$.

Gravitation

Newton's Law of Universal Gravitation

Every mass attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

• Formula: $F = G\frac{m_1 m_2}{d^2}$ where $G = 6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2$ is the universal gravitational constant.

Distinction: $G$ is the universal constant; $g$ is the acceleration due to gravity at Earth's surface ($g \approx 9.80\ \text{m/s}^2$).

• Weight at Earth's Surface: $F = mg = G\frac{M_e m}{R_e^2}$ where $M_e$ is Earth's mass and $R_e$ is Earth's radius.

• At height $h$ above Earth's surface: $g(h) = G\frac{M_e}{(R_e + h)^2}$

Summary Table: Key Rotational and Gravitational Quantities

Quantity	Symbol	Formula	Unit
Work	W	$W = Fd$	Joule (J)
Kinetic Energy	KE	$KE = \frac{1}{2}mv^2$	Joule (J)
Potential Energy	U	$U = mgh$	Joule (J)
Torque	$\tau$	$\tau = rF$	N·m
Rotational Inertia	I	Depends on mass distribution	kg·m2
Angular Momentum	L	$L = I\omega$	kg·m2/s
Centripetal Force	$F_c$	$F_c = \frac{mv^2}{r}$	Newton (N)
Gravitational Force	F	$F = G\frac{m_1 m_2}{d^2}$	Newton (N)

Additional info: Some equations and explanations have been expanded for clarity and completeness. Examples and applications have been added to illustrate key concepts.