Physics Exam Review: Momentum, Work & Energy, Rotational Motion, Gravitation, and Kinematics

Chapter 6: Momentum

Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is directly proportional to both the mass and velocity of the object.

• Definition: The product of mass and velocity.

• Formula: $p = mv$

• Units: kg·m/s

• Example: A 2 kg ball moving at 3 m/s has a momentum of $p = 2 \times 3 = 6$ kg·m/s.

Impulse

Impulse quantifies the effect of a force acting over a period of time, resulting in a change in momentum.

• Definition: The result of a force acting on an object for an elapsed time.

• Formula: $F t = \Delta p$

• Units: N·s (Newton-seconds)

• Example: If a force of 10 N acts for 2 seconds, the impulse is $10 \times 2 = 20$ N·s.

Conservation of Momentum

If no external forces act on a system, the total momentum of the system remains constant.

• Formula: $\sum p_{\text{initial}} = \sum p_{\text{final}}$

• Application: Used to analyze collisions and explosions.

Collisions

Collisions are interactions between two or more objects where momentum is transferred.

• Elastic Collisions: Objects rebound without deformation or heat generation.

• Inelastic Collisions: Objects deform, generate heat, and may stick together.

• Example: Billiard balls colliding (elastic); clay balls sticking together (inelastic).

Work, Energy, and Power

Work

Work is the process of energy transfer to an object via a force causing displacement.

• Definition: The effort required to change the energy of an object.

• Formula: $W = F d$

• Units: Joules (J)

• Example: Lifting a 5 kg box 2 meters: $W = 5 \times 9.8 \times 2 = 98$ J.

Work-Energy Theorem

The net work done on an object is equal to the change in its kinetic energy.

• Formula: $W = \Delta KE$

Power

Power measures the rate at which work is done or energy is transferred.

• Formula: $\text{Power} = \frac{\text{work done}}{\text{time interval}}$

• Units: Watts (W)

Energy

Energy is the capacity to do work. It exists in various forms, such as kinetic and potential energy.

• Potential Energy (PE): Energy stored due to position. For gravitational potential energy: $PE = mgh$

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

Conservation of Energy

Energy cannot be created or destroyed; it can only be transformed from one form to another.

• Key Point: The total amount of energy in a closed system remains constant.

Machines and Efficiency

Machines are devices that change the direction or magnitude of a force. Efficiency measures how much input energy is converted into useful work.

• Formula: $\text{efficiency} = \frac{\text{useful energy output}}{\text{total energy input}} \times 100\%$

Linear and Rotational Motion

Comparison Table: Linear vs. Rotational Motion

This table compares analogous quantities in linear and rotational motion.

Linear	Rotational (R is radial distance)
Position, x	Angle, θ
Displacement, Δx	Rotational displacement, Δθ
Velocity (tangential), v = Δx/Δt	Rotational velocity, ω = Δθ/Δt
Acceleration (tangential), a = Δv/Δt	Rotational acceleration, α = Δω/Δt
Force, F	Torque, τ = F × r (perpendicular lever arm)
Inertia (based on mass)	Rotational inertia, I (based on mass, shape, and rotational axis)
Momentum, p = mv	Angular momentum, L = Iω (for small objects compared to radius, L = pR = mvR)

Centripetal Force

Centripetal force is the force directed toward the center of a circular path, keeping an object in circular motion.

• Formula: $F_{cp} = \frac{mv^2}{R}$

Conservation of Angular Momentum

Angular momentum is conserved unless acted upon by a net external torque.

• Key Point: $L_{\text{initial}} = L_{\text{final}}$ if net torque is zero.

Center of Mass (CM)

The center of mass is the average position of the mass of an object. The CM moves as if all forces act at that point.

Gravitation

Law of Universal Gravitation

Every body attracts every other body with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Formula: $F = G \frac{m_1 m_2}{d^2}$

• Where: $G = 6.67 \times 10^{-11} \text{N} \cdot \text{m}^2 / \text{kg}^2$

Weight

Weight is the force exerted by gravity on an object.

• Formula: $W = mg$

Weightlessness

Weightlessness occurs when there is no support force acting on a body, such as in free fall.

Tides

Tides are caused by differences in gravitational forces from the moon and sun on Earth's oceans, leading to periodic changes in ocean levels.

Gravitational Fields

A gravitational field is a region of space where a mass experiences a force due to gravity.

• Formula: $g_1 = \frac{F_{12}}{m_2}$

Kinematics and Kepler's Laws

Kinematics Equations

Kinematics describes motion under constant acceleration. These equations can be applied to one or more dimensions independently.

• Acceleration due to gravity: $g = 9.8 \frac{m}{s^2}$

• Displacement: $d_{\text{traveled}} = \frac{1}{2} a t^2$

• Velocity: $v = v_0 + a t$

• Average velocity: $\text{average velocity} = \frac{v + v_0}{2}$

• Speed: $\text{speed} = \frac{\text{distance traveled}}{\text{time elapsed}}$

• Velocity (general): $\vec{v} = \frac{\Delta x}{\Delta t}$

• Acceleration (general): $\text{acceleration} = \frac{\text{change in velocity}}{\text{change in time}}$

Kepler's Laws

Kepler's Laws describe planetary motion in the solar system.

1. The path of each planet around the Sun is an ellipse with the Sun at one focus.

2. The line from the Sun to any planet sweeps out equal areas in equal time intervals.

3. The square of the orbital period of a planet is directly proportional to the cube of the average distance from the Sun ($T^2 \propto r^3$ for all planets).