Review Study Notes: Linear Momentum, Collisions, Rotational Motion, and Gravity (Chapters 9–12)

Linear Momentum & Collisions

Conservation of Linear Momentum

Linear momentum is a fundamental concept in physics, defined as the product of an object's mass and velocity. In an isolated system, the total linear momentum remains constant unless acted upon by external forces.

• Definition: Linear momentum, p, is given by $p = m v$.

• Conservation Law: In the absence of external forces, $\sum p_{\text{initial}} = \sum p_{\text{final}}$.

• Application: Used to analyze collisions and explosions.

• Example: Two blocks colliding on a frictionless surface; total momentum before equals total momentum after.

Types of Collisions

Collisions are classified based on whether kinetic energy is conserved.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

• Perfectly Inelastic Collision: Colliding objects stick together after collision.

• Equation for Elastic Collisions: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$ $\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

• Example: Determining the character of a collision using masses and velocities before and after.

Impulse

Impulse is the change in momentum resulting from a force applied over a time interval.

• Definition: $J = \Delta p = F \Delta t$

• Application: Used to calculate the effect of forces during collisions.

• Example: Calculating the impulse delivered to a block during a collision.

Rotational Kinematics and Dynamics

Rotational Motion

Rotational motion involves objects spinning about an axis. Key quantities include angular velocity, angular acceleration, and moment of inertia.

• Angular Velocity: $\omega = \frac{\Delta \theta}{\Delta t}$

• Moment of Inertia: $I = \sum m_i r_i^2$ (for discrete masses)

• Rotational Kinetic Energy: $K_{\text{rot}} = \frac{1}{2} I \omega^2$

• Example: Calculating the angular speed of a wheel given its diameter and linear speed.

Torque and Equilibrium

Torque is the rotational equivalent of force, causing objects to rotate. Static equilibrium occurs when the sum of torques and forces on an object is zero.

• Torque: $\tau = r F \sin \theta$

• Equilibrium Condition: $\sum F = 0$, $\sum \tau = 0$

• Application: Used to analyze balances, levers, and rotating systems.

• Example: Determining the mass required to balance a meter stick.

Rotational Impulse and Angular Momentum

Angular momentum is conserved in the absence of external torques.

• Angular Momentum: $L = I \omega$

• Conservation Law: $L_{\text{initial}} = L_{\text{final}}$

• Example: A child jumping onto a merry-go-round and calculating the final angular speed.

Gravity

Newton's Law of Universal Gravitation

Gravity is the attractive force between two masses, described by Newton's law.

• Equation: $F_g = G \frac{m_1 m_2}{r^2}$

• Gravitational Acceleration: $g = \frac{G M}{r^2}$

• Application: Used to calculate the force between planets, or the acceleration due to gravity on different celestial bodies.

• Example: Comparing gravitational acceleration on Venus and Earth.

Orbits and Circular Motion

Objects in orbit experience a centripetal force provided by gravity.

• Centripetal Force: $F_c = \frac{m v^2}{r}$

• Orbital Speed: $v = \sqrt{\frac{G M}{r}}$

• Example: Calculating the orbital speed of a planet or satellite.

Sample Table: Types of Collisions

Type	Momentum Conserved?	Kinetic Energy Conserved?	Objects Stick Together?
Elastic	Yes	Yes	No
Inelastic	Yes	No	Sometimes
Perfectly Inelastic	Yes	No	Yes

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Equations and examples are based on standard physics curriculum for chapters 9–12.