Fundamental Concepts and Problems in Classical Mechanics

Motion in One Dimension

Kinematic Equations

Motion in one dimension describes the movement of objects along a straight line, characterized by position, velocity, and acceleration.

• Position as a function of time: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity as a function of time: $v = v_0 + a t$

• Velocity as a function of position: $v^2 = v_0^2 + 2a(x - x_0)$

• Constant acceleration: Acceleration $a$ remains unchanged over time.

Example: A car accelerates from rest at $2\,\text{m/s}^2$ for $5$seconds. Its final velocity is$v = 0 + 2 \times 5 = 10\,\text{m/s}$.

Circular Motion and Centripetal Acceleration

Centripetal Acceleration

Objects moving in a circle experience an acceleration directed toward the center of the circle, called centripetal acceleration.

• Formula: $a_c = \frac{v^2}{r}$

• Where: $v$ is the speed of the object, $r$ is the radius of the circle.

Example: A ball moving at $4\,\text{m/s}$ in a circle of radius $2\,\text{m}$ has $a_c = \frac{16}{2} = 8\,\text{m/s}^2$.

Quadratic Equation

The quadratic equation is used to solve for variables in kinematic equations and other physics problems.

• General form: $ax^2 + bx + c = 0$

• Solution: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Spring Force

Hooke's Law

The force exerted by a spring is proportional to its displacement from equilibrium.

• Formula: $F = -kx$

• Where: $k$ is the spring constant, $x$ is the displacement.

Gravitational Force

Newton's Law of Universal Gravitation

Describes the attractive force between two masses.

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

• Where: $G$ is the gravitational constant, $m_1$ and $m_2$ are masses, $r$ is the distance between centers.

Center of Mass

The center of mass of a system is the weighted average of the positions of all the masses.

• Formula: $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

Rotational Dynamics

Rotational Form of Newton's Second Law

Relates torque to angular acceleration.

• Formula: $\tau = I \alpha$

• Where: $\tau$ is torque, $I$ is moment of inertia, $\alpha$ is angular acceleration.

Rolling Without Slipping

Describes the motion of objects that roll without sliding.

• Relationship: $v = r \omega$

• Where: $v$ is linear speed, $r$ is radius, $\omega$ is angular speed.

Tensile/Compressive Stress

Stress is the force per unit area applied to a material.

• Formula: $\text{Stress} = \frac{F}{A}$

• Where: $F$ is force, $A$ is cross-sectional area.

Selected Physics Problems and Solutions

Orbital Mechanics and Gravitational Force

Problems involving gravitational force, orbital radius, and mass calculations for celestial bodies.

Body	Mass	Radius	Orbital radius	Orbital period
Moon A	4.0 × 1020 kg	unknown	2.0 × 108 m	4.0 × 105 s
Moon B	1.5 × 1020 kg	2.0 × 105 m	3.0 × 108 m	unknown
Mithra	unknown	unknown	3.0 × 108 m	unknown

Application: Use $F = G \frac{m_1 m_2}{r^2}$ and orbital period formulas to solve for unknowns.

Rotational Motion and Torque

Problems involving angular acceleration, torque, and moment of inertia.

• Torque: $\tau = r F \sin \theta$

• Angular acceleration: $\alpha = \frac{\tau}{I}$

Spring and Elasticity Problems

Problems involving spring stretching and Young's modulus.

• Spring stretch: $F = kx$

• Young's modulus: $Y = \frac{F/A}{\Delta L/L}$

Statics and Equilibrium

Problems involving forces, tension, and equilibrium of rigid bodies.

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

• Application: Used to solve for unknown forces and tensions in signs, ladders, and beams.

Sample Problem: Ladder Against Wall

A ladder of length $3.00\,\text{m}$ and mass $20\,\text{kg}$ is supported horizontally. Find the tension in the supporting wire.

• Solution: Use torque equilibrium about the wall attachment point.

Sample Problem: Teeter-Totter

A child sits on a teeter-totter. Find the minimum distance for equilibrium given a force applied on the opposite side.

• Solution: Set torques about the pivot equal for equilibrium.

Additional info:

• Some problems involve conversion between angular and linear quantities: $v = r \omega$.

• Stress and strain concepts are applied in elasticity problems.

• All equations are standard for introductory college physics.