Comprehensive Study Guide: Key Topics for the Physics Final Exam

Oscillations and Simple Harmonic Motion

Oscillations of a Spring

• hed to a spring with $k = 200$ N/m oscillates with $T = 2\pi\sqrt{0.5/200} \approx 0.314$ s.

General Simple Harmonic Motion (Kinematics and Dynamics)

• Displacement: $x(t) = A\cos(\omega t + \phi)$

• Velocity: $v(t) = -A\omega\sin(\omega t + \phi)$

• Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi)$

• Energy in SHM: $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$

Pendulums

• Simple Pendulum: A mass $m$ on a string of length $L$ oscillates with period $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles).

• Physical Pendulum: $T = 2\pi\sqrt{\frac{I}{mgd}}$, where $I$ is the moment of inertia and $d$ is the distance from pivot to center of mass.

Gravitation and Orbital Motion

Universal Gravitational Force

• Newton's Law of Universal Gravitation: $F = G\frac{m_1 m_2}{r^2}$

• Gravitational Constant: $G = 6.674 \times 10^{-11}$ N·m$^2$/kg$^2$

Orbital Motion

• Circular Orbit Speed: $v = \sqrt{\frac{GM}{r}}$

• Orbital Period: $T = 2\pi\sqrt{\frac{r^3}{GM}}$

Third Kepler’s Law

• Kepler's Third Law: $\frac{T^2}{r^3} = \text{constant}$ for all planets orbiting the same star.

• Application: Used to compare orbital periods and radii of planets.

Apparent Weight

• Definition: The normal force exerted on an object, which may differ from true weight due to acceleration.

• Formula: $N = m(g \pm a)$, where $a$ is the acceleration of the system.

• Example: In an elevator accelerating upward, apparent weight increases.

Rotational Motion and Equilibrium

Static Equilibrium

An object is in static equilibrium if the net force and net torque on it are zero.

• Conditions: $\sum \vec{F} = 0$, $\sum \tau = 0$

• Application: Used to analyze structures, beams, and supports.

Newton's Second Law for Torque

• Rotational Analog: $\sum \tau = I\alpha$

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

Conservation of Angular Momentum

• Law: $L = I\omega$ is conserved if net external torque is zero.

• Formula: $I_1\omega_1 = I_2\omega_2$

• Example: Figure skater spins faster by pulling in arms (reducing $I$ increases $\omega$).

Rolling Without Slipping

• Condition: $v_{cm} = R\omega$

• Application: Used for wheels, cylinders, and spheres rolling on surfaces.

Rotational Kinematics

• Equations:

  • $\omega = \omega_0 + \alpha t$

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Moment of Inertia

• Definition: $I = \sum m_ir_i^2$ (discrete masses), $I = \int r^2 dm$ (continuous mass)

• Examples:

  • Solid cylinder: $I = \frac{1}{2}MR^2$

  • Thin rod (center): $I = \frac{1}{12}ML^2$

Parallel Axis Theorem

• Theorem: $I = I_{cm} + Md^2$, where $d$ is the distance from the center of mass axis to the new axis.

• Application: Used to find $I$ about axes not through the center of mass.

Linear Momentum and Collisions

1D Collisions and Linear Momentum Conservation

• Law of Conservation of Momentum: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$

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

• Inelastic Collisions: Only momentum is conserved.

Work and Energy

Law of Mechanical Energy Conservation

• Statement: In the absence of non-conservative forces, total mechanical energy is conserved.

• Formula: $K_i + U_i = K_f + U_f$

• Where: $K$ is kinetic energy, $U$ is potential energy.

Work-Energy Theorem

• Theorem: The net work done on an object equals its change in kinetic energy.

• Formula: $W_{net} = \Delta K = K_f - K_i$

Forces and Newton's Laws

Newton's Laws for Forces

• First Law (Inertia): An object remains at rest or in uniform motion unless acted on by a net force.

• Second Law: $\sum \vec{F} = m\vec{a}$

• Third Law: For every action, there is an equal and opposite reaction.

• Free-Body Diagrams (FBDs): Essential for analyzing forces acting on a body.

Additional info: Students are expected to show all steps, start from fundamental formulas, and include diagrams such as FBDs in their solutions.