Physics 7A: Core Concepts and Problem-Solving Guide

Units, Kinematics, and Dynamics

Units and Measurements

Physics relies on precise measurements and the use of standard units. The SI system is the most widely used, with base units such as meters (m) for length, kilograms (kg) for mass, and seconds (s) for time.

• Key Point: Always include units in calculations and final answers.

• Key Point: Dimensional analysis helps verify the correctness of equations.

One- and Two-Dimensional Kinematics

Kinematics describes the motion of objects without considering the forces causing the motion. It includes displacement, velocity, and acceleration in one or more dimensions.

• Displacement: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity: $v = v_0 + a t$

• Projectile motion: Involves both horizontal and vertical components, with gravity acting downward.

• Circular motion: Objects moving in a circle experience centripetal acceleration $a_c = \frac{v^2}{r}$.

Forces, Energy, and Momentum

Newton's Laws of Motion

Newton's laws describe the relationship between forces and motion.

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

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

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

Work, Energy, and Power

Work is done when a force causes displacement. Energy is the capacity to do work, and power is the rate of doing work.

• Work: $W = \vec{F} \cdot \vec{d}$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy (gravity): $U = mgh$

• Conservation of Energy: $E_{\text{initial}} = E_{\text{final}}$ (if no non-conservative forces)

• Power: $P = \frac{dW}{dt}$

Momentum and Collisions

Momentum is a measure of an object's motion, and is conserved in isolated systems.

• Linear Momentum: $\vec{p} = m \vec{v}$

• Impulse: $\vec{J} = \Delta \vec{p}$

• Conservation of Momentum: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

Rotational Motion and Oscillations

Rotational Kinematics and Dynamics

Rotational motion involves angular displacement, velocity, and acceleration. The rotational analogs of Newton's laws apply.

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

• Angular Velocity: $\omega = \omega_0 + \alpha t$

• Moment of Inertia: $I = \sum m_i r_i^2$ (depends on mass distribution)

• Torque: $\tau = I \alpha$

• Rotational Kinetic/

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

Simple Harmonic Motion (SHM)

SHM describes oscillatory motion, such as a mass on a spring or a pendulum.

• Equation of Motion: $x(t) = A \cos(\omega t + \phi)$

• Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$

• Period: $T = \frac{2\pi}{\omega}$

• Mechanical Energy in SHM: $E = \frac{1}{2} k A^2$

Fluid Dynamics

Properties of Fluids and Bernoulli's Equation

Fluids are substances that flow and take the shape of their container. Fluid dynamics studies the motion of fluids and the forces on them.

• Continuity Equation: $A_1 v_1 = A_2 v_2$ (for incompressible fluids)

• Bernoulli's Equation: $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

• Pressure in a Fluid: $P = P_0 + \rho g h$

Waves and Sound

Standing Waves and Resonance

Standing waves are formed by the interference of two waves traveling in opposite directions. Resonance occurs at specific frequencies, depending on boundary conditions.

• Open-Open Tube: $f_n = n \frac{v}{2L}$, $n = 1, 2, 3, ...$

• Open-Closed Tube: $f_n = n \frac{v}{4L}$, $n = 1, 3, 5, ...$

• Wave Speed: $v = f \lambda$

Gravitation and Universal Constants

Newton's Law of Universal Gravitation

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

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

• Gravitational Constant: $G$ (measured using a torsion balance)

Key Equations and Reference Table

Summary Table of Key Equations

Topic	Equation	Description
Kinematics	$x = x_0 + v_0 t + \frac{1}{2} a t^2$	Position as a function of time
Newton's 2nd Law	$\sum \vec{F} = m \vec{a}$	Force and acceleration
Work	$W = \vec{F} \cdot \vec{d}$	Work done by a force
Kinetic Energy	$K = \frac{1}{2} m v^2$	Energy of motion
Potential Energy	$U = mgh$	Gravitational potential energy
Momentum	$\vec{p} = m \vec{v}$	Linear momentum
Impulse	$\vec{J} = \Delta \vec{p}$	Change in momentum
Torque	$\tau = r F \sin \theta$	Rotational force
Moment of Inertia	$I = \sum m_i r_i^2$	Rotational inertia
SHM Frequency	$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$	Oscillator frequency
Bernoulli's Eq.	$P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$	Fluid energy conservation
Gravitation	$F = G \frac{m_1 m_2}{r^2}$	Universal gravitation

Rotational Inertia Reference

Object	Moment of Inertia $I$
Hoop or cylindrical shell	$MR^2$
Hollow cylinder or disk	$M \frac{(R_1^2 + R_2^2)}{2}$
Solid cylinder or disk	$\frac{1}{2} MR^2$
Rectangular plate (axis at edge)	$\frac{1}{3} ML^2$
Long thin rod (axis at center)	$\frac{1}{12} ML^2$
Long thin rod (axis at end)	$\frac{1}{3} ML^2$
Solid sphere	$\frac{2}{5} MR^2$
Thin spherical shell	$\frac{2}{3} MR^2$

Sample Applications and Problem Types

• Satellite in Orbit: Use conservation of angular momentum and energy to analyze elliptical orbits and calculate speeds and torques.

• Standing Waves in Tubes: Identify boundary conditions (open/closed), calculate frequencies, and relate harmonics.

• Torsion Balance: Apply rotational equilibrium and oscillation concepts to measure the gravitational constant $G$.

• Fluid Flow and Manometers: Use Bernoulli's equation and continuity to find velocities and pressure differences; apply hydrostatics for manometer readings.

• Simple Harmonic Oscillator: Analyze mass-spring systems for frequency, energy, and equations of motion.

• Rolling Motion and Energy Conservation: For rolling disks, use rotational inertia and conservation of energy to find speeds and trajectories.

• Variable Mass Systems: For systems like a water cannon, apply momentum conservation and differential equations to describe motion as mass changes.

Additional info:

• This guide is based on a final exam covering core topics in introductory physics, including kinematics, dynamics, energy, momentum, rotation, fluids, waves, and gravitation.

• Equation sheets and reference tables are essential for solving quantitative problems efficiently.