Physics Final Exam Study Guide: Kinematics, Dynamics, Energy, and Rotational Motion

Introduction to Units

Unit Conversion and Decimal Representation

Physics measurements often use prefixes to denote powers of ten. Converting between units and expressing values in decimal form is essential for clarity and calculation.

• Key Point: 1 micrometer (μm) = $1 imes 10^{-6}$ meters (m).

• Example: 9.8 μm = $9.8 imes 10^{-6}$ m = 0.0000098 m.

Motion in One Dimension

Vertical Motion and Free Fall

Objects in free fall experience constant acceleration due to gravity. When released from a moving object, their initial velocity must be considered.

• Key Point: The position after time $t$ is $y = y_0 + v_0 t - rac{1}{2}gt^2$.

• Key Point: The speed after time $t$ is $v = v_0 - gt$.

• Example: A bag dropped from a balloon moving upward at 10 m/s from 50 m above ground: after 0.5 s, calculate position and speed using above formulas.

Representing Motion

Position-Time Graphs & Instantaneous Velocity

Position-time graphs show how an object's position changes over time. The slope at any point gives the instantaneous velocity.

• Key Point: Instantaneous velocity at time $t$ is the slope of the tangent to the curve at $t$.

• Formula: $v = rac{dx}{dt}$

• Example: If the graph is flat at $t=15$ s, velocity is zero; if sloping upward, velocity is positive; if downward, negative.

Circular Motion, Orbits & Gravity

Uniform Circular Motion and Radial Acceleration

Objects moving in a circle experience radial (centripetal) acceleration. For a planet, if this acceleration exceeds gravity, objects can escape.

• Key Point: Radial acceleration: $a_r = rac{4 heta^2 R}{T^2}$, where $R$ is radius, $T$ is period.

• Key Point: For escape, $a_r > g$.

• Example: Calculate the period $T$ for Mars so that $a_r > 3.7$ m/s$^2$.

Vectors and Motion in Two Dimensions

Projectile Motion from Moving Vehicles

When objects are dropped from moving vehicles, their initial horizontal velocity must be considered. The time to fall depends on vertical distance.

• Key Point: Time to fall: $t = rac{h}{g}$ (for vertical drop).

• Key Point: Horizontal distance: $d = v_{x} t$.

• Example: Helicopter flying at 100 km/h at 150 m altitude: find $t$ and $d$.

Acceleration in 2D

Acceleration and velocity in two dimensions are vector quantities. Their magnitude and direction can be found using vector addition and trigonometry.

• Key Point: Velocity vector: $\vec{v}(t) = \left( \frac{t^2}{2} \hat{i} + (10t - 1.6t^2) \hat{j} \right)$

• Key Point: Magnitude: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$; Direction: $\theta = \tan^{-1}(v_y/v_x)$

Forces & Newton's Laws of Motion

Newton's Third Law & Action-Reaction Pairs

Newton's Third Law states that for every action, there is an equal and opposite reaction. Forces between two objects are equal in magnitude and opposite in direction.

• Key Point: $F_{A ightarrow B} = -F_{B ightarrow A}$

• Example: SUV pushes truck with 2200 N; truck pushes SUV with equal force.

Equilibrium & Elasticity

Equilibrium in 2D and Magnetic Forces

Equilibrium occurs when the sum of forces and torques on a system is zero. Magnetic forces can cause objects to swing or rotate.

• Key Point: For equilibrium: $\sum F = 0$, $\sum \tau = 0$

• Example: Bar magnets repel, causing suspended magnet to swing to a certain angle; use torque balance to find force.

Equilibrium with Multiple Supports

When an object is supported at multiple points, the distribution of forces and torques determines stability.

• Key Point: To prevent toppling, the center of mass must remain within the support base.

• Example: Dog walking on plank: calculate maximum distance before plank tips.

Rotational Motion

Moment of Inertia of Systems

The moment of inertia quantifies an object's resistance to rotational acceleration about an axis. It depends on mass distribution.

• Key Point: For a thin rod of length $L$ and mass $m$ about center: $I = \frac{1}{12}mL^2$

• Key Point: For composite shapes, use the parallel axis theorem: $I = I_{cm} + Md^2$

• Example: Rod bent into V-shape: calculate $I$ about vertex.

Oscillations

Springs and Hooke's Law

Springs obey Hooke's Law, which relates force and displacement. The spring constant $k$ characterizes stiffness.

• Key Point: $F = -kx$

• Key Point: Potential energy stored: $U = \frac{1}{2}kx^2$

• Example: Block compresses spring: find $k$ from force and displacement.

Momentum

Impulse and Conservation of Momentum

Impulse is the change in momentum due to a force over time. In collisions, total momentum is conserved.

• Key Point: $J = F \Delta t = \Delta p$

• Key Point: Conservation: $m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$ (for perfectly inelastic collision)

• Example: Bullet embeds in block: use conservation to find initial speed.

Angular Momentum & Newton's Second Law for Rotation

Angular momentum is conserved in the absence of external torques. The rate of change of angular momentum equals the net torque.

• Key Point: $L = r \times p$

• Key Point: $\tau = \frac{dL}{dt}$

• Example: Marble launched from building: calculate $L$ and $\tau$.

Energy & Work

Elastic Potential Energy

Energy stored in a compressed or stretched spring is elastic potential energy.

• Key Point: $U = \frac{1}{2}kx^2$

• Example: Block dropped onto spring: set gravitational potential energy equal to elastic potential energy to find maximum compression.

Conservation of Energy in Rotational Systems

Work done on a rotating system changes its kinetic energy. For orbital motion, changing radius affects speed and energy.

• Key Point: $W = \Delta K = \frac{1}{2}m(v_f^2 - v_i^2)$

• Example: Space station module pulled inward: calculate work required for new orbit.

Summary Table: Key Formulas

Topic	Formula	Description
Free Fall	$y = y_0 + v_0 t - \frac{1}{2}gt^2$	Vertical position after time $t$
Velocity (Graph)	$v = \frac{dx}{dt}$	Slope of position-time graph
Circular Motion	$a_r = \frac{4\pi^2 R}{T^2}$	Radial acceleration
Projectile Motion	$d = v_x t$	Horizontal distance
Hooke's Law	$F = -kx$	Spring force
Impulse	$J = F \Delta t$	Change in momentum
Moment of Inertia (Rod)	$I = \frac{1}{12}mL^2$	About center
Elastic Potential Energy	$U = \frac{1}{2}kx^2$	Energy stored in spring

Additional info: Some context and formulas have been expanded for completeness and clarity.