Physics Final Exam Study Guide: Vectors, Kinematics, Forces, Energy, Momentum, and Rotation

Final Exam Overview

Exam Structure and Preparation

• Comprehensive Coverage: The exam covers all major topics in introductory physics, with an equal distribution of questions across topics.

• Format: Multiple choice, 80–90 questions.

• Preparation Tips: Create your own formula sheet and practice problems for each formula.

Topic Emphasis

Key Topics for Review

More Important	Less Important
Vectors	Rotation
Force Diagrams	Moment of Inertia
Equations of Motion in 1D	Waves
Energy	2-D Motion
Momentum and Impulse
Torque

Vectors

Definition and Properties

• Vectors have both magnitude and direction.

• In two dimensions, a vector is described by two components (e.g., $v_x$, $v_y$).

Trigonometric Relationships

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Analytical Method for Vector Addition

• Find components for each vector:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

  • $B_x = B \cos \theta$

  • $B_y = B \sin \theta$

• Add components:

  • $C_x = A_x + B_x$

  • $C_y = A_y + B_y$

• Magnitude: $|\vec{C}| = \sqrt{C_x^2 + C_y^2}$

Example

• If $A_x = -5.0$ and $A_y = 12.6$, then $|\vec{A}| = \sqrt{(-5.0)^2 + (12.6)^2} = 13.6$

Kinematics

Equations of Motion in 1D

Relationship	Equation
Velocity-Time	$v_f = v_0 + a t$
Displacement-Time	$x = x_0 + v_0 t + \frac{1}{2} a t^2$
Velocity-Displacement	$v_f^2 = v_0^2 + 2a(x - x_0)$

Graphical Analysis

• Slope of x vs. t graph: Represents velocity.

• Slope of v vs. t graph: Represents acceleration.

• Area under v vs. t graph: Represents displacement.

Example

• Dropping a ball from a tower: Use $x = \frac{1}{2} g t^2$ to find distance fallen in 3.0 s.

Projectile and 2-D Motion

Principles

• Projectile motion involves constant velocity in the horizontal direction and constant acceleration (gravity) in the vertical direction.

• Vertical and horizontal motions are independent.

Example

• At the top of a projectile's trajectory, the vertical velocity is zero, but the horizontal velocity remains constant.

Force Diagrams

Free-Body Diagrams (FBD)

• Show all forces acting on an object.

• Apply Newton's Second Law: $\sum F = m a$

• For a box on a table: $N = w = m g$ (normal force equals weight)

System of Blocks Example

• For each block, write equations for forces in x and y directions.

• Example:

  • Block 2: $\sum F_x = T = m_2 a$, $\sum F_y = N_2 - w_2 = 0$

  • Block 1: $\sum F_x = F - T = m_1 a$, $\sum F_y = N_1 - w_1 = 0$

Friction

Types of Friction

• Static friction prevents objects from sliding: $f_s \leq \mu_s N$

• Kinetic friction slows down sliding objects: $f_k = \mu_k N$

Period and Frequency

Uniform Circular Motion

• Period (T): Time for one complete revolution.

• Frequency (f): Number of revolutions per unit time: $f = \frac{1}{T}$

• Speed: $v = \frac{2 \pi r}{T} = 2 \pi r f$

• Angular velocity: $|\omega| = \frac{v}{r} = 2 \pi f$

Example

• Earth's angular speed: $|\omega| = \frac{2 \pi \text{ rad}}{86400 \text{ s}} = 7.3 \times 10^{-5} \text{ rad/s}$

Work and Energy

Work Done by a Constant Force

• Work: $W = F \Delta r \cos \theta$

• Work can be positive, negative, or zero depending on the angle between force and displacement.

• Normal force does no work if perpendicular to displacement.

Example

• Lifting a chest: $W = 1400 \text{ N} \times 1.0 \text{ m} \times \cos 0 = 1400 \text{ J}$

Total Work

• Total work is the sum of work done by all forces: $W_{\text{total}} = W_1 + W_2 + \cdots + W_n$

• For a rigid object: $W_{\text{total}} = F_{\text{net}} \Delta r \cos \theta$

Kinetic Energy

• Translational kinetic energy: $K = \frac{1}{2} m v^2$

• Work-kinetic energy theorem: $W_{\text{total}} = \Delta K$

Momentum and Impulse

Definitions and Applications

• Momentum: $p = m v$

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

• Impulse changes momentum; increasing the time over which a force acts reduces the force (e.g., airbags in cars).

Example

• Car crash: $p = 65 \text{ kg} \times 20.0 \text{ m/s} = 1300 \text{ kg} \cdot \text{m/s}$

• Force without seatbelt: $F = \frac{1300}{0.003} = 4.3 \times 10^5 \text{ N}$

• Force with airbag: $F = \frac{1300}{0.03} = 4.3 \times 10^4 \text{ N}$

Torque and Moment of Inertia

Rotational Equilibrium and Comparison

• Torque: The tendency of a force to rotate an object about an axis.

• Moment of inertia: A measure of an object's resistance to changes in rotational motion.

• Comparing masses and lever arms affects rotational balance.

Example

• Balancing a rod: If cut at the center, the masses are equal; if cut elsewhere, the lever arm affects balance.

Additional info:

• Some questions and examples are provided in multiple-choice format to reinforce concepts.

• Students should practice interpreting graphs and diagrams, as these are common in physics exams.