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

Final Exam Overview

This study guide summarizes key topics for a comprehensive introductory college physics final exam. The exam covers a broad range of fundamental concepts, with emphasis on vectors, kinematics, force diagrams, energy, momentum, impulse, and torque. Both conceptual understanding and problem-solving skills are required.

Topic Emphasis

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

Vectors

Definition and Properties

Vectors are quantities that have both magnitude and direction. In two dimensions, a vector is fully described by its components along the x and y axes.

• Vector Notation: v (bold) denotes a vector; |v| or v denotes its magnitude.

• Vectors in 2D require two pieces of information: magnitude and direction (or x and y components).

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 the x and y 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 the components:

  • $C_x = A_x + B_x$

  • $C_y = A_y + B_y$

• Magnitude of resultant: $C = \sqrt{C_x^2 + C_y^2}$

Example:

If $A_x = 5.0$ and $A_y = 12.0$, then $|\vec{A}| = \sqrt{5.0^2 + 12.0^2} = 13.0$.

Kinematics in One Dimension

Kinematic Equations

These equations describe motion with constant acceleration:

Relationship	Equation
velocity-time	$v_x = v_{0x} + a_x t$
displacement-time	$x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$
velocity-displacement	$v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$

Example:

You drop a ball from rest. How far does it fall in 2.0 s? Use $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$ with $a_x = -9.8$ m/s2, $v_{0x} = 0$:

$x = 0 + 0 + \frac{1}{2}(-9.8)(2.0)^2 = -19.6$ m

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:

Area under a speed-time curve from 0 to 4 s (triangle): $\frac{1}{2} \times 4 \times 8 = 16$ m

Projectile and 2-D Motion

Projectile Motion

Projectile motion involves constant velocity in the horizontal direction ($a_x = 0$) and constant acceleration in the vertical direction ($a_y = -g$).

• Horizontal and vertical motions are independent.

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

Force Diagrams (Free-Body Diagrams)

Drawing and Analyzing Forces

• Represent all forces acting on an object as arrows.

• Common forces: weight ($w = mg$), normal force ($N$), tension ($T$), friction ($f$).

• Newton's 2nd Law: $\sum F = ma$

Example:

For a box on a table: $N = w = mg$ (normal force balances weight).

Friction

Friction is a contact force parallel to the surfaces in contact.

• Static friction prevents motion: $f_s \leq \mu_s N$

• Kinetic friction opposes sliding: $f_k = \mu_k N$

Systems of Forces

Equations for Multiple Objects

• Draw a free-body diagram for each object.

• Apply Newton's 2nd Law to each:

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

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

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 in a circle: $v = \frac{2\pi r}{T} = 2\pi r f$

• Angular velocity: $|\omega| = \frac{v}{r} = 2\pi f$ (in rad/s)

Example:

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

Work and Energy

Work Done by a Constant Force

• $W = F \Delta r \cos \theta$ (θ is the angle between force and displacement)

• Work can be positive, negative, or zero.

• Normal force does no work if perpendicular to displacement.

Example:

Lifting a 1400 N chest 1.0 m vertically: $W = 1400 \times 1.0 \times \cos 0 = 1400$ J

Total Work

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

• For a rigid object with no rotation: $W_{\text{total}} = F_{\text{net}} \Delta r \cos \theta$

Kinetic Energy and the Work-Energy Theorem

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

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

Example:

If the total work done on an object is positive, its speed increases.

Momentum and Impulse

Impulse-Momentum Theorem

• Impulse: $J = F \Delta t$

• Change in momentum: $\Delta p = m \Delta v$

• Impulse equals change in momentum: $F \Delta t = m \Delta v$

Example:

A 65 kg passenger in a car moving at 20.0 m/s comes to rest in 0.0030 s (no seat belt):

• Initial momentum: $p = 65 \times 20.0 = 1300$ kg·m/s

• Force: $F = \frac{\Delta p}{\Delta t} = \frac{1300}{0.0030} = 4.3 \times 10^5$ N

• With airbag (0.030 s): $F = \frac{1300}{0.030} = 4.3 \times 10^4$ N

Extending the time of impact reduces the force experienced.

Torque and Moment of Inertia

Torque

• Torque (τ): The rotational equivalent of force, causing objects to rotate.

• Depends on force magnitude, direction, and distance from pivot.

Moment of Inertia

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

• Depends on mass distribution relative to the axis of rotation.

Example:

Comparing masses on a balance: If a rod is cut and balanced, the side with the smaller mass must be farther from the pivot to balance the torque.

Additional info: Some slides included multiple-choice questions and answers for practice. The notes above expand on the main concepts and provide context for exam preparation.