PHYS 2204 – Final Exam Study Notes: Mechanics, Energy, and Rotational Motion

Vectors and Vector Operations

Vector Addition and Magnitude

Vectors are quantities that have both magnitude and direction. The sum of two vectors is found using the parallelogram rule or by resolving into components.

• Magnitude of a Vector: For a vector V with components Vx and Vy:

$|\vec{V}| = \sqrt{V_x^2 + V_y^2}$

• Example: If $|\vec{Z}| = 16.0$ units and $|\vec{A}| = 7.0$ units, and $\vec{Z} = \vec{A} + \vec{C}$, use vector subtraction and the Pythagorean theorem to find $|\vec{C}|$ if the vectors are perpendicular.

Scalar (Dot) Product

The scalar product of two vectors A and B is:

$\vec{A} \cdot \vec{B} = |A||B|\cos\theta$

• Application: Used to find work done by a force or to determine orthogonality of vectors.

Motion in One and Two Dimensions

Constant and Variable Acceleration

• Constant Acceleration Equations:

$v = v_0 + at$ $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Variable Acceleration: If $a(t)$ is a function of time, integrate to find velocity:

$v(t) = v_0 + \int_0^t a(t') dt'$

Projectile Motion

• Horizontal and vertical motions are independent (neglecting air resistance).

• Horizontal velocity is constant; vertical velocity changes due to gravity.

Newton's Laws of Motion

Force and Free-Body Diagrams

• Newton's Second Law:

$\sum \vec{F} = m \vec{a}$

• Draw all forces acting on an object to analyze its motion.

Friction

• Static Friction: $f_s \leq \mu_s N$

• Kinetic Friction: $f_k = \mu_k N$

Work, Energy, and Power

Work Done by a Force

• Work is the product of force and displacement in the direction of the force:

$W = \vec{F} \cdot \vec{d}$

Kinetic and Potential Energy

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

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

• Conservation of Energy: Total mechanical energy is conserved in the absence of non-conservative forces.

Momentum and Collisions

Linear Momentum

• Definition: $\vec{p} = m\vec{v}$

• Impulse: $\vec{J} = \Delta \vec{p}$

Collisions

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

• Inelastic Collision: Only momentum is conserved.

Rotational Motion

Angular Kinematics and Dynamics

• Angular Velocity: $\omega = \frac{d\theta}{dt}$

• Angular Acceleration: $\alpha = \frac{d\omega}{dt}$

• Moment of Inertia: $I = \sum m_ir_i^2$

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

• Torque: $\tau = rF\sin\theta$

• Newton's Second Law for Rotation: $\sum \tau = I\alpha$

Equilibrium and Elasticity

Conditions for Equilibrium

• First Condition: $\sum F_x = 0$, $\sum F_y = 0$

• Second Condition: $\sum \tau = 0$

Gravitation

Newton's Law of Universal Gravitation

• $F = G\frac{m_1 m_2}{r^2}$

• Used to calculate gravitational force between two masses.

Sample Table: Types of Collisions

Type	Momentum Conserved?	Kinetic Energy Conserved?
Elastic	Yes	Yes
Inelastic	Yes	No
Perfectly Inelastic	Yes	No (objects stick together)

Example Problems and Applications

• Finding the acceleration of an object with time-dependent acceleration: Integrate $a(t)$ over the given time interval.

• Work done by friction: $W_{friction} = f_k d$

• Torque and equilibrium for extended objects: Sum torques about a pivot point and set equal to zero for equilibrium.

• Energy conservation in rotational systems: Equate loss in potential energy to gain in kinetic energy (translational + rotational).

Additional info: These notes cover the main topics addressed in the exam, including vectors, kinematics, Newton's laws, energy, momentum, rotational motion, equilibrium, and gravitation. The problems provided are representative of standard introductory college physics topics.