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

Units, Physical Quantities & Vectors

Vector Addition and Magnitude

Vectors are quantities that have both magnitude and direction. They can be added using the parallelogram rule or by resolving into components.

• Vector Components: Any vector V can be written as Vx (horizontal) and Vy (vertical) components.

• Magnitude of a Vector: $|\vec{V}| = \sqrt{V_x^2 + V_y^2}$

• Example: If $\vec{C} = \vec{A} - \vec{B}$, find components of each, subtract, then use the magnitude formula.

Motion Along a Straight Line

Velocity and Acceleration

Describes how position and velocity change with time for objects moving in one dimension.

• Constant Acceleration: $v = v_0 + at$

• Variable Acceleration: Integrate acceleration function to find velocity.

• Example: If $a(t) = 3.00\,\text{m/s}^3 t$, then $v(t) = \int a(t) dt$

Motion in Two or Three Dimensions

Projectile Motion

Projectile motion involves both horizontal and vertical components, with gravity acting downward.

• Horizontal Velocity: Remains constant (if air resistance is neglected).

• Vertical Velocity: Changes due to gravity: $v_y = v_{0y} - gt$

• Key Point: The horizontal and vertical motions are independent.

Newton's Laws of Motion & Applications

Newton's Second Law

Relates the net force on an object to its acceleration: $\sum \vec{F} = m\vec{a}$

• Free-Body Diagrams: Essential for analyzing forces acting on objects.

• Friction: $f_s \leq \mu_s N$ (static), $f_k = \mu_k N$ (kinetic)

• Inclined Planes: Resolve forces parallel and perpendicular to the surface.

Work, Energy, and Conservation

Work and Kinetic Energy

Work is done when a force causes displacement. Kinetic energy is the energy of motion.

• Work: $W = \vec{F} \cdot \vec{d}$

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

• Work-Energy Theorem: $W_{\text{net}} = \Delta K$

Potential Energy and Conservation of Energy

• Gravitational Potential Energy: $U = mgh$

• Conservation of Mechanical Energy: $K_i + U_i = K_f + U_f$ (if no non-conservative forces)

Momentum, Impulse, and Collisions

Linear Momentum and Impulse

• Momentum: $\vec{p} = m\vec{v}$

• Impulse: $\vec{J} = \Delta \vec{p} = \int \vec{F} dt$

• Conservation of Momentum: In the absence of external forces, total momentum is conserved.

Collisions

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

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

Rotation of Rigid Bodies & Dynamics of Rotational Motion

Rotational Kinematics

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

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

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

Rotational Dynamics

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

• Work in Rotation: $W = \tau \theta$

• Kinetic Energy of Rotation: $K = \frac{1}{2}I\omega^2$

Equilibrium & Elasticity

Conditions for Equilibrium

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

• Second Condition: $\sum \tau = 0$ (sum of torques must be zero)

• Applications: Ladders, beams, and objects supported by ropes or surfaces.

Sample Table: Forces and Torques in Equilibrium Problems

Quantity	Symbol	Equation	Description
Force	F	$F = ma$	Net force causes acceleration
Torque	$\tau$	$\tau = rF\sin\theta$	Rotational effect of a force
Moment of Inertia	I	$I = \sum m_ir_i^2$	Rotational analog of mass

Additional Topics Covered

• Friction on Inclined Planes: $f = \mu N$

• Work Done by Friction: $W_f = f d$

• Energy in Rotational Systems: Conservation of energy applies to both translational and rotational motion.

• Systems of Particles: Center of mass, conservation of momentum in collisions.

Examples and Applications

• Finding the Mass of an Object Using Friction: Use $f_s = \mu_s N$ and resolve forces to solve for mass.

• Rotational Kinetic Energy: For a flywheel, $K = \frac{1}{2}I\omega^2$

• Pulley Systems: Apply Newton's laws to each mass and the pulley, relate linear and angular acceleration.

• Collisions: Use conservation of momentum and, if elastic, conservation of kinetic energy to solve for final velocities.

Additional info: These notes are based on a comprehensive final exam covering core topics in introductory mechanics, including vectors, kinematics, Newton's laws, energy, momentum, rotation, equilibrium, and applications to real-world systems such as pulleys, inclined planes, and collisions.