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 V = V_x \hat{i} + V_y \hat{j}.

• Magnitude: The magnitude of a vector \vec{C} = \vec{A} + \vec{B} is found using the Pythagorean theorem if the vectors are perpendicular.

Formula:

$|{C}|=\sqrt{C_x^2 + C_y^2}$

Example: If \vec{A} and \vec{B} are given with their magnitudes and directions, resolve into components, add, and find the resultant magnitude.

Motion Along a Straight Line

Velocity, Acceleration, and Graphs

Describing motion involves position, velocity, and acceleration as functions of time.

• Constant Velocity: Position changes linearly with time.

• Acceleration: The rate of change of velocity. If acceleration is a function of time, integrate to find velocity.

Formula:

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

Example: If $a(t) = 3.00 t^2$, then $v(t) = v_0 + \int_0^t 3.00 t'^2 dt' = v_0 + t^3$.

Motion in Two or Three Dimensions

Projectile Motion

Projectile motion involves two-dimensional motion under gravity, with horizontal and vertical components analyzed separately.

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

• Vertical Velocity: Changes due to gravity.

Formula:

$y(t) = y_0 + v_{0y} t - \frac{1}{2} g t^2$

Example: A hockey puck launched at an angle; resolve initial velocity into components and use kinematic equations to find time of flight.

Newton's Laws of Motion & Applications

Newton's Second Law

Newton's Second Law relates the net force on an object to its acceleration:

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

• Friction: Static friction prevents motion up to a maximum value $f_s = \mu_s N$.

• Kinetic Friction: $f_k = \mu_k N$ acts when objects slide.

Example: Calculating the minimum mass needed to move a box given a force and coefficient of static friction.

Work, Energy, and Conservation

Work and Kinetic Energy

Work is done when a force causes displacement. The work-energy theorem relates work to the change in kinetic energy:

$W = \Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$

• Potential Energy: Energy stored due to position, e.g., gravitational potential energy $U = mgh$.

• Conservation of Energy: Total mechanical energy (kinetic + potential) is conserved in the absence of non-conservative forces.

Example: Calculating the work done by friction or the change in kinetic energy as an object moves along a path.

Momentum, Impulse, and Collisions

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant.

$\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

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

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

Example: Two marbles colliding on a surface; use conservation laws to find final velocities.

Rotation of Rigid Bodies & Dynamics of Rotational Motion

Rotational Kinematics and Dynamics

Rotational motion is described by angular displacement, velocity, and acceleration. Newton's Second Law for rotation:

$\sum \tau = I \alpha$

• Moment of Inertia (I): Measures resistance to angular acceleration.

• Torque (\tau): The rotational equivalent of force.

Example: Calculating the angular acceleration of a grinding wheel or the torque due to forces on a plate.

Equilibrium & Elasticity

Conditions for Equilibrium

An object is in equilibrium if the sum of forces and the sum of torques acting on it are zero.

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

• Second Condition: $\sum \tau = 0$

Example: Calculating the forces on a ladder leaning against a wall or the tensions in ropes supporting a traffic light.

Sample Table: Comparison of Static and Kinetic Friction

Type of Friction	Symbol	Formula	When It Acts
Static	$f_s$	$f_s \leq \mu_s N$	Before motion starts
Kinetic	$f_k$	$f_k = \mu_k N$	During motion

Additional Topics Covered

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

• Work by Friction: $W_{fric} = f_k d$

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

• Energy Conservation in Rotational Systems: Potential energy lost equals kinetic energy gained (translational + rotational).

Example Problems and Applications

• Finding the magnitude of a resultant vector given components.

• Calculating the time for a puck to stop due to friction.

• Determining the work done by friction as a box is pushed across a surface.

• Analyzing the equilibrium of a ladder with forces and torques.

• Solving for the final velocities in a two-body collision using conservation laws.

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, and equilibrium. The problems are representative of standard college-level physics exams.