PHYS 220A – Fall 2023 Final Exam Study Notes

Vectors and Linear Motion

Vector Magnitude and Scalar Product

Vectors are quantities that have both magnitude and direction. The magnitude of a vector V with components Vx and Vy is given by:

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

• Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

Example: For $\vec{A} = 3\hat{i} + 2\hat{j}$ and $\vec{B} = 1\hat{i} + 4\hat{j}$, the dot product is $3 \times 1 + 2 \times 4 = 11$.

Linear Motion and Kinematics

Velocity and Acceleration

Linear motion describes the movement of objects along a straight path. The velocity is the rate of change of position, and acceleration is the rate of change of velocity.

• Velocity: $v = \frac{dx}{dt}$

• Acceleration: $a = \frac{dv}{dt}$

• For constant acceleration: $v = v_0 + at$

Example: If $a(t) = 10.0t$ m/s2, then $v(t) = \int a(t) dt = 5.0t^2 + C$.

Projectile and Satellite Motion

Projectile Motion

Projectile motion involves two-dimensional motion under gravity. The horizontal and vertical components are treated independently.

• Horizontal velocity is constant (no air resistance).

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

Example: A ball thrown horizontally from a height will have a constant horizontal velocity and a vertically increasing speed due to gravity.

Newton's Laws of Motion

Newton's Second Law

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

• $\sum \vec{F} = m\vec{a}$

Example: If two forces act on an object, the net force is the vector sum, and the acceleration is $a = \frac{F_{net}}{m}$.

Work, Energy, and Power

Kinetic and Potential Energy

Energy is the ability to do work. Kinetic energy is due to motion, and potential energy is due to position.

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

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

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

Example: If a box is pushed across a surface, the work done by friction is $W = F_{fric} \cdot d$.

Momentum and Collisions

Conservation of Momentum

Momentum is conserved in isolated systems. For two objects colliding:

• $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$

Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum.

Rotational Motion

Angular Kinematics and Dynamics

Rotational motion involves objects rotating about an axis. Key quantities include angular velocity ($\omega$), angular acceleration ($\alpha$), and moment of inertia ($I$).

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

• Torque: $\tau = I\alpha$

Example: A grinding wheel slowing down due to friction experiences angular deceleration, $\alpha = \frac{\Delta \omega}{\Delta t}$.

Statics and Equilibrium

Conditions for Equilibrium

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

• $\sum F_x = 0$, $\sum F_y = 0$, $\sum \tau = 0$

Example: A ladder leaning against a wall is analyzed using force and torque equilibrium conditions.

Tables

Below is an example of a table comparing types of collisions:

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?
Elastic	Yes	Yes
Inelastic	Yes	No

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Topics covered correspond to: Vectors, Linear Motion, Newton's Laws, Work and Energy, Momentum, Rotational Motion, and Statics, matching the provided chapter list.