Physics 140/211 Midterm 1 Equation Sheet: Core Concepts and Formulas

Units, Physical Quantities & Vectors

Physical Quantities and SI Units

Physics relies on precise measurement of physical quantities, which are expressed in terms of fundamental SI units.

• Displacement ($\vec{r}$): Vector quantity representing change in position.

• Velocity ($\vec{v}$): Rate of change of displacement.

• Acceleration ($\vec{a}$): Rate of change of velocity.

• SI Units: Meter (m), Kilogram (kg), Second (s).

Vector Addition: $\vec{A} + \vec{B}$ is performed component-wise.

Magnitude of a Vector: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$

Motion Along a Straight Line

Kinematic Equations for Constant Acceleration

Describes the motion of objects moving in one dimension under constant acceleration.

• Displacement: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity: $v = v_0 + a t$

• Velocity-Displacement Relation: $v^2 = v_0^2 + 2a(x - x_0)$

Average Velocity: $\bar{v} = \frac{\Delta x}{\Delta t}$

Instantaneous Velocity: $v = \frac{dx}{dt}$

Motion in Two or Three Dimensions

Projectile Motion

Describes the motion of objects launched into the air, subject only to gravity.

• Horizontal Displacement: $x = v_{0x} t$

• Vertical Displacement: $y = v_{0y} t - \frac{1}{2} g t^2$

• Time of Flight: $t = \frac{2 v_{0y}}{g}$ (for symmetric trajectory)

Relative Velocity: $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$

Newton's Laws of Motion

Fundamental Laws Governing Motion

Newton's laws describe the relationship between forces and motion.

• First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net force.

• Second Law: $\vec{F}_{net} = m \vec{a}$

• Third Law: For every action, there is an equal and opposite reaction.

Weight: $W = mg$

Friction: $f = \mu N$

Applying Newton's Laws

Forces in Equilibrium and Dynamics

Application of Newton's laws to solve for unknown forces and accelerations.

• Normal Force: $N$ is perpendicular to the contact surface.

• Tension: Force transmitted through a string or rope.

• Inclined Plane: $F_{parallel} = mg \sin \theta$, $F_{perpendicular} = mg \cos \theta$

Work & Kinetic Energy

Work and Energy Relationships

Work is the transfer of energy via force acting over a distance.

• Work: $W = \vec{F} \cdot \vec{d} = Fd \cos \theta$

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

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

Potential Energy & Conservation

Conservation of Mechanical Energy

Energy can be stored as potential energy and converted to kinetic energy.

• Gravitational Potential Energy: $U_g = mgh$

• Elastic Potential Energy: $U_s = \frac{1}{2} k x^2$

• Conservation of Energy: $E_{mech} = K + U$ (remains constant if only conservative forces act)

Momentum, Impulse, and Collisions

Linear Momentum and Impulse

Momentum is a measure of motion, and impulse is the change in momentum.

• Momentum: $\vec{p} = m \vec{v}$

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

• Conservation of Momentum: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$ (in absence of external forces)

Elastic Collision: Both momentum and kinetic energy are conserved.

Inelastic Collision: Only momentum is conserved.

Rotation of Rigid Bodies

Rotational Kinematics and Dynamics

Describes the motion of objects rotating about a fixed axis.

• Angular Displacement: $\theta$ (radians)

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

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

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

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

Dynamics of Rotational Motion

Torque and Rotational Equilibrium

Torque causes rotational acceleration, analogous to force in linear motion.

• Torque: $\tau = r F \sin \theta$

• Newton's Second Law for Rotation: $\tau_{net} = I \alpha$

• Angular Momentum: $L = I \omega$

• Conservation of Angular Momentum: $L_{initial} = L_{final}$ (if $\tau_{ext} = 0$)

Equilibrium & Elasticity

Conditions for Equilibrium

An object is in equilibrium if the net force and net torque on it are zero.

• Translational Equilibrium: $\sum \vec{F} = 0$

• Rotational Equilibrium: $\sum \tau = 0$

• Elasticity: $F = k \Delta L$ (Hooke's Law)

Summary Table: Key Equations

Topic	Equation	Description
Kinematics	$x = x_0 + v_0 t + \frac{1}{2} a t^2$	Displacement under constant acceleration
Newton's Second Law	$\vec{F}_{net} = m \vec{a}$	Force and acceleration
Work	$W = Fd \cos \theta$	Work done by a force
Kinetic Energy	$K = \frac{1}{2} m v^2$	Energy of motion
Momentum	$\vec{p} = m \vec{v}$	Linear momentum
Impulse	$\vec{J} = \int \vec{F} dt$	Change in momentum
Rotational Motion	$\tau = I \alpha$	Rotational analog of Newton's law
Elasticity	$F = k \Delta L$	Hooke's Law