PHYS 201 Equation Sheet: Chapters 1–11 (Mechanics, Rotational Motion, and Periodic Motion)

Mechanics and Motion

Constant Acceleration Equations

These equations describe the motion of objects under constant acceleration, fundamental for analyzing straight-line motion.

• Velocity as a function of time: $v_x = v_{0x} + a_x t$

• Position as a function of time: $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$

• Velocity-position relation: $v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$

• Displacement using average velocity: $x - x_0 = \left(\frac{v_{0x} + v_x}{2}\right) t$

• Quadratic formula (for solving kinematic equations): $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: If a car accelerates from rest at $2\,\text{m/s}^2$ for $5\,\text{s}$, its velocity is $v = 0 + 2 \times 5 = 10\,\text{m/s}$.

Newton's Laws and Forces

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

• Weight: $w = mg$ (where $g = 9.80\,\text{m/s}^2$)

• Force components: $\sum F_x = m a_x$, $\sum F_y = m a_y$

• Kinetic friction: $f_k = \mu_k n$

• Static friction: $f_s \leq \mu_s n$

• Spring force (Hooke's Law): $F_{\text{spr}} = k \Delta L$

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

Example: A block on a horizontal surface with $\mu_k = 0.2$ and $n = 50\,\text{N}$ has $f_k = 0.2 \times 50 = 10\,\text{N}$.

Circular Motion & Gravitation

Circular Motion

Objects moving in circles experience centripetal acceleration and forces.

• Radial acceleration: $a_{\text{rad}} = \frac{v^2}{R}$

• Speed in a circle: $v = \frac{2\pi R}{T}$

Gravitational Force

Newton's law of universal gravitation describes the force between two masses.

• Gravitational force: $F_g = G \frac{m_1 m_2}{r^2}$

• Gravitational constant: $G = 6.67 \times 10^{-11}\,\text{Nm}^2/\text{kg}^2$

• Orbital period: $T = 2\pi \frac{r^{3/2}}{\sqrt{GM}}$

Example: The force between two $1\,\text{kg}$ masses $1\,\text{m}$ apart is $F_g = 6.67 \times 10^{-11}\,\text{N}$.

Work, Energy, and Power

Work and Energy

Work is the product of force and displacement; energy is the capacity to do work.

• Work: $W = F_{\parallel} s = (F \cos \phi) s$

• Total work: $W_{\text{total}} = K_f - K_i = \Delta K$

• Gravitational potential energy: $U_{\text{grav}} = mgy$

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

• Elastic potential energy: $U_{\text{el}} = \frac{1}{2} k x^2$

• Conservation of energy: $K_i + U_i = K_f + U_f - W_{\text{NC}}$

Power

• Average power: $P_{\text{avg}} = \frac{W}{\Delta t}$

• Instantaneous power: $P = F_{\parallel} v$

Momentum and Impulse

Linear Momentum

Momentum is the product of mass and velocity; impulse is the change in momentum.

• Momentum: $\vec{p} = m \vec{v}$

• Impulse: $\Delta \vec{p} = \vec{F}_{\text{av}} (t_f - t_i) = \vec{J}$

Rotational Motion

Rotational Kinematics

Describes angular motion analogous to linear kinematics.

• Angular velocity: $\omega = \omega_0 + \alpha t$

• Angular displacement: $\theta - \theta_0 = \omega_0 t + \frac{1}{2} \alpha t^2$

• Angular velocity squared: $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

• Average angular velocity: $\theta - \theta_0 = \left(\frac{\omega + \omega_0}{2}\right) t$

• Arc length: $s = r \theta$

• Tangential velocity: $v = r \omega$

• Tangential acceleration: $a_{\text{tan}} = r \alpha$

• Radial acceleration: $a_{\text{rad}} = \frac{v^2}{r} = r \omega^2$

Rotational Energy and Inertia

• Rotational kinetic energy: $K_{\text{rot}} = \frac{1}{2} I \omega^2$

• Moment of inertia (point mass): $I_{\text{point}} = m r^2$

• Total kinetic energy: $K_{\text{tot}} = \frac{1}{2} M v_{\text{cm}}^2 + \frac{1}{2} I_{\text{cm}} \omega^2$

• Potential energy (center of mass): $U = M g y_{\text{cm}}$

Torque and Angular Momentum

• Torque: $\tau = F_{\perp} r = F l = F r \sin(\phi)$

• Sum of torques: $\sum \tau = I \alpha$

• Work by torque: $\Delta W = \tau \Delta \theta$

• Power: $P = \tau \omega$

• Angular momentum: $L = I \omega$

• Change in angular momentum: $\sum \tau = \frac{\Delta L}{\Delta t}$

• Angular momentum (point mass): $L = m v l$

Equilibrium Conditions

For an object to be in equilibrium, both net force and net torque must be zero.

• $\sum F_x = 0$

• $\sum F_y = 0$

• $\sum \tau = 0$ (about any axis)

Periodic Motion

Simple Harmonic Motion (SHM)

SHM describes oscillatory motion where restoring force is proportional to displacement.

• Restoring force: $F_x = -k x$

• Acceleration: $a_x = -\frac{k}{m} x$

• Angular frequency: $\omega = 2\pi f$

• Frequency: $f = \frac{1}{T}$

• Elastic potential energy: $U_{\text{el}} = \frac{1}{2} k x^2$

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

• Position as a function of time: $x = A \cos(\omega t)$

• Velocity as a function of time: $v_x = -\omega A \sin(\omega t)$

• Angular frequency (spring): $\omega = \sqrt{\frac{k}{m}}$

• Frequency (spring): $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

• Period (spring): $T = 2\pi \sqrt{\frac{m}{k}}$

• Angular frequency (pendulum): $\omega = \sqrt{\frac{g}{L}}$

• Frequency (pendulum): $f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$

• Period (pendulum): $T = 2\pi \sqrt{\frac{L}{g}}$

• Total energy in SHM: $E = \frac{1}{2} m v_x^2 + \frac{1}{2} k x^2$

• Velocity as a function of position: $v_x(x) = \pm \sqrt{\frac{k}{m}} \sqrt{A^2 - x^2}$

Example: A $0.5\,\text{kg}$ mass on a spring with $k = 200\,\text{N/m}$ has $\omega = \sqrt{200/0.5} = 20\,\text{rad/s}$.

Summary Table: Key Equations

Topic	Equation	Description
Linear Kinematics	$v_x = v_{0x} + a_x t$	Velocity under constant acceleration
Newton's Second Law	$\sum F_x = m a_x$	Force and acceleration
Work	$W = F_{\parallel} s$	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
Rotational Kinematics	$\omega = \omega_0 + \alpha t$	Angular velocity under constant angular acceleration
Torque	$\tau = F_{\perp} r$	Rotational effect of a force
SHM Frequency (spring)	$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$	Oscillation frequency for a spring-mass system
SHM Frequency (pendulum)	$f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$	Oscillation frequency for a simple pendulum

Additional info: This equation sheet covers the core concepts and formulas from introductory mechanics, rotational motion, and periodic motion, suitable for exam preparation in a college physics course.