PHY2004 Exam 1 Formula Sheet: Physics with Calculus Study Notes

Measurement Prefixes and Physical Constants

SI Prefixes

SI prefixes are used to denote multiples or fractions of units in the metric system. Understanding these prefixes is essential for interpreting measurements in physics.

• femto- (f): $\times 10^{-15}$

• pico- (p): $\times 10^{-12}$

• nano- (n): $\times 10^{-9}$

• micro- (\mu): $\times 10^{-6}$

• milli- (m): $\times 10^{-3}$

• centi- (c): $\times 10^{-2}$

• kilo- (k): $\times 10^{3}$

• mega- (M): $\times 10^{6}$

• giga- (G): $\times 10^{9}$

Example: 1 nanometer (nm) = $1 \times 10^{-9}$ meters.

Kinematics

Basic Kinematic Equations

Kinematics describes the motion of objects without considering the forces that cause the motion. The following equations are fundamental for one-dimensional motion with constant acceleration.

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

• Velocity: $v = v_0 + a t$

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

Example: If a car starts from rest ($v_0 = 0$) and accelerates at $2\,\text{m/s}^2$ for $5$seconds, its displacement is$x = 0 + 0 \times 5 + \frac{1}{2} \times 2 \times 5^2 = 25\,\text{m}$.

Dynamics & Conservation Laws

Newton's Laws and Forces

Dynamics involves the study of forces and their effect on motion. Newton's laws form the foundation of classical mechanics.

• Newton's First Law: An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: $F_{\text{net}} = m a$

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

Example: If a 10 kg object experiences a net force of 20 N, its acceleration is $a = \frac{F_{\text{net}}}{m} = \frac{20}{10} = 2\,\text{m/s}^2$.

Conservation of Momentum

Momentum is conserved in isolated systems. The total momentum before an event equals the total momentum after.

• Linear Momentum: $p = m v$

• Conservation Law: $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

Example: In a collision, the sum of the momenta of the objects before impact equals the sum after impact.

Work, Power, & Energy

Work and Energy

Work is done when a force causes displacement. Energy is the capacity to do work.

• Work: $W = F d \cos \theta$

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

• Potential Energy (gravitational): $PE = m g h$

• Conservation of Energy: $E_{\text{total}} = KE + PE$ (in absence of non-conservative forces)

Example: Lifting a 5 kg object 2 m high: $PE = 5 \times 9.8 \times 2 = 98\,\text{J}$.

Power

Power is the rate at which work is done or energy is transferred.

• Power: $P = \frac{W}{t}$

Example: If 100 J of work is done in 5 seconds, $P = \frac{100}{5} = 20\,\text{W}$.

Rotational Basics I

Rotational Kinematics

Rotational motion involves objects rotating about an axis. The following equations are analogous to linear kinematics.

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

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

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

Example: If a wheel starts from rest and accelerates at $1\,\text{rad/s}^2$ for $3$seconds, its angular displacement is$\theta = 0 + 0 \times 3 + \frac{1}{2} \times 1 \times 3^2 = 4.5\,\text{rad}$.

Rotational Dynamics

Rotational dynamics deals with the effect of torques and rotational inertia.

• Torque: $\tau = r F \sin \theta$

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

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

Example: For a solid disk of mass $m$ and radius $R$, $I = \frac{1}{2} m R^2$.

Center of Mass

Definition and Calculation

The center of mass is the point where the mass of a system is concentrated. It is important for analyzing motion and stability.

• Center of Mass (one dimension): $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$

• Center of Mass (two dimensions): $\vec{r}_{\text{cm}} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$

Example: For two masses, $m_1 = 2\,\text{kg}$ at $x_1 = 0$ and $m_2 = 3\,\text{kg}$ at $x_2 = 4\,\text{m}$, $x_{\text{cm}} = \frac{2 \times 0 + 3 \times 4}{2 + 3} = \frac{12}{5} = 2.4\,\text{m}$.

Forces & Momentum

Impulse and Momentum

Impulse is the change in momentum resulting from a force applied over time.

• Impulse: $J = F \Delta t$

• Impulse-Momentum Theorem: $J = \Delta p$

Example: A force of $10\,\text{N}$ applied for $2\,\text{s}$ gives $J = 10 \times 2 = 20\,\text{N} \cdot \text{s}$.

Physical Constants

Common Constants

Physical constants are values that are universally recognized and used in calculations.

• Acceleration due to gravity: $g = 9.8\,\text{m/s}^2$

• Pi: $\pi \approx 3.1416$

• Speed of light: $c = 3.00 \times 10^8\,\text{m/s}$

• Planck's constant: $h = 6.626 \times 10^{-34}\,\text{J} \cdot \text{s}$

Summary Table: SI Prefixes

Prefix	Symbol	Multiplier
femto	f	$10^{-15}$
pico	p	$10^{-12}$
nano	n	$10^{-9}$
micro	\mu	$10^{-6}$
milli	m	$10^{-3}$
centi	c	$10^{-2}$
kilo	k	$10^{3}$
mega	M	$10^{6}$
giga	G	$10^{9}$

Additional info: Some formula symbols and notations in the original file were unclear or fragmented. Standard equations and context were inferred based on typical Physics with Calculus curriculum and the listed topics.