• Object: Single rigid body
• System: Collection of objects that can move together or separately
• Can treat system as single object located at center of mass
• Internal forces cancel out (Newton's 3rd Law))

• $x_{cm} = \frac{m_1x_1 + m_2x_2 + ...}{m_1 + m_2 + ...}$
• $y_{cm} = \frac{m_1y_1 + m_2y_2 + ...}{m_1 + m_2 + ...}$
• For continuous objects: $x_{cm} = \frac{1}{M}\int x\,dm$
• System behaves as if all mass concentrated at center of mass

• $F_{net,ext} = Ma_{cm}$ (Newton's 2nd Law for systems)
• Center of mass moves as if all external force applied there
• Internal forces don't affect center of mass motion
• Crucial for connected objects problems

• Contact forces: Require physical contact
• Normal force, tension, friction, spring force
• Field forces: Act at a distance
• Gravitational force, electric force, magnetic force
• Both types included in free-body diagrams

1. Isolate the object of interest
2. Identify all forces acting ON the object (not BY the object)
3. Draw arrows from center of mass in direction of force
4. Label forces with standard notation
5. Choose coordinate system
6. Include magnitudes if known or use variables

• Normal force ($N$): Perpendicular to contact surface
• Prevents interpenetration of objects
• Self-adjusting: magnitude adjusts as needed
• Not equal to weight (common misconception)
• Tension ($T$): Pulling force along rope/string
• Transmits force through rope
• Constant throughout ideal rope (massless)
• Direction along rope away from object

• For every action, there is an equal and opposite reaction
• Forces act on different objects (never on same object)
• Equal in magnitude, opposite in direction
• Same type of force (force on A by B, force on B by A)
• Example: Earth pulls you down, you pull Earth up (equal and opposite)

• Forces always occur in pairs
• Each force in pair has different object experiencing it
• Cannot cancel each other (act on different objects)
• Essential for analyzing connected objects and collisions

• Inertia: Resistance to change in motion
• Proportional to mass
• Quantified by Newton's First Law
• Mass: measure of inertia (scalar)
• Greater mass = greater inertia

• Translational equilibrium: $\sum \vec{F} = 0$
• Vector sum of all forces = zero
• No net force on object
• Velocity constant (could be zero or non-zero)

• Static equilibrium: Object at rest ($v = 0$)
• Dynamic equilibrium: Object moving with constant velocity
• Both satisfy $\sum \vec{F} = 0$
• Both are equilibrium states

$\vec{F}_{net} = m\vec{a}$

• Net force causes acceleration
• Acceleration is directly proportional to net force
• Acceleration is inversely proportional to mass
• Vector equation (direction matters)
• Component form: $F_x = ma_x$, $F_y = ma_y$

• Weight components:
• Parallel to incline: $mg\sin\theta$
• Perpendicular to incline: $mg\cos\theta$
• Normal force: $N = mg\cos\theta$ (flat incline)
• Friction acts parallel to surface (opposes motion)
• Acceleration down incline: $a = g\sin\theta - \mu_k g\cos\theta$

• Two masses connected by string over pulley
• Same tension throughout string
• Acceleration: $a = \frac{(m_1 - m_2)g}{m_1 + m_2}$
• Tension: $T = \frac{2m_1m_2g}{m_1 + m_2}$
• Heavier mass accelerates downward, lighter upward

• Normal force perceived by person in accelerating reference frame
• Elevator accelerating upward: $N_{apparent} = m(g + a)$
• Elevator accelerating downward: $N_{apparent} = m(g - a)$
• Zero apparent weight during free fall

$F_g = G\frac{m_1 m_2}{r^2}$

• $G = 6.67 \times 10^{-11}$ N - m2/kg2
• Inverse square law
• Attractive force between any two masses
• Explains planetary motion

• $g = \frac{F_g}{m} = \frac{GM}{r^2}$
• Gravitational acceleration near Earth's surface: $g = 9.8$ m/s2
• Varies with altitude and planet
• Independent of falling object's mass

• Inertial mass: Resistance to acceleration ($m = F/a$)
• Gravitational mass: Strength of gravitational attraction
• Experimentally found to be equivalent (Equivalence Principle)
• All objects fall with same acceleration in vacuum

• $f_s \leq \mu_s N$
• Static friction prevents relative motion
• Can range from zero to maximum value
• Adjusts to match applied force (until limit reached)
• $\mu_s$: coefficient of static friction

• $f_k = \mu_k N$
• Kinetic friction opposes sliding motion
• Constant magnitude while sliding
• $\mu_k$: coefficient of kinetic friction
• Always $\mu_k < \mu_s$

• Dimensionless quantity (0 to 1, typically)
• Depends on materials in contact
• $\mu_s$: static coefficient
• $\mu_k$: kinetic coefficient
• Ice on ice: $\mu \approx 0.1$, rubber on concrete: $\mu \approx 1.0$

$F_{spring} = -kx$

• Restoring force proportional to displacement
• Direction opposite to displacement
• $k$: spring constant (N/m)
• $x$: displacement from equilibrium
• Valid for elastic deformation (within elastic limit)
• Negative sign indicates restoring force

• Massless spring (no inertia)
• Hooke's law perfectly obeyed
• No damping
• Restoring force only depends on displacement
• Simplifies calculations

• Net force directed toward center of circular path
• Causes change in direction of velocity
• Not a new force, but name for net force
• Examples: tension, gravity, normal force, friction
• $F_c = \frac{mv^2}{r} = ma_c$

• $a_c = \frac{v^2}{r} = \omega^2r$
• Directed toward center of circle
• Perpendicular to velocity
• Constant magnitude for uniform circular motion
• Required for circular motion

• No friction needed for correct speed
• $\tan\theta = \frac{v^2}{rg}$
• Design speed for given bank angle
• At design speed: $N\sin\theta = mv^2/r$ (provides centripetal force)
• Different speeds require friction

• Top of loop: minimum speed $v_{min} = \sqrt{gR}$
• Tension + weight provides centripetal force
• At minimum speed: tension = 0, weight alone sufficient
• Bottom of loop: maximum tension
• Tension - weight provides centripetal force
• $T = m\left(g + \frac{v^2}{r}\right)$