Define what objects are included in the system.

Internal forces: Forces between objects within the system External forces: Forces from objects outside the system

Why it matters:

• Internal forces cancel by Newton's Third Law
• Only external forces affect system motion
• Critical for momentum and energy analysis

Choice of system:

• Include all objects that interact significantly
• May exclude "fixed" objects (Earth, walls)
• Different system boundaries for different problems

Discrete system (finite number of particles):

$\vec{r}_{cm} = \frac{\sum_{i=1}^{N} m_i \vec{r}_i}{\sum_{i=1}^{N} m_i}$

In components: $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$

Continuous system (distributed mass):

$\vec{r}_{cm} = \frac{\int \vec{r} \, dm}{\int dm} = \frac{\int \vec{r} \rho(\vec{r}) \, dV}{\int \rho(\vec{r}) \, dV}$

Where $\rho(\vec{r})$ = mass density at position $\vec{r}$

Properties:

• Center of mass moves as if all mass concentrated there
• External forces cause $\vec{a}_{cm} = \vec{F}_{ext}/M$
• Internal forces don't affect $\vec{r}_{cm}$ motion

Weight (gravitational force): $\vec{W} = m\vec{g} = -mg\hat{j}$

Always acts downward (toward Earth's center).

Normal force (N):

• Perpendicular to contact surface
• Prevents penetration
• Magnitude determined by other forces

Friction (f):

• Opposes relative motion or impending motion
• Parallel to contact surface
• Related to normal force: $f \le \mu N$

Tension (T):

• Force along rope/string/cable
• Pulls on both connected objects
• Usually constant along ideal massless rope

Applied forces: External pushes, pulls, etc.

Free-Body Diagram (FBD) procedure:

1. Isolate object: Draw separate diagram for each object
2. Represent as particle: Draw object as dot (simplifies vector addition)
3. Identify forces: List all external forces acting on object
4. Draw vectors: Draw each force as arrow originating from object
5. Label clearly: Use physics names (mg, N, T, f, not "gravity force")
6. Choose coordinates: Align coordinate system to simplify problem

Common mistakes to avoid:

• Including internal forces
• Drawing forces ON instead of BY the object
• Missing forces (especially normal forces)
• Incorrect directions (especially friction)

For every action force, there is an equal and opposite reaction force.

$\vec{F}_{AB} = -\vec{F}_{BA}$

Where:

• $\vec{F}_{AB}$ = force on A by B
• $\vec{F}_{BA}$ = force on B by A

Key characteristics:

• Equal magnitude: $|\vec{F}_{AB}| = |\vec{F}_{BA}|$
• Opposite direction
• Act on different objects
• Same type of force
• Simultaneous (no delay)

Tension in rope:

• Rope pulls equally on both ends
• $T_{on A by rope} = -T_{on rope by A}$

Contact forces:

• Object A pushes on B; B pushes back on A
• $\vec{N}_{A \to B} = -\vec{N}_{B \to A}$

Important:

• Action-reaction pairs DON'T cancel because they act on different objects
• Forces on SAME object can cancel (equilibrium)
• Don't include reaction forces on single FBD

Object at rest or moving with constant velocity.

Condition: $\vec{F}_{net} = 0$

Or component-wise: $\sum F_x = 0, \quad \sum F_y = 0$

Characteristics:

• No acceleration
• Velocity constant (could be zero)
• All forces balance

Applications:

• Structures (bridges, buildings)
• Hanging objects
• Stationary systems

Object moves with constant velocity (no acceleration).

Condition: Same as static equilibrium: $\sum \vec{F} = 0$

Examples:

• Terminal velocity (when drag = weight)
• Constant velocity motion on horizontal surface
• Object in orbit (circular orbit: F provides centripetal acceleration, not equilibrium)

When $\vec{F}_{net} = 0$, velocity is constant.

$\vec{v} = \text{constant}$

Note: In circular motion, $F_{net} \neq 0$ because object is accelerating (changing direction).

Vector sum of all forces acting on object.

$\vec{F}_{net} = \sum_{i=1}^{N} \vec{F}_i$

Component form: $F_{net,x} = \sum F_x, \quad F_{net,y} = \sum F_y$

Scalar measure of object's resistance to acceleration.

Properties:

• Inertial mass (resistance to change in motion)
• Gravitational mass (determines weight)
• Equivalence principle: inertial mass = gravitational mass

Units: kilograms (kg)

Rate of change of velocity.

$\vec{a} = \frac{d\vec{v}}{dt}$

Units: m/s2

Units: m/s2

Applying SigmaF = ma in 1D and 2D:

1D application: $\sum F = ma$

Choose coordinate axis; use sign conventions carefully.

2D application (component form): $\sum F_x = ma_x, \quad \sum F_y = ma_y$

Object on incline at angle θ to horizontal.

Forces:

• Weight components:
• Parallel: $W_{\parallel} = mg\sin\theta$ (down incline)
• Perpendicular: $W_{\perp} = mg\cos\theta$ (into incline)
• Normal force: $N = mg\cos\theta$
• Friction (opposes motion): $f \le \mu N = \mu mg\cos\theta$

Acceleration (if sliding down): $a = g\sin\theta - \mu_k g\cos\theta$

Critical angle (when object just begins to slide): $\theta_c = \arctan(\mu_s)$

Multiple objects connected by ropes over pulleys.

Atwood machine (two masses, frictionless pulley): $a = \frac{(m_1 - m_2)g}{m_1 + m_2}, \quad T = \frac{2m_1 m_2 g}{m_1 + m_2}$

Heavier mass accelerates down, lighter up.

Procedure:

1. Draw FBD for each mass
2. Assume acceleration directions
3. Write Newton's Second Law for each
4. Use constraint (tension same in rope, accelerations related)
5. Solve system of equations

Weight measured in accelerating reference frame.

Elevator accelerating upward with a: $W_{apparent} = m(g + a)$

Elevator accelerating downward with a: $W_{apparent} = m(g - a)$

Free fall (a = g): $W_{apparent} = 0$ (weightlessness)

Physical meaning:

• Normal force from scale = apparent weight
• Scale reads m(g ± a)

Friction when object is stationary but sliding is impending.

$f_s \le \mu_s N$

Where:

• $f_s$ = static friction force
• $\mu_s$ = coefficient of static friction
• N = normal force

Characteristics:

• Opposes impending motion
• Can be any value from 0 to $\mu_s N$
• Maximum at threshold of motion: $f_{s,max} = \mu_s N$
• Generally $\mu_s > \mu_k$

Finding if object moves: Compare applied force to maximum static friction.

Friction when object is sliding.

$f_k = \mu_k N$

Where $\mu_k$ = coefficient of kinetic friction.

Characteristics:

• Constant magnitude (independent of speed)
• Opposes direction of motion
• Usually $\mu_k < \mu_s$
• $f_k \le f_{s,max}$

Force opposing motion through air.

Linear drag (slow speeds): $\vec{F}_d = -b\vec{v}$

Where b = drag coefficient.

Quadratic drag (high speeds): $\vec{F}_d = -bv^2\hat{v}$

Characteristics:

• Opposes velocity direction
• Increases with speed
• Non-conservative force

Constant velocity when drag equals driving force.

For falling object: $mg - bv_{term} = 0$

$v_{term} = \frac{mg}{b}$ (linear drag)

$v_{term} = \sqrt{\frac{mg}{b}}$ (quadratic drag)

Characteristics:

• Net force = 0 at terminal velocity
• Acceleration = 0
• Maximum speed for falling object

Equations relating motions of connected objects.

Types:

• String constraints: objects connected have same acceleration magnitude
• Contact constraints: separation distance constant
• Rolling constraints: $v = R\omega$

For ideal massless, inextensible string:

• Tension is constant throughout string
• Forces balance at each connection point

For real strings with mass:

• Tension varies along string
• Must account for string weight

Connected objects accelerate as system.

System mass: $M = \sum m_i$

Net external force: $\vec{F}_{net,ext}$

$\vec{a}_{system} = \frac{\vec{F}_{net,ext}}{M}$

Internal forces cancel: Don't include tension between connected objects in $\vec{F}_{net,ext}$