Inclined Plane Formulas for AP Physics

Axis Orientation:
- $x$ -axis: Oriented parallel to the ramp surface.
- $y$ -axis: Oriented perpendicular to the ramp surface.
Essential Free Body Diagram (FBD) Forces:
- Weight ( $mg$ ) directed straight down.
- Normal force ( $N$ ) perpendicular to the ramp surface.
- Friction force ( $F_f$ ) opposing motion parallel to the surface.
- Tension ( $T$ ) along the ramp (if a rope or cable is attached).

Gravity Components:
- Parallel to the ramp (downward pull along surface): $F_{g,x} = mg \sin(\theta)$
- Perpendicular to the ramp (into the surface): $F_{g,y} = mg \cos(\theta)$
Normal Force ( $N$ ):
- In equilibrium along the perpendicular axis: $N = mg \cos(\theta)$
- If acceleration occurs perpendicular to the ramp: $\Sigma F_y = ma$
Friction Force ( $F_f$ ):
- Kinetic friction (object moving): $F_f = \mu_k N$
- Static friction (object stationary): $F_f \le \mu_s N$
Newton's Second Law:
- $\Sigma F = ma$ applied independently in both $x$ - and $y$ -directions.

Acceleration ( $a$ ):
- Equation along the incline: $mg \sin(\theta) - F_f = ma$
Coefficient of Friction ( $\mu$ ):
- $\mu = \frac{F_f}{N}$
Mass of the Block ( $m$ ):
- $m = \frac{mg \sin(\theta) - F_f}{a}$
Ramp Angle ( $\theta$ ):
- Calculated from gravity components: $\sin(\theta) = \frac{F_{g,x}}{mg}$ or $\cos(\theta) = \frac{F_{g,y}}{mg}$
Atwood / Pulley System ( $m_1$ and $m_2$ ):
- Acceleration with frictionless surface and pulley: $a = \frac{m_1 g - m_2 g}{m_1 + m_2}$
Tension in Rope ( $T$ ):
- Solved using the hanging mass: $T = mg - ma$ or $T = ma + mg$

Free Body Diagram Setup: Always draw an FBD, set tilted coordinate axes, and resolve the weight force ( $mg$ ) into $mg \sin(\theta)$ and $mg \cos(\theta)$ .
Sign Consistency: Establish a defined positive direction for each object and follow it across all equations.
Coupled Systems: For multi-body pulley setups, write separate equations for each mass and add them to eliminate tension ( $T$ ).