Formula Sheet: Force and Newton's laws of Motion

Table of Contents
1. Fundamental Concepts and Definitions
2. Newton's Laws of Motion
3. Types of Forces
4. Inclined Planes
5. Connected Objects and Systems
6. Equilibrium
7. Free Body Diagrams (FBD)
8. Problem-Solving Strategy
9. Special Cases and Important Relationships
10. Important Distinctions and Concepts
11. Key Assumptions for MCAT Problems
12. Units and Conversions
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Fundamental Concepts and Definitions

Force

• Force: A push or pull on an object; a vector quantity
• Unit: Newton (N) = kg⋅m/s²
• Net Force (F_net): Vector sum of all forces acting on an object

Mass and Weight

• Mass (m): Measure of inertia; scalar quantity; Unit: kilogram (kg)
• Weight (W or F_g): Gravitational force on an object; vector quantity

Where:

• \(W\) = weight (N)
• \(m\) = mass (kg)
• \(g\) = acceleration due to gravity ≈ 9.8 m/s² (often approximated as 10 m/s² on MCAT)

Newton's Laws of Motion

Newton's First Law (Law of Inertia)

Statement: An object at rest stays at rest, and an object in motion stays in motion with constant velocity, unless acted upon by a net external force.

• Condition for equilibrium: \(\vec{F}_{net} = 0\)
• Consequence: \(\vec{a} = 0\) (object is either at rest or moving with constant velocity)

Newton's Second Law

Where:

• \(\vec{F}_{net}\) = net force (N)
• \(m\) = mass (kg)
• \(\vec{a}\) = acceleration (m/s²)

Alternative forms:

Component form:

• \(F_{net,x} = ma_x\)
• \(F_{net,y} = ma_y\)
• \(F_{net,z} = ma_z\)

Newton's Third Law (Action-Reaction)

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

Where:

• \(\vec{F}_{AB}\) = force exerted by object A on object B
• \(\vec{F}_{BA}\) = force exerted by object B on object A
• Note: Action-reaction pairs act on different objects

Types of Forces

Gravitational Force

Where:

• \(F_g\) = gravitational force (N)
• \(G\) = gravitational constant = 6.67 × 10^-11 N⋅m²/kg²
• \(m_1, m_2\) = masses of the two objects (kg)
• \(r\) = distance between centers of mass (m)

Normal Force (F_N or N)

• Definition: Perpendicular contact force exerted by a surface on an object
• On horizontal surface: \(F_N = mg\) (when no other vertical forces present)
• On inclined plane: \(F_N = mg\cos\theta\)
• Note: Normal force adjusts to prevent objects from passing through surfaces

Friction

Static Friction (f_s)

Prevents: Objects from starting to move

Maximum static friction:

Where:

• \(f_s\) = static friction force (N)
• \(\mu_s\) = coefficient of static friction (dimensionless)
• \(F_N\) = normal force (N)

Kinetic Friction (f_k)

Acts: When object is already moving

Where:

• \(f_k\) = kinetic friction force (N)
• \(\mu_k\) = coefficient of kinetic friction (dimensionless)
• \(F_N\) = normal force (N)
• Note: \(\mu_k < \mu_s\)="" (kinetic="" friction="" is="" typically="" less="" than="" maximum="" static="">
• Direction: Always opposes motion or intended motion

Tension (T or F_T)

• Definition: Force transmitted through a rope, string, cable, or wire when pulled
• Direction: Always pulls away from the object
• Massless rope assumption: Tension is the same throughout the rope
• Note: For rope with mass, tension varies along its length

Spring Force (Hooke's Law)

Or magnitude only:

Where:

• \(F_s\) = spring force (N)
• \(k\) = spring constant (N/m)
• \(x\) = displacement from equilibrium position (m)
• Negative sign: Indicates restoring force (opposes displacement)

Applied Force (F_app)

• Definition: External force applied to an object by a person or another agent
• Note: Can be at any angle; resolve into components if necessary

Inclined Planes

Force Components on Incline

Parallel to incline (down the slope):

Perpendicular to incline:

Normal force on incline:

Where:

• \(\theta\) = angle of incline measured from horizontal
• \(m\) = mass of object (kg)
• \(g\) = acceleration due to gravity (m/s²)

Motion on Inclined Plane

Without friction:

With friction (sliding down):

With friction (sliding up):

Connected Objects and Systems

Atwood Machine (Two masses connected by rope over pulley)

Acceleration of system:

Where \(m_2 > m_1\) (m₂ accelerates downward)

Tension in rope:

Objects Connected Horizontally

System acceleration (no friction, applied force F on system):

Tension between objects:

Where F is applied to m₁ and m₂ is behind m₁

Equilibrium

Static Equilibrium

Conditions:

• \(\sum \vec{F} = 0\) (net force equals zero)
• \(\vec{a} = 0\) (no acceleration)
• \(\vec{v} = 0\) (object at rest)

Component form:

• \(\sum F_x = 0\)
• \(\sum F_y = 0\)

Dynamic Equilibrium

Conditions:

• \(\sum \vec{F} = 0\) (net force equals zero)
• \(\vec{a} = 0\) (no acceleration)
• \(\vec{v} = \text{constant}\) (object moving with constant velocity)

Free Body Diagrams (FBD)

Steps for Creating FBD

1. Isolate the object of interest
2. Draw all forces acting ON the object as arrows from center of mass
3. Label each force clearly
4. Choose coordinate system (usually x-horizontal, y-vertical)
5. Resolve forces into components if necessary

Common Forces to Include

• Weight (mg): Always points downward
• Normal force (F_N): Perpendicular to contact surface
• Friction (f): Parallel to contact surface, opposes motion
• Tension (T): Along rope/string, pulls away from object
• Applied force (F_app): In direction applied

Problem-Solving Strategy

General Approach

1. Draw free body diagram for each object
2. Choose appropriate coordinate system
3. Resolve forces into components
4. Apply Newton's Second Law: \(\vec{F}_{net} = m\vec{a}\)
5. Write equations for x and y components separately
6. Solve system of equations

Sign Conventions

• Horizontal: Right is positive (+), left is negative (-)
• Vertical: Up is positive (+), down is negative (-)
• Inclines: Down the slope often chosen as positive for convenience
• Consistency: Must maintain same convention throughout problem

Special Cases and Important Relationships

Apparent Weight

In elevator accelerating upward:

In elevator accelerating downward:

In free fall:

Where:

• \(F_N\) = apparent weight (normal force) (N)
• \(a\) = magnitude of elevator's acceleration (m/s²)

Drag Force (Air Resistance)

Where:

• \(F_d\) = drag force (N)
• \(C\) = drag coefficient (dimensionless)
• \(\rho\) = fluid density (kg/m³)
• \(A\) = cross-sectional area (m²)
• \(v\) = velocity (m/s)
• Note: Detailed drag calculations rarely appear on MCAT; conceptual understanding more important

Terminal Velocity

Condition: When drag force equals weight

• \(a = 0\) (no further acceleration)
• Result: Object falls at constant velocity

Important Distinctions and Concepts

Mass vs Weight

• Mass: Intrinsic property; doesn't change with location; scalar
• Weight: Force due to gravity; varies with gravitational field; vector

Inertial vs Gravitational Mass

• Inertial mass: Resistance to acceleration (from F = ma)
• Gravitational mass: Response to gravitational field (from F_g = mg)
• Note: These are equivalent (equivalence principle)

Contact vs Non-Contact Forces

• Contact forces: Normal, friction, tension, applied force
• Non-contact forces: Gravitational, electric, magnetic

Key Assumptions for MCAT Problems

• Massless ropes/pulleys: Unless stated otherwise
• Frictionless surfaces: Unless friction coefficient given
• Rigid bodies: Objects don't deform under force
• Point masses: For simple problems, objects treated as points
• g ≈ 10 m/s²: Common approximation for easier calculation
• Air resistance negligible: Unless specifically mentioned

Units and Conversions

Force

• SI unit: Newton (N) = kg⋅m/s²
• Other units: 1 pound (lb) ≈ 4.45 N

Mass

• SI unit: kilogram (kg)
• Other units: 1 pound mass ≈ 0.454 kg

Acceleration

• SI unit: m/s²
• Standard gravity: g = 9.8 m/s² ≈ 10 m/s²