Laws of Motion

Newton's 3 laws (1687, Principia):

1st Law (Inertia): A body remains at rest or in uniform motion unless acted on by external force.

• Mass is a measure of inertia.
• Defines inertial frames.

2nd Law: F = ma (or more general: F = dp/dt where p = mv).

• 1 N = 1 kg·m/s².
• Impulse: J = F·Δt = Δp.

3rd Law: For every action, there is equal and opposite reaction. Acts on different bodies (so they don't cancel).

FREE-BODY DIAGRAM (FBD) — workflow:

1. Isolate the body.
2. Draw all forces acting on it (weight, normal, friction, applied, tension).
3. Choose axes (often along motion).
4. Write F = ma in each axis.

TYPES OF FORCES:

• Weight: W = mg, downward.
• Normal force (N): perpendicular to contact surface, contact constraint.
• Tension (T): along string, same throughout if string massless and frictionless.
• Friction (f): opposes relative motion / tendency of motion.
• Spring force: F = −kx (Hooke's law).

FRICTION

• Static friction: fs ≤ μs·N. Adjusts up to μs·N.
• Kinetic friction: fk = μk·N. (μk < μs.)
• Coefficient of friction is dimensionless.
• Angle of repose (θᵣ): tan θᵣ = μs.
• Angle of friction (θf): tan θf = μs (= angle of repose).
• Rolling friction << kinetic friction.

SYSTEMS WITH PULLEYS (Atwood machine):

For masses m₁ > m₂ over a frictionless pulley:

• Acceleration a = (m₁ − m₂)g / (m₁ + m₂).
• Tension T = 2 m₁ m₂ g / (m₁ + m₂).

CIRCULAR MOTION:

For an object in uniform circular motion (radius r, speed v):

• Centripetal acceleration aᶜ = v²/r = ω²r, directed inward.
• Centripetal force Fᶜ = mv²/r.
• Source can be: tension (string), gravity (orbits), friction (car turning), normal (banked road).

Banked road: for no friction, tan θ = v²/(rg). With friction: tan θ ± μ multiplier.

Conical pendulum: tension component supplies centripetal force.

PSEUDO FORCES (non-inertial frames):

When observer is accelerating with acceleration a, we add a pseudo force F = −ma on every body to use Newton's laws.

• Centrifugal force = mv²/r outward (in rotating frame).
• In a lift accelerating up at a: apparent weight = m(g+a).
• In free fall: apparent weight = 0 (weightlessness).

LAW OF CONSERVATION OF MOMENTUM:

In absence of external force, total momentum is conserved.

• Used in: collisions, rocket propulsion, recoil.
• Recoil of gun: m_gun · V_gun = m_bullet · V_bullet.

COLLISIONS:

1D elastic collision (equal masses): bodies exchange velocities.

Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂). e=1 elastic; e=0 perfectly inelastic.

EXAM HOOKS:

• F = dp/dt — useful when mass varies (rockets).
• Apparent weight in lift: m(g±a).
• Banked road frictionless: tan θ = v²/(rg).
• Pseudo force always opposes acceleration of frame.
• In Atwood, if m₁ = m₂, system in equilibrium.
• Conservation of momentum applies even when KE isn't (inelastic).

The popular form F = ma is only true when mass is constant. The general form is:

F = dp/dt, where p = mv is linear momentum.

This means force is the rate of change of momentum. When mass is constant, dp/dt = m(dv/dt) = ma, and you recover F = ma.

Why this matters. For systems where mass changes — rockets ejecting fuel, raindrops growing as they fall — F = ma gives wrong answers. You must use the momentum form.

Worked example: pulling a chain. A chain of mass M and length L lies on a frictionless table. You pull one end up with constant velocity v. The force needed isn't just (M/L) × v² — you have to account for the mass being accelerated into motion at the link being lifted. The momentum form handles this naturally; F = ma does not.

Units. 1 newton = 1 kg·m/s². If you ever see "kg-force", multiply by g (≈ 9.8) to convert to newtons.

90% of "Laws of Motion" problems become trivial once you draw a correct free body diagram (FBD).

Steps:

1. Pick one object. Draw it as a dot or box.
2. Draw every force acting on it as an arrow from the dot. Common forces: weight (mg, down), normal (perpendicular to surface), tension (along string), friction (parallel to surface, opposing relative motion).
3. Choose axes — usually one along motion, one perpendicular.
4. Write ΣF = ma along each axis.

Worked example. A 5 kg block on a 30° incline. Weight = 5 × 9.8 = 49 N straight down. Decompose: along incline = mg sin 30° = 24.5 N (down the slope); perpendicular = mg cos 30° = 42.4 N (into surface). Normal force N balances the perpendicular component. If frictionless, the block accelerates down the slope at a = g sin 30° = 4.9 m/s².

Common mistake: people mix forces on the object with forces by the object. Newton's third law pairs always act on different objects — never include both members of a pair in the same FBD.

Newton's 3 laws (1687, Principia):

1st Law (Inertia): A body remains at rest or in uniform motion unless acted on by external force.

• Mass is a measure of inertia.
• Defines inertial frames.

2nd Law: F = ma (or more general: F = dp/dt where p = mv).

• 1 N = 1 kg·m/s².
• Impulse: J = F·Δt = Δp.

3rd Law: For every action, there is equal and opposite reaction. Acts on different bodies (so they don't cancel).

FREE-BODY DIAGRAM (FBD) — workflow:

1. Isolate the body.
2. Draw all forces acting on it (weight, normal, friction, applied, tension).
3. Choose axes (often along motion).
4. Write F = ma in each axis.

TYPES OF FORCES:

• Weight: W = mg, downward.
• Normal force (N): perpendicular to contact surface, contact constraint.
• Tension (T): along string, same throughout if string massless and frictionless.
• Friction (f): opposes relative motion / tendency of motion.
• Spring force: F = −kx (Hooke's law).

FRICTION

• Static friction: fs ≤ μs·N. Adjusts up to μs·N.
• Kinetic friction: fk = μk·N. (μk < μs.)
• Coefficient of friction is dimensionless.
• Angle of repose (θᵣ): tan θᵣ = μs.
• Angle of friction (θf): tan θf = μs (= angle of repose).
• Rolling friction << kinetic friction.

SYSTEMS WITH PULLEYS (Atwood machine):

For masses m₁ > m₂ over a frictionless pulley:

• Acceleration a = (m₁ − m₂)g / (m₁ + m₂).
• Tension T = 2 m₁ m₂ g / (m₁ + m₂).

CIRCULAR MOTION:

For an object in uniform circular motion (radius r, speed v):

• Centripetal acceleration aᶜ = v²/r = ω²r, directed inward.
• Centripetal force Fᶜ = mv²/r.
• Source can be: tension (string), gravity (orbits), friction (car turning), normal (banked road).

Banked road: for no friction, tan θ = v²/(rg). With friction: tan θ ± μ multiplier.

Conical pendulum: tension component supplies centripetal force.

PSEUDO FORCES (non-inertial frames):

When observer is accelerating with acceleration a, we add a pseudo force F = −ma on every body to use Newton's laws.

• Centrifugal force = mv²/r outward (in rotating frame).
• In a lift accelerating up at a: apparent weight = m(g+a).
• In free fall: apparent weight = 0 (weightlessness).

LAW OF CONSERVATION OF MOMENTUM:

In absence of external force, total momentum is conserved.

• Used in: collisions, rocket propulsion, recoil.
• Recoil of gun: m_gun · V_gun = m_bullet · V_bullet.

COLLISIONS:

1D elastic collision (equal masses): bodies exchange velocities.

Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂). e=1 elastic; e=0 perfectly inelastic.

EXAM HOOKS:

• F = dp/dt — useful when mass varies (rockets).
• Apparent weight in lift: m(g±a).
• Banked road frictionless: tan θ = v²/(rg).
• Pseudo force always opposes acceleration of frame.
• In Atwood, if m₁ = m₂, system in equilibrium.
• Conservation of momentum applies even when KE isn't (inelastic).

Newton's 3 laws (1687, Principia):

1st Law (Inertia): A body remains at rest or in uniform motion unless acted on by external force.

• Mass is a measure of inertia.
• Defines inertial frames.

2nd Law: F = ma (or more general: F = dp/dt where p = mv).

• 1 N = 1 kg·m/s².
• Impulse: J = F·Δt = Δp.

3rd Law: For every action, there is equal and opposite reaction. Acts on different bodies (so they don't cancel).

FREE-BODY DIAGRAM (FBD) — workflow:

1. Isolate the body.
2. Draw all forces acting on it (weight, normal, friction, applied, tension).
3. Choose axes (often along motion).
4. Write F = ma in each axis.

TYPES OF FORCES:

• Weight: W = mg, downward.
• Normal force (N): perpendicular to contact surface, contact constraint.
• Tension (T): along string, same throughout if string massless and frictionless.
• Friction (f): opposes relative motion / tendency of motion.
• Spring force: F = −kx (Hooke's law).

FRICTION

• Static friction: fs ≤ μs·N. Adjusts up to μs·N.
• Kinetic friction: fk = μk·N. (μk < μs.)
• Coefficient of friction is dimensionless.
• Angle of repose (θᵣ): tan θᵣ = μs.
• Angle of friction (θf): tan θf = μs (= angle of repose).
• Rolling friction << kinetic friction.

SYSTEMS WITH PULLEYS (Atwood machine):

For masses m₁ > m₂ over a frictionless pulley:

• Acceleration a = (m₁ − m₂)g / (m₁ + m₂).
• Tension T = 2 m₁ m₂ g / (m₁ + m₂).

CIRCULAR MOTION:

For an object in uniform circular motion (radius r, speed v):

• Centripetal acceleration aᶜ = v²/r = ω²r, directed inward.
• Centripetal force Fᶜ = mv²/r.
• Source can be: tension (string), gravity (orbits), friction (car turning), normal (banked road).

Banked road: for no friction, tan θ = v²/(rg). With friction: tan θ ± μ multiplier.

Conical pendulum: tension component supplies centripetal force.

PSEUDO FORCES (non-inertial frames):

When observer is accelerating with acceleration a, we add a pseudo force F = −ma on every body to use Newton's laws.

• Centrifugal force = mv²/r outward (in rotating frame).
• In a lift accelerating up at a: apparent weight = m(g+a).
• In free fall: apparent weight = 0 (weightlessness).

LAW OF CONSERVATION OF MOMENTUM:

In absence of external force, total momentum is conserved.

• Used in: collisions, rocket propulsion, recoil.
• Recoil of gun: m_gun · V_gun = m_bullet · V_bullet.

COLLISIONS:

1D elastic collision (equal masses): bodies exchange velocities.

Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂). e=1 elastic; e=0 perfectly inelastic.

EXAM HOOKS:

• F = dp/dt — useful when mass varies (rockets).
• Apparent weight in lift: m(g±a).
• Banked road frictionless: tan θ = v²/(rg).
• Pseudo force always opposes acceleration of frame.
• In Atwood, if m₁ = m₂, system in equilibrium.
• Conservation of momentum applies even when KE isn't (inelastic).

Two distinct frictional forces show up in problems:

• Static friction (f_s) acts between surfaces not sliding relative to each other. It is not a fixed value — it adjusts to whatever is needed to prevent motion, up to a maximum: f_s ≤ μ_s × N.

• Kinetic friction (f_k) acts between surfaces sliding relative to each other. It is a fixed value: f_k = μ_k × N.

Important: μ_s > μ_k (almost always). That's why pushing a heavy box gets easier the moment it starts sliding.

The mistake. Students often write f = μN even when the object isn't moving. Wrong — f only equals μ_s × N at the threshold of slipping, not before.

Worked example. A 10 kg box on the floor, μ_s = 0.4, μ_k = 0.3. You push with 30 N horizontally. Maximum static friction = 0.4 × 10 × 9.8 = 39.2 N. Since 30 < 39.2, the box stays still and friction = 30 N (matching your push exactly). Now push with 50 N: 50 > 39.2, so the box slides. Friction is now kinetic = 0.3 × 10 × 9.8 = 29.4 N. Net force = 50 − 29.4 = 20.6 N. Acceleration = 2.06 m/s².

Newton's 3 laws (1687, Principia):

1st Law (Inertia): A body remains at rest or in uniform motion unless acted on by external force.

• Mass is a measure of inertia.
• Defines inertial frames.

2nd Law: F = ma (or more general: F = dp/dt where p = mv).

• 1 N = 1 kg·m/s².
• Impulse: J = F·Δt = Δp.

3rd Law: For every action, there is equal and opposite reaction. Acts on different bodies (so they don't cancel).

FREE-BODY DIAGRAM (FBD) — workflow:

1. Isolate the body.
2. Draw all forces acting on it (weight, normal, friction, applied, tension).
3. Choose axes (often along motion).
4. Write F = ma in each axis.

TYPES OF FORCES:

• Weight: W = mg, downward.
• Normal force (N): perpendicular to contact surface, contact constraint.
• Tension (T): along string, same throughout if string massless and frictionless.
• Friction (f): opposes relative motion / tendency of motion.
• Spring force: F = −kx (Hooke's law).

FRICTION

• Static friction: fs ≤ μs·N. Adjusts up to μs·N.
• Kinetic friction: fk = μk·N. (μk < μs.)
• Coefficient of friction is dimensionless.
• Angle of repose (θᵣ): tan θᵣ = μs.
• Angle of friction (θf): tan θf = μs (= angle of repose).
• Rolling friction << kinetic friction.

SYSTEMS WITH PULLEYS (Atwood machine):

For masses m₁ > m₂ over a frictionless pulley:

• Acceleration a = (m₁ − m₂)g / (m₁ + m₂).
• Tension T = 2 m₁ m₂ g / (m₁ + m₂).

CIRCULAR MOTION:

For an object in uniform circular motion (radius r, speed v):

• Centripetal acceleration aᶜ = v²/r = ω²r, directed inward.
• Centripetal force Fᶜ = mv²/r.
• Source can be: tension (string), gravity (orbits), friction (car turning), normal (banked road).

Banked road: for no friction, tan θ = v²/(rg). With friction: tan θ ± μ multiplier.

Conical pendulum: tension component supplies centripetal force.

PSEUDO FORCES (non-inertial frames):

When observer is accelerating with acceleration a, we add a pseudo force F = −ma on every body to use Newton's laws.

• Centrifugal force = mv²/r outward (in rotating frame).
• In a lift accelerating up at a: apparent weight = m(g+a).
• In free fall: apparent weight = 0 (weightlessness).

LAW OF CONSERVATION OF MOMENTUM:

In absence of external force, total momentum is conserved.

• Used in: collisions, rocket propulsion, recoil.
• Recoil of gun: m_gun · V_gun = m_bullet · V_bullet.

COLLISIONS:

1D elastic collision (equal masses): bodies exchange velocities.

Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂). e=1 elastic; e=0 perfectly inelastic.

EXAM HOOKS:

• F = dp/dt — useful when mass varies (rockets).
• Apparent weight in lift: m(g±a).
• Banked road frictionless: tan θ = v²/(rg).
• Pseudo force always opposes acceleration of frame.
• In Atwood, if m₁ = m₂, system in equilibrium.
• Conservation of momentum applies even when KE isn't (inelastic).

Newton's 3 laws (1687, Principia):

1st Law (Inertia): A body remains at rest or in uniform motion unless acted on by external force.

• Mass is a measure of inertia.
• Defines inertial frames.

2nd Law: F = ma (or more general: F = dp/dt where p = mv).

• 1 N = 1 kg·m/s².
• Impulse: J = F·Δt = Δp.

3rd Law: For every action, there is equal and opposite reaction. Acts on different bodies (so they don't cancel).

FREE-BODY DIAGRAM (FBD) — workflow:

1. Isolate the body.
2. Draw all forces acting on it (weight, normal, friction, applied, tension).
3. Choose axes (often along motion).
4. Write F = ma in each axis.

TYPES OF FORCES:

• Weight: W = mg, downward.
• Normal force (N): perpendicular to contact surface, contact constraint.
• Tension (T): along string, same throughout if string massless and frictionless.
• Friction (f): opposes relative motion / tendency of motion.
• Spring force: F = −kx (Hooke's law).

FRICTION

• Static friction: fs ≤ μs·N. Adjusts up to μs·N.
• Kinetic friction: fk = μk·N. (μk < μs.)
• Coefficient of friction is dimensionless.
• Angle of repose (θᵣ): tan θᵣ = μs.
• Angle of friction (θf): tan θf = μs (= angle of repose).
• Rolling friction << kinetic friction.

SYSTEMS WITH PULLEYS (Atwood machine):

For masses m₁ > m₂ over a frictionless pulley:

• Acceleration a = (m₁ − m₂)g / (m₁ + m₂).
• Tension T = 2 m₁ m₂ g / (m₁ + m₂).

CIRCULAR MOTION:

For an object in uniform circular motion (radius r, speed v):

• Centripetal acceleration aᶜ = v²/r = ω²r, directed inward.
• Centripetal force Fᶜ = mv²/r.
• Source can be: tension (string), gravity (orbits), friction (car turning), normal (banked road).

Banked road: for no friction, tan θ = v²/(rg). With friction: tan θ ± μ multiplier.

Conical pendulum: tension component supplies centripetal force.

PSEUDO FORCES (non-inertial frames):

When observer is accelerating with acceleration a, we add a pseudo force F = −ma on every body to use Newton's laws.

• Centrifugal force = mv²/r outward (in rotating frame).
• In a lift accelerating up at a: apparent weight = m(g+a).
• In free fall: apparent weight = 0 (weightlessness).

LAW OF CONSERVATION OF MOMENTUM:

In absence of external force, total momentum is conserved.

• Used in: collisions, rocket propulsion, recoil.
• Recoil of gun: m_gun · V_gun = m_bullet · V_bullet.

COLLISIONS:

1D elastic collision (equal masses): bodies exchange velocities.

Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂). e=1 elastic; e=0 perfectly inelastic.

EXAM HOOKS:

• F = dp/dt — useful when mass varies (rockets).
• Apparent weight in lift: m(g±a).
• Banked road frictionless: tan θ = v²/(rg).
• Pseudo force always opposes acceleration of frame.
• In Atwood, if m₁ = m₂, system in equilibrium.
• Conservation of momentum applies even when KE isn't (inelastic).