L-3: Laws of Motion

Why Do Things Move? — Starting Point

In the previous lesson you learnt to describe motion using displacement, velocity, and acceleration. But a deeper question remains: what makes an object move? Why does a ball rolling on the ground eventually stop? From everyday experience we know that pushing or pulling is needed to change an object's position. A football must be kicked, a cricket ball must be hit hard — muscular activity is involved and its effect is visible.

Yet many causes of motion are not visible. What makes raindrops fall? What makes Earth orbit the Sun? In this lesson you discover that force causes motion. Newton showed that force and motion are intimately connected. The laws of motion are fundamental — they help us understand walking, driving, collisions, rocket propulsion, and countless everyday phenomena.

NIOS learning objectives: explain inertia; state and illustrate Newton's three laws; explain conservation of momentum; understand equilibrium of concurrent forces; define coefficients of friction; distinguish static, kinetic, and rolling friction; suggest methods of reducing friction; and apply Newton's laws using free body diagrams.

3.1 Concepts of Force and Inertia

Stationary objects remain where placed unless forced to move. Similarly, an object moving with constant velocity must be forced to change its state of motion. The property by which an object resists change in its state of rest or uniform motion in a straight line is called inertia. Think of it as laziness built into matter — objects prefer to keep doing what they are already doing.

Mass is the measure of inertia. A heavy truck at low speed is harder to stop than a light cricket ball at high speed because the truck has far more mass. Inertia is a remarkable property. Without it, your books could mingle with your sibling's notes and your wardrobe could drift to a friend's house. However, rest and uniform motion are not absolute: an object at rest for one observer may appear in motion to another (Lesson 1). Observations confirm that change in velocity occurs only when a net force acts on the body.

3.1.1 Force and Its Effects

Force is familiar from daily life — pulling, pushing, kicking, hitting. Though invisible, its effects are seen. Forces can: (a) change the shape and size of an object (a balloon deforms); (b) influence motion — start motion, stop motion, or change direction or speed; (c) rotate a body about an axis (covered in Lesson 7).

Force is a vector quantity. When several forces act simultaneously, a net equivalent force is found by vector addition (Lesson 1). Motion is characterised by displacement and velocity. Velocity may increase (free fall) or decrease (ball on horizontal surface). A net non-zero force is required to change state of motion.

• If net force is parallel to velocity → speed increases.
• If net force is opposite to velocity → speed decreases.
• If net force is perpendicular to velocity → speed stays constant but direction changes (uniform circular motion).

Student tip: A body does NOT always move in the direction of net force. If it already has velocity, force may only bend the path. Acceleration always aligns with net force; velocity may not.

3.1.2 Galileo's Experiments and the First Law

When you roll a marble on a smooth floor it stops — friction reduces velocity to zero. To keep it moving at constant velocity you must continuously apply force. Similarly, a trolley at constant velocity must be continuously pushed or pulled to balance friction.

Galileo showed that without external force, a body continues in rest or uniform straight-line motion. On an inclined plane moving downward, a body accelerates; moving upward, it retards. On a horizontal plane with no friction, it would move with uniform speed forever.

In his thought experiment, a ball rolls down plane PQ and rises on plane RS to nearly the same height. As RS becomes less inclined, the ball travels farther. When RS is horizontal, the ball keeps moving indefinitely if friction is absent.

Newton's First Law: A body continues in a state of rest or of uniform motion in a straight line unless acted upon by a net external force.

Because motion is relative, measurements must be made with respect to a frame of reference. An inertial frame is one in which a body in translatory motion has constant velocity when no net external force acts. A frame fixed to Earth is inertial for practical purposes.

3.2 Concept of Momentum

A fielder struggles to stop a fast cricket ball (small mass, large velocity) but also struggles with a slow truck (large mass, small velocity). Both mass and velocity matter when studying force and motion.

The product of mass m and velocity v is linear momentum p:

Momentum changes when magnitude, direction, or both change. Example: a freely falling 2 kg object has zero momentum at t = 0; at t = 1 s, v = 9.8 m/s downward so p = 19.6 kg·m/s downward; at t = 2 s, p = 39.2 kg·m/s downward. Gravitational force causes this continuous increase.

When a 0.2 kg ball strikes a wall at 10 m/s and rebounds at 10 m/s, magnitude of momentum is unchanged but direction reverses. If initial momentum is +2 kg·m/s, final is −2 kg·m/s, so change Δp = −4 kg·m/s. The wall exerts force on the ball.

3.3 Newton's Second Law of Motion

A body at constant velocity has constant momentum; the first law implies no net external force. For a freely falling body, momentum increases with time — connecting force, time, and momentum change.

Second Law (momentum form): The rate of change of momentum is directly proportional to the net force, and change occurs in the direction of the net force.

Writing p = mv and assuming constant mass:

Example: A 0.4 kg ball rolling at 20 m/s stops in 10 s. |F| = m|Δv/Δt| = 0.4 × 20/10 = 0.8 N opposite to motion.

Example: A 10 kg body at 10 m/s is stopped by 50 N opposing force. t = m(v₀ − v)/F = 10 × 10/50 = 2 s.

Exam note: For rockets where mass changes with time, use F = d(mv)/dt — the more general form. F = ma alone is for constant mass only.

3.4 Forces in Pairs — Third Law and Impulse

When Earth pulls an object, does the object pull Earth? When you push an almirah, does it push back? Actions between two bodies are always mutual. Forces exist in pairs — both are real interaction forces.

Newton's Third Law: When two objects interact, the force exerted by one on the other is equal in magnitude and opposite in direction to the force exerted by the second on the first.

A book on a table exerts downward force F₁ = mg on the table; the table exerts upward F₂ on the book. F₁ and F₂ do not cancel because they act on different bodies. Action and reaction exist simultaneously, not one after the other.

Impulse

Action-Reaction Concept Map

                    NEWTON'S THIRD LAW
                           |
            +--------------+--------------+
            |              |              |
     Forces exist      Equal          Opposite
       in PAIRS       magnitude       direction
            |              |              |
            v              v              v
    +---------------+  F₁₂ = -F₂₁   Act on DIFFERENT
    |  Examples:    |              bodies (no cancel)
    +---------------+
    | Man kicks     |  Action: foot on ball
    |  football     |  Reaction: ball on foot
    +---------------+
    | Earth-Moon    |  Action: Earth pulls moon
    |               |  Reaction: moon pulls Earth
    +---------------+
    | Ball hits     |  Action: ball on wall
    |  wall         |  Reaction: wall on ball
    +---------------+
    | High jumper   |  Action: jumper pushes ground
    |               |  Reaction: ground pushes jumper UP
    +---------------+

3.5 Conservation of Momentum

When two bodies interact and mutual interaction is the only force, the vector sum of their momenta remains unchanged. For a closed (isolated) system — no external forces — total momentum is constant. Individual momenta may change due to internal mutual forces.

Derivation: From second law Δp = FΔt. For bodies A and B: F_AB = −F_BA, so Δp_A/Δt = −Δp_B/Δt, hence Δp_A + Δp_B = 0.

(a) Gun recoil: mv₁ + Mv₂ = 0, so v₂ = −(m/M)v₁. Since m ≪ M, recoil is much slower than the bullet.

(b) Collision: Two trolleys (mass m each) at v hit three stationary trolleys. 2mv = 5mv′ → v′ = (2/5)v.

(c) Explosion: Bomb at rest has zero momentum; fragments fly opposite ways so vector sum stays zero.

(d) Rocket propulsion: Escaping gases provide thrust via momentum conservation.

3.5.3 Equilibrium of Concurrent Forces

Forces acting simultaneously at one point are concurrent forces. They are in equilibrium when their resultant is zero: F₁ + F₂ + F₃ = 0. Graphically, the vector sum of any two must equal and oppose the third. For a point object in static equilibrium, the vector sum of all forces must be zero.

3.6 Friction

When a batsman hits a ball along the ground, it eventually stops — friction opposes motion and changes momentum. Friction is a contact force, parallel to surfaces and opposite to attempted or actual motion. Friction is a necessary evil: it wastes energy yet lets us walk, drive, and brake.

3.6.1 Static and Kinetic Friction

A block on a horizontal surface does not move until applied force F_ext exceeds a limit. While at rest, static friction f_s matches F_ext up to maximum f_s(max). Beyond this, the block slides and kinetic friction f_k acts. Starting motion needs more force than maintaining it: f_s(max) > f_k.

Example: 2 kg block, μ_s = 0.25 → f_s(max) = 0.25 × 2 × 9.8 = 4.9 N.

Example: 5 kg block, μ_k = 0.1, pulled by 10 N. f_k = 4.9 N; F_net = 5.1 N; a = 1.02 m/s².

3.6.2 Rolling Friction and Reducing Friction

Rolling friction is much smaller than sliding friction (~1/100 for steel on steel). Methods to reduce friction: wheels and ball bearings; lubricants (grease, oil); compressed air between surfaces; streamlined shapes for fluid friction. Cars are most fuel-efficient around 40–45 km/h due to air resistance.

3.7 Free Body Diagram Technique

Newton's laws become easier with free body diagrams (FBD) — diagrams showing all forces on an isolated object.

         HOW TO SOLVE FBD PROBLEMS
         =========================
                    |
                    v
         [1] Draw neat diagram of system
                    |
                    v
         [2] ISOLATE the object of interest
             (this is the "free body")
                    |
                    v
         [3] Mark ALL external forces with arrows
             touching the body; show line of action
                    |
                    v
         [4] Apply Newton's 2nd Law:
             ΣF = ma   (or ΣFx = max, ΣFy = may)
                    |
                    v
         [5] Count unknowns vs independent equations
             Need equal number for complete solution
                    |
                    v
         [6] Solve + CHECK extreme cases
             (e.g. m1 = m2, m1 >> m2 for pulleys)

Example — Two blocks on smooth surface: Blocks m₁ and m₂ connected by string; m₂ pulled by F. a = F/(m₁ + m₂). Tension T = m₁F/(m₁ + m₂).

Example — Atwood machine: m₁ > m₂ over frictionless pulley. a = (m₁ − m₂)g/(m₁ + m₂); T = 2m₁m₂g/(m₁ + m₂).

Example — Trolley with friction: M = 10 kg, m = 2 kg, μ_k = 0.02 gives a = 1.47 m/s², T = 16.66 N.

3.8 Inertial and Non-Inertial Frames

An inertial frame is stationary or moves at constant velocity; Newton's laws hold directly. A non-inertial frame accelerates — add a pseudo force (−ma). Centrifugal force appears in rotating frames. Water tilts in a starting train because effective gravity in the train frame is g − a.

Summary

• Inertia resists change in rest or uniform motion; mass measures inertia.
• First Law: Uniform motion or rest continues unless net external force acts.
• Second Law: F = Δp/Δt; for constant mass F = ma.
• Third Law: F₁₂ = −F₂₁ on different bodies.
• Conservation: Isolated system → total momentum constant.
• Friction: f_s(max) = μ_s F_N; f_k = μ_k F_N; rolling ≪ kinetic.
• FBD before every numerical — the most reliable exam strategy.