Overview
This chapter introduces 'Force and Laws of Motion' — a foundational topic that explains how and why objects move. It begins with the concept of force as a push or pull that can change the state of motion of an object, defines balanced and unbalanced forces, and explains inertia as the tendency of objects to resist changes in motion. The chapter develops Newton's three laws of motion: the first (law of inertia), the second (quantitative relation F = ma), and the third (action–reaction pairs). Important related concepts such as mass versus weight, unit of force (newton), momentum, impulse, conservation of momentum in collisions, and the role of friction are covered. Practical activities and examples illustrate how these laws operate in everyday life (seat belts, vehicle collisions, sporting actions) and in experiments (trolley collisions, friction tests). Importance: Understanding these laws builds the basis for dynamics in higher classes, helps analyse real-world problems involving forces and motion, and trains students to apply quantitative reasoning and simple experiments. Key themes: definition and measurement of force, inertia and types of inertia, Newton's three laws with…
Learning Objectives
- Define force, mass and inertia.
- State Newton's three laws of motion.
- Explain the concept of inertia and distinguish inertia of rest, inertia of motion and inertia of direction.
- Apply Newton's second law to calculate acceleration, net force or mass in numerical problems (F = ma).
- Derive the relation F = dp/dt and show how it reduces to F = ma for constant mass.
- Calculate linear momentum and impulse and use the impulse–momentum theorem to solve problems.
- Explain conservation of linear momentum and solve one-dimensional collision problems (elastic and inelastic).
- Describe action–reaction pairs with clear examples and analyze interactions using Newton's third law.
Topics in this chapter
14 topics · tap a topic title to jump straight to it.
Force
Definition: A force is a push or a pull that can change the state of motion or shape of an object. It is a vector quantity (has magnitude and direction).
SI unit: Newton (N). 1 N is the force that gives a mass of 1 kg an acceleration of 1 m/s2.
Representation: Forces are represented by arrows (lines of action). The length of the arrow shows magnitude and the arrowhead shows direction.
Types of forces:
- Contact forces: applied force (push/pull), normal reaction, friction, tension, spring force.
- Non-contact forces: gravitational force (weight), electrostatic force, magnetic force.
Effects of a force: A force can (i) change the shape of an object (deformation), (ii) change the speed or direction of an object (produce acceleration), or (iii) keep an object in equilibrium (when resultant force is zero).
Balanced and unbalanced forces: If the resultant (net) force on an object is zero, forces are balanced and there is no change in motion. If resultant is non-zero (unbalanced), the object accelerates in the direction of the resultant.
Resultant and resolution: Multiple forces acting on a body can be replaced by a single resultant force using vector addition. A force can be resolved into perpendicular components (usually horizontal and vertical).
Measurement: Forces are commonly measured using a spring (spring balance) or force sensor. Hooke's law (for an elastic spring) states that extension is proportional to applied force up to the elastic limit.
Connection to Newton's laws (brief): Newton's Second Law relates force to motion: the net force on an object equals mass times acceleration (F_net = ma). Newton's First Law describes equilibrium (no net force → constant velocity). Newton's Third Law: forces occur in action–reaction pairs (equal and opposite).
- Pushing a door to open it (applied force produces rotation and motion).
- A book resting on a table: weight (gravity) downwards and normal reaction upwards (balanced forces).
- Pulling a trolley: tension in the rope causes the trolley to accelerate.
- Friction when braking a bicycle: frictional force opposes motion and slows the bicycle down.
- Magnet attracting a paperclip: non-contact magnetic force causing motion.
- Stretching a spring: elastic force opposes the pull (Hooke's law region).
- Newton's 2nd law: F_net = m · a (vector form).
- Weight (gravitational force): W = m · g (g ≈ 9.8 m/s² near Earth's surface).
- Hooke's law (spring): F = -k · x (k is spring constant, x is extension).
- Friction (approximate): f_k = μ_k · N and f_s(max) = μ_s · N (μ = coefficient of friction, N = normal force).
- Resultant of two perpendicular forces: R = √(F1² + F2²).
- Components: Fx = F · cosθ, Fy = F · sinθ (for a force F at angle θ).
Types of Forces
What is a force? A force is a push or pull on an object that can cause a change in its state of motion or shape. Force is a vector quantity — it has magnitude and direction. The SI unit of force is the newton (N).
Main classification
- Contact forces: Forces that arise from physical contact between objects. Common contact forces include:
- Frictional force — resists relative motion between surfaces (static and kinetic friction).
- Normal reaction — perpendicular contact force from a surface on an object placed on it.
- Tension — force transmitted through a string, rope or cable when it is pulled tight.
- Spring (elastic) force — force exerted by a stretched or compressed spring (obeys Hooke's law within elastic limit).
- Applied (muscular) force — any push or pull applied by a person or machine.
- Air resistance / drag — frictional force experienced by objects moving through a fluid (air or water).
- Buoyant force — upward force exerted by a fluid on a submerged or floating body.
- Non-contact (action-at-a-distance) forces: Forces that act without physical contact:
- Gravitational force — attraction between masses; near Earth it gives objects their weight.
- Electrostatic force — attraction/repulsion between charged bodies.
- Magnetic force — force between magnets or between magnets and magnetic materials.
Balanced and unbalanced forces: If the vector sum of all forces on an object is zero, forces are balanced and there is no change in motion (constant velocity). If the net force is non-zero (unbalanced), the object accelerates.
Representation: Forces are shown by arrows in free-body diagrams; arrow length represents magnitude and arrow direction shows the direction of the force.
Link to Newton's laws: Newton's second law quantitatively relates force, mass and acceleration (F = ma). Many types of forces appear as terms in equations of motion.
- Gravitational force: An apple falling from a tree toward the Earth; the weight of a person standing on the ground (W = mg).
- Frictional force: Brakes slowing a bicycle, or shoes providing grip when you walk.
- Tension: A chandelier hanging from a ceiling by a chain; the chain transmits tension.
- Normal reaction: A book resting on a table experiences an upward normal force from the table.
- Spring force: A spring balance showing weight; the spring pulls back when stretched.
- Air resistance: A parachute slowing the descent of a skydiver.
- Newton's second law: F_net = m · a (Net force equals mass times acceleration)
- Weight (gravitational force near Earth): W = m · g (g ≈ 9.8 m/s²)
- Hooke's law (spring, within elastic limit): F_spring = −k · x (k = spring constant, x = extension/compression)
- Friction (approximate): f = μ · N (μ = coefficient of friction, N = normal force). Static μs > kinetic μk
- Impulse (change in momentum): J = Δp = F_avg · Δt (area under force–time curve)
- Resultant force (vector sum): ΣF = F1 + F2 + ... (use vector addition or components)
Balanced and Unbalanced Forces
What is a force? A force is a push or pull that can change the state of rest or motion of an object. It is a vector quantity (has magnitude and direction). SI unit: newton (N).
Balanced forces
- When two or more forces acting on an object produce a net (resultant) force of zero, they are called balanced forces.
- Result: No change in the object's state of motion. If the object was at rest, it remains at rest; if it was moving, it continues to move with constant velocity (Newton's first law).
- Condition: ΣF = 0 (sum of all vector forces equals zero).
Unbalanced forces
- If the vector sum of forces on an object is not zero, the forces are unbalanced.
- Result: The object accelerates (its speed and/or direction changes). The acceleration is in the direction of the net force (Newton's second law).
- Condition: ΣF ≠ 0, and a = F_net / m.
How to identify
- Draw a free-body diagram showing all forces with directions (gravity, normal reaction, friction, applied force, tension, etc.).
- Resolve forces along a chosen axis and add vectorially. If the resultant is zero, forces are balanced.
Simple example calculation
- Mass m = 5 kg, resultant force F_net = 10 N to the right. Acceleration a = F_net / m = 10 / 5 = 2 m/s² to the right.
Important points
- Balanced forces do not necessarily mean zero forces — they mean the forces cancel each other.
- An object in dynamic equilibrium moves with constant velocity (balanced forces) — there is no acceleration.
- Unbalanced forces change the velocity (magnitude or direction).
- Balanced: A book lying on a table — gravity downward is balanced by the table's normal force upward (net force = 0).
- Balanced (dynamic): A car moving at constant speed on a straight, level road when driving force equals resistive forces (engine thrust = friction + air resistance).
- Unbalanced: Pushing a stationary box with a greater force than friction — box starts to move (net force ≠ 0).
- Unbalanced: A car accelerating when the engine thrust exceeds resistive forces (net forward force produces acceleration).
- Unbalanced: A ball thrown upward — gravity (unbalanced) slows it down, reverses direction, and accelerates it downward.
- Balanced (tug-of-war tie): Two teams pull with equal force in opposite directions so the rope doesn’t move.
- Net force: F_net = ΣF (vector sum of all forces acting on the object)
- Newton's 2nd law: F_net = m · a
- Acceleration from net force: a = F_net / m
- Equilibrium (balanced forces) condition: ΣF = 0 → a = 0
- Units: Force in newton (N), mass in kilogram (kg), acceleration in m/s² (1 N = 1 kg·m/s²)
Inertia
Definition: Inertia is the property of a body to resist any change in its state of motion (whether at rest or moving). A body at rest tends to remain at rest, and a body in motion tends to remain in motion with the same speed and direction unless acted upon by an external force. This idea is expressed by Newton's First Law of Motion (the law of inertia).
Types of inertia:
- Inertia of rest: tendency to remain at rest (e.g., a book stays on a table unless pushed).
- Inertia of motion: tendency to continue moving (e.g., a rolling ball keeps rolling unless friction slows it).
- Inertia of direction: tendency to continue moving in the same direction (e.g., passengers lean outward when a car turns sharply).
Mass as measure of inertia: The greater the mass of an object, the greater its inertia. Mass is the quantitative measure of inertia — heavy objects resist changes in motion more than light objects. In other words, for the same force, a larger mass produces a smaller acceleration (Newton's second law).
Relation to laws of motion: Newton's First Law states the principle of inertia. Newton's Second Law (F = ma) gives the quantitative relationship: a body’s acceleration (change of motion) produced by a net force is inversely proportional to its mass (measure of inertia). Thus to change the motion of a more massive object requires a larger force.
Important points:
- Inertia is a passive property; it does not cause motion but resists change of motion.
- Inertia depends only on mass, not on velocity or the presence of friction.
- Every object has inertia, including those at rest and those moving.
- When a bus starts suddenly, passengers are pushed backwards — inertia of rest (their bodies try to remain at rest while the bus moves forward).
- When a bus brakes suddenly, passengers lurch forward — inertia of motion (their bodies tend to keep moving).
- The tablecloth trick: a quick pull removes the cloth but dishes remain nearly in place — inertia resists sudden change of motion.
- A heavy shopping trolley is harder to start or stop than an empty one — more mass means more inertia.
- Coins on a card: flick the card quickly and the coins drop into the glass — inertia keeps the coins almost at rest while the card is removed.
- A bicycle rider leans into a turn to change direction because the bicycle and rider tend to continue in a straight line (inertia of direction).
- Newton's First Law (verbal): A body remains at rest or in uniform motion in a straight line unless acted on by an external force (law of inertia).
- Newton's Second Law: F = m a (Net force F produces acceleration a in a body of mass m).
- Acceleration from a given force: a = F / m (shows a ∝ 1/m for same force — larger mass → smaller acceleration).
- Momentum (related concept): p = m v (more mass → greater momentum for same velocity).
- Extension (rotational inertia, advanced): I = Σ m r^2 (moment of inertia for point masses at distance r from axis; used in rotation problems).
Newton's First Law of Motion (Law of Inertia)
Statement: A body at rest will remain at rest, and a body in motion will continue to move with constant velocity in a straight line unless acted upon by an external unbalanced force. This is Newton's First Law of Motion, also called the Law of Inertia.
What it means: Motion does not change by itself. To change the state of motion (to start, stop, speed up, slow down or change direction) an unbalanced external force is required.
Inertia: Inertia is the property of matter by which it resists any change in its state of motion. Mass is the quantitative measure of inertia — larger mass means greater inertia and more resistance to change in motion.
Types of inertia:
- Inertia of rest — tendency to remain at rest.
- Inertia of motion — tendency to continue moving.
- Inertia of direction — tendency to maintain direction of motion.
Connection with other laws (brief): Using Newton's Second Law, F = ma. If the net external force ΣF = 0, then a = 0, so the velocity remains constant (which is exactly the First Law).
Important notes: The First Law is valid in inertial frames of reference (frames that are not accelerating). In a non-inertial (accelerating) frame, apparent forces (like centrifugal force) appear and must be accounted for.
- When a moving car suddenly stops, passengers lunge forward — their bodies tend to remain in motion (hence seatbelts are needed).
- A book on a table remains at rest until you push it — inertia of rest.
- If you pull a card quickly from under a coin, the coin remains almost at rest and drops into the glass — inertia resists change of motion.
- On sudden acceleration of a bus, passengers are pushed backwards — inertia of motion relative to the bus.
- A tablecloth pulled quickly from under dishes leaves dishes nearly undisturbed if friction is small — demonstrates inertia.
- A ball in a moving cart continues to move in a straight line if the cart suddenly stops; relative motion makes the ball appear to lurch forward.
- Net force zero: ΣF = 0 ⇒ acceleration a = 0 (constant velocity or rest)
- Newton's second law (related): ΣF = ma
- If a = 0 then v = constant (including v = 0 for rest)
- Qualitative relation: inertia ∝ mass (greater mass ⇒ greater inertia)
Mass and Weight
Mass: Mass is the amount of matter in a body. It is a scalar quantity and does not change with location. SI unit: kilogram (kg). Mass is measured using a beam balance (compares two masses) and is independent of local gravity.
Weight: Weight is the gravitational force exerted on a mass by a massive body (for example, the Earth). It is a vector quantity, directed towards the centre of the attracting body. SI unit: newton (N). Weight is measured using a spring balance (it measures force through extension).
Relation between mass and weight:
- Weight W = m g, where m is mass and g is the acceleration due to gravity at that location (typical g at Earth’s surface ≈ 9.8 m/s2).
- The acceleration due to gravity g itself is g = GM/r2, where G is the universal gravitational constant, M is the mass of the attracting body (Earth), and r is the distance from its centre.
Key differences:
- Mass is constant for a body everywhere; weight depends on local g and therefore changes with location (height, latitude, planet).
- Mass is measured in kg; weight in N.
- Mass is measured by comparing with standard masses (beam balance); weight is measured by the force a scale/spring experiences (spring balance).
Apparent weight and weightlessness:
- If an object of mass m is in a non-inertial frame (for example, an elevator accelerating upward with acceleration a), the reading of a spring balance (apparent weight) becomes Wapp = m(g + a) when acceleration is upward, and Wapp = m(g − a) when downward.
- In free fall (a = g) Wapp = 0: objects appear weightless (astronauts in orbit are in continuous free fall around Earth and feel weightless).
Measurement notes:
- Beam balance compares masses, so it gives the same mass reading on Earth, Moon or in a lift in free-fall (until the balance itself is affected). A spring balance reads weight (force) and thus varies with g and acceleration.
- Weight varies slightly with altitude and latitude because g varies with r and Earth's rotation.
- A 5 kg mass on Earth (g = 9.8 m/s2) has weight W = m g = 5 × 9.8 = 49 N.
- Same 5 kg mass on the Moon (g ≈ 1.63 m/s2) has weight W = 5 × 1.63 ≈ 8.15 N — mass remains 5 kg but weight is smaller.
- A person standing in an elevator accelerating upward at 2 m/s2: apparent weight Wapp = m(g + 2). If m = 70 kg, Wapp = 70 × (9.8 + 2) = 826 N (greater than normal).
- In a freely falling lift (acceleration downward equal to g), the spring balance reads zero (weightless), though the mass is unchanged.
- A beam balance gives the same mass reading on Earth and on the Moon because it compares masses; a spring balance gives different readings because it measures weight.
- Weight: W = m g (W in newton, m in kg, g in m/s2)
- Acceleration due to gravity: g = G M / r2 (G = universal gravitational constant)
- Gravitational force between two masses: F = G m1 m2 / r2
- Apparent weight in accelerating frame (vertical): Wapp = m (g + a) for upward acceleration a
- Apparent weight in accelerating frame: Wapp = m (g − a) for downward acceleration a; Wapp = 0 when a = g (free fall)
Newton's Second Law of Motion
Statement: Newton's Second Law of Motion states that the acceleration produced in an object is directly proportional to the net force acting on it and inversely proportional to its mass. In symbol form: F_net = m a.
Meaning and key points:
- Net force: F_net is the vector sum of all forces acting on the body. Only the net (resultant) force determines acceleration.
- Proportionalities: a ∝ F_net (for constant mass) and a ∝ 1/m (for constant force).
- Vector nature: Since force and acceleration are vectors, the direction of acceleration is the same as the direction of the net force.
- Units: Force in SI units is newton (N). 1 N = 1 kg·m/s2. Mass is in kilograms (kg) and acceleration in m/s2.
- Special case (zero net force): If F_net = 0, acceleration is zero; the body either remains at rest or moves with constant velocity (Newton's First Law).
More general form (momentum): Newton's second law can be written as F_net = dp/dt, where p = m v is momentum. For constant mass this reduces to F_net = m (dv/dt) = m a. The momentum form is useful when mass changes (e.g., rockets).
Impulse: The change in momentum produced by a force applied for a short time Δt is given by Impulse J = F_avg Δt = Δp. Impulse explains effects of short strong forces (e.g., collisions).
Experimental verification (simple classroom setup): Using a low-friction trolley: (1) Apply the same force to trolleys of different masses—measured acceleration decreases as mass increases. (2) For a fixed mass, apply different forces—acceleration increases proportionally with force. Plotting force vs acceleration gives a straight line through the origin; slope = mass.
Applications: vehicle acceleration and braking, pushing shopping carts, sports (kicking/throwing), rocket motion (using momentum form), seatbelts (reducing impulse by increasing time of collision), and many mechanical systems.
- A 5 kg box is pushed with a net force of 20 N. Find its acceleration. a = F/m = 20 / 5 = 4 m/s^2.
- Two horizontal forces, 8 N to the right and 3 N to the left, act on a 2 kg object. Net force = 8 - 3 = 5 N (right). Acceleration = 5 / 2 = 2.5 m/s^2 (to the right).
- A car of mass 1000 kg accelerates from 0 to 20 m/s in 10 s. Average acceleration = (20 - 0)/10 = 2 m/s^2. Required net force ≈ m a = 1000 × 2 = 2000 N (forward).
- Kicking a stationary ball (mass 0.5 kg) gives it a velocity of 10 m/s instantly (idealised). Change in momentum Δp = 0.5 × 10 = 5 kg·m/s. If the kick lasted 0.05 s, average force ≈ Δp/Δt = 5 / 0.05 = 100 N.
- A rocket loses mass while expelling exhaust. Use F_net = dp/dt (momentum form) to account for thrust even though mass changes.
- During a sudden stop, seatbelts increase the stopping time so that the average force (impulse/time) on passengers is smaller, reducing injury.
- F_net = m a (net force = mass × acceleration)
- a = F_net / m (acceleration = net force divided by mass)
- F_net = dp/dt (force equals rate of change of momentum)
- Impulse J = F_avg Δt = Δp (impulse equals change in momentum)
- Weight: W = m g (a special case of force due to gravity, g ≈ 9.8 m/s^2)
Momentum
What is momentum?
Momentum is a physical quantity that measures the amount of motion an object has. For a particle moving with velocity v and having mass m, momentum (denoted by p) is defined as the product of mass and velocity: p = m v. Momentum is a vector quantity — it has both magnitude and direction (same as the velocity).
SI unit: kilogram metre per second (kg·m/s).
Key ideas
- Formula: p = m v. For constant mass, change in momentum Δp = m Δv.
- Impulse: Impulse is the effect of a force acting for a time. If a force F acts for a short time Δt, the impulse J = F Δt. Impulse changes momentum: J = Δp (this follows from Newton’s second law).
- Conservation of momentum: In an isolated system (no external net force), total momentum before an interaction equals total momentum after: for two bodies, m1 v1 + m2 v2 = m1 v1' + m2 v2'.
- Relation with force: F = dp/dt. A large force acting briefly can produce the same change in momentum as a smaller force acting longer.
Types of interactions: Elastic collisions conserve both momentum and kinetic energy; in inelastic collisions momentum is conserved but some kinetic energy is transformed (for example into heat or deformation).
Why momentum is useful: Momentum helps analyse collisions and motions where objects exchange motion, and explains effects such as recoil, why seatbelts and airbags reduce injury (they increase the time over which the body’s momentum is brought to zero, reducing force), and why heavier moving objects are harder to stop.
- A car of mass 1000 kg moving at 20 m/s has momentum p = 1000 × 20 = 20 000 kg·m/s.
- When a ball is caught, the catcher’s hands give an impulse that changes the ball’s momentum to zero. Bending the arms increases Δt and reduces the force on the hands.
- Recoil of a gun: the bullet gains forward momentum and the gun gains equal backward momentum so total momentum is conserved.
- Two billiard balls collide: momentum is transferred from one ball to the other during the short impact.
- A child on a skateboard pushes off the ground: the child+skateboard system gains momentum; if no external horizontal force acts, momentum is conserved.
- Momentum: p = m v (vector).
- Change of momentum: Δp = m Δv (for constant mass).
- Impulse: J = F Δt (for approximately constant force).
- Impulse–momentum theorem: J = Δp.
- Newton’s 2nd law (general form): F = dp/dt.
- Conservation (two-body): m1 v1 + m2 v2 = m1 v1' + m2 v2'.
Impulse
What is impulse? Impulse is a measure of the effect of a force acting over a short time interval. It is defined as the change in momentum of an object produced by a force acting for a time interval.
Mathematical definition and derivation: From Newton's second law, F = dp/dt. Integrating both sides over the time interval t1 to t2 gives
I = \int_{t1}^{t2} F(t) dt = p(t2) - p(t1) = \Delta p.
Thus impulse I equals the change in momentum Δp. For constant mass, Δp = m(v_f - v_i).
Special cases and average force: If the force is constant during the interval Δt = t2 − t1, then I = F Δt. For a variable force we often use the average force F_avg = Δp / Δt so that I = F_avg Δt.
Direction and units: Impulse is a vector and has the same direction as the change in momentum. SI unit: newton-second (N·s), which is equivalent to kg·m/s.
Physical meaning and applications: A large impulse can be produced by a large force acting for a short time (big F, small Δt) or by a smaller force acting for a longer time (smaller F, larger Δt) as long as the area under the force–time curve is the same. This principle explains why safety devices (airbags, cushioned mats) reduce injury: they increase the time over which momentum is brought to zero, reducing the average force on the body.
Impulse–momentum theorem (summary): The impulse delivered to an object equals the change in its momentum: I = Δp. This is widely used to analyze collisions and impacts.
- Kicking a football: the foot applies a force for a short time, changing the ball's momentum (impulse = change in ball's momentum).
- Catching a fast ball with a gloved hand: the glove increases the time of contact, increasing Δt and reducing the average force felt by the hand.
- Car airbags: during a crash the airbag increases the time over which the passenger's momentum is reduced, lowering the force on the passenger.
- Jumping off the ground: your legs exert a force on the ground over the time of push-off; impulse changes your momentum upward.
- Bouncing vs. sticking collisions: in a bounce (elastic or partially elastic) the change in momentum is larger (reversal of velocity) so impulse is larger than in a perfectly inelastic collision where objects stick together.
- Using brakes to stop a bicycle: applying brakes over a longer distance/time reduces average force compared with an abrupt stop (same change in momentum).
- Impulse I = Δp (change in momentum)
- \[I = ∫_{t1}^{t2} F(t) dt (general definition\]\[area under F–t curve)\]
- For constant force: I = F Δt
- For constant mass: I = m(v_f - v_i)
- Average force: F_avg = Δp / Δt
- SI unit: 1 N·s = 1 kg·m/s
Law of Conservation of Momentum
Definition: The law of conservation of momentum states that if no external force acts on a closed (isolated) system of interacting objects, the total momentum of the system remains constant in time.
Momentum (brief): Momentum of a body is defined as the product of its mass and velocity: p = m v. It is a vector quantity (has direction).
Why it is true (simple derivation for two bodies):strong>
- Consider two bodies 1 and 2 that interact with each other. Let internal force on 1 by 2 be F12 and on 2 by 1 be F21.
- By Newton's third law, F12 = −F21.
- Rate of change of momentum: dp1/dt = F12, dp2/dt = F21. Adding, dp1/dt + dp2/dt = F12 + F21 = 0.
- So d/dt (p1 + p2) = 0 ⇒ total momentum ptotal = constant.
Meaning and consequences:
- In an isolated system, momentum before interaction (collision or explosion) equals momentum after interaction: total p(before) = total p(after).
- Individual momenta can change, but their vector sum stays constant.
- Impulse relates force and momentum: impulse = change in momentum.
Types of collisions (brief):
- Elastic collision: both momentum and kinetic energy are conserved.
- Inelastic collision: momentum is conserved but some kinetic energy is converted to other forms (heat, deformation).
- Perfectly (completely) inelastic collision: the colliding bodies stick together after collision; they move with a common velocity.
Important notes for Class 9: Always check whether external forces (like friction, external pushes) are negligible. If they are negligible over the short interaction time (e.g., two colliding balls), treat the system as isolated and use momentum conservation.
- Recoil of a gun: When a bullet is fired forward, the gun recoils backward. Total momentum before firing (zero) equals total momentum after: m_bullet·v_bullet + m_gun·v_gun = 0.
- Ice skaters pushing apart: Two skaters initially at rest push each other. They move in opposite directions such that the vector sum of their momenta is zero.
- Billiard balls collision: When one ball strikes another, momentum is transferred; the total momentum of the two-ball system (neglecting friction) remains constant.
- Car crash (inelastic): In a collision where cars stick together, momentum is conserved even though kinetic energy is not conserved.
- Explosion/firework: A firework shell explodes into fragments; the vector sum of fragment momenta equals the momentum of the shell just before explosion.
- Simple numeric example (perfectly inelastic): A 2 kg object moving at 3 m/s collides and sticks to a 3 kg object at rest. Common velocity v = (2·3 + 3·0)/(2+3) = 6/5 = 1.2 m/s.
- Momentum of a body: p = m v (vector).
- Conservation of total momentum (two-body): m1·v1 + m2·v2 = m1·v1' + m2·v2' (if no external force).
- Impulse: J = Δp = F_avg · Δt (impulse = change in momentum; area under force–time curve).
- Common velocity after perfectly inelastic (stick together): v_common = (m1·v1 + m2·v2) / (m1 + m2).
- One-dimensional elastic collision (final velocities): v1' = [(m1 - m2)/(m1 + m2)]·v1 + [2m2/(m1 + m2)]·v2 v2' = [2m1/(m1 + m2)]·v1 + [(m2 - m1)/(m1 + m2)]·v2 (useful when kinetic energy is also conserved).
- If net external force F_ext = 0 then d/dt (Σ p) = 0 ⇒ Σ p = constant.
Newton's Third Law of Motion
Statement: For every action there is an equal and opposite reaction.
Meaning: When body A exerts a force on body B (action), body B simultaneously exerts a force of equal magnitude but opposite direction on body A (reaction). These two forces form an interaction pair (action–reaction pair).
Key points:
- The two forces are equal in magnitude and opposite in direction (F_AB = −F_BA).
- They act on two different bodies, never on the same body, so they do not cancel each other.
- Action and reaction are simultaneous — one cannot exist without the other.
- This law applies to both contact forces (push, pull, normal force, friction) and non-contact forces (gravitational, electrical).
Why they don't cancel: Because forces that make up the action–reaction pair act on different objects. Net force on any one object is the vector sum of forces acting on that object alone.
Consequences: Newton's third law underlies conservation of momentum. In an isolated two-body system the forces are equal and opposite, so the total momentum of the system remains constant.
Simple illustration (qualitative): If a book rests on a table, the book exerts a downward gravitational/weight force on the table (action), and the table exerts an equal upward normal force on the book (reaction). Both forces are equal but act on different bodies (book and table).
- Walking: foot pushes backward on ground (action); ground pushes foot forward (reaction) making you move.
- Jumping: legs push down on ground (action); ground pushes you upward (reaction) lifting you into the air.
- Recoil of a gun: bullet is pushed forward (action); gun is pushed backward with equal momentum change (reaction).
- Rocket propulsion: hot gases pushed backward out of nozzle (action); rocket is pushed forward (reaction).
- Swimming: swimmer pushes water backwards with hands (action); water pushes swimmer forward (reaction).
- Balloon rocket: air rushes out backward (action); balloon moves forward (reaction).
- Vector form: F_AB = −F_BA (force on B by A equals minus force on A by B)
- Magnitudes: |F_AB| = |F_BA|
- If only the pair acts on two bodies: m1 a1 = − m2 a2 (so m1 v1 + m2 v2 = constant → conservation of momentum)
- Impulse form: Δp1 = −Δp2 (change in momentum of one body equals negative change in momentum of the other)
- Impulse relation: ∫F_AB dt = − ∫F_BA dt (equal and opposite impulses over the interaction interval)
Applications of Newton's Laws
Introduction
Newton's three laws explain how forces affect the motion of objects. These laws help us analyze everyday situations (vehicles, sports, machines) and design safety devices (seat belts, brakes).
First Law (Law of Inertia)
An object remains at rest or in uniform motion in a straight line unless acted on by a net external force. Inertia is the tendency to resist change in motion.
- Application: Seat belts — when a car stops suddenly, passengers tend to continue moving forward; the belt provides the unbalanced force to stop them safely.
- Application: Tablecloth trick — a quick pull leaves dishes nearly undisturbed because their inertia resists the sudden change.
Second Law (F = ma)
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass: F_net = m·a. This law quantifies how forces change motion.
- Application: Vehicle acceleration — for a given engine force, a lighter car accelerates more (larger a) than a heavier car.
- Application: Braking distance — larger braking force produces larger deceleration (negative a), reducing stopping distance.
Third Law (Action–Reaction)
For every action force, there is an equal and opposite reaction force. Forces always occur in pairs acting on different bodies.
- Application: Walking — your foot pushes the ground backward (action) and the ground pushes your foot forward (reaction), moving you ahead.
- Application: Rocket propulsion — expelled gas pushes backward; the rocket is pushed forward by an equal and opposite force.
Combined and practical applications
Many devices and phenomena rely on combinations of the laws: vehicle design (suspension, brakes, crumple zones use inertia and forces), sports techniques (follow-through uses action–reaction), and machinery (pulleys and engines use force–acceleration relations). Friction and normal reaction are forces that modify outcomes predicted by the laws (e.g., static friction prevents slipping until a threshold).
Important notes for problem solving
1) Draw free-body diagrams showing all forces (gravity, normal, friction, tension, applied). 2) Use F_net = m·a along chosen axes. 3) Remember action–reaction pairs act on different bodies and do not cancel each other for a single object.
- Seat belts in cars (First law + Second law): Prevent passengers continuing forward during sudden stops by providing an unbalanced force.
- Pushing a loaded cart (Second law): For the same push (force), the empty cart (smaller mass) accelerates more than the loaded cart.
- Recoil of a gun (Third law): Bullet is pushed forward while the gun is pushed backward with equal and opposite momentum change.
- Walking or running (Third law): Feet push ground backwards; ground pushes you forwards.
- Rocket launch (Third law): Exhaust gases expelled backward push the rocket forward.
- Tablecloth trick (First law): Quick pull leaves objects almost undisturbed due to their inertia.
- Newton's second law: F_net = m · a (net force = mass × acceleration)
- Weight: W = m · g (g ≈ 9.8 m/s² near Earth's surface)
- Friction (approximate): f ≤ μ_s · N (static), f_k = μ_k · N (kinetic); N = normal force
- Momentum (useful related quantity): p = m · v
- Impulse (related concept): J = Δp = F_avg · Δt (area under force–time graph)
Problem Solving and Numerical Examples
Overview
Problem solving in the chapter "Force and Laws of Motion" means applying Newton's laws (mainly the second law) and the definitions of momentum and impulse to compute forces, accelerations, velocities and changes of motion. The standard approach is:
- Draw a clear free-body diagram (FBD) showing all forces (gravity, normal, tension, friction, applied forces).
- Choose a convenient axis (often along the direction of motion) and assign signs.
- Write Newton's second law: ΣF = ma (sum of forces along the axis = mass × acceleration).
- If collision or sudden change is involved, use impulse-momentum: Impulse J = Δp = F_avg · Δt.
- Check units (kg, m, s, N) and the special cases: ΣF = 0 → a = 0 (balanced forces), or isolated system → use conservation of momentum when no external impulse acts.
Common steps & tips
- Always list given data and what you must find.
- Use FBD to identify normal force and friction; friction (if given) is f = μN or given directly.
- When multiple forces act, add them vectorially along chosen axis (treat opposite directions with opposite signs).
- For collisions where objects stick together, use (m1 v1 + m2 v2) = (m1 + m2) v_final.
- For stopping problems, average force = (change in momentum)/(stopping time). The sign gives direction of the force.
- Estimate reasonableness: if computed acceleration is huge for reasonable force/mass, re-check arithmetic or units.
Key concepts to apply: Newton's three laws (especially F = ma), momentum p = mv, impulse J = Δp, frictional force when coefficient μ is given (f = μN), and conservation of linear momentum for isolated systems.
- Example 1 — Basic F = ma: A force of 10 N acts on a block of mass 2 kg on a frictionless surface. Find the acceleration. Solution: a = F/m = 10 / 2 = 5 m/s².
- Example 2 — Net force with friction: A 5 kg box is pulled horizontally with a force of 20 N. Friction opposing motion is 6 N. Find acceleration. Solution: Net force = 20 − 6 = 14 N; a = 14 / 5 = 2.8 m/s².
- Example 3 — Impulse & stopping force: A ball of mass 0.20 kg moving at 10 m/s is stopped by a glove in 0.05 s. Find the average force exerted by the glove. Solution: Initial momentum p_i = 0.20×10 = 2.0 kg·m/s; final p_f = 0; Δp = −2.0 kg·m/s; F_avg = Δp/Δt = −2.0 / 0.05 = −40 N (magnitude 40 N, direction opposite motion).
- Example 4 — Inelastic collision (momentum conservation): A bullet of mass 0.02 kg traveling at 200 m/s embeds in a stationary wooden block of mass 2.00 kg. Find the velocity of the block+bullet after impact. Solution: Total initial momentum = 0.02×200 + 2.00×0 = 4.0 kg·m/s. Total mass = 2.02 kg. v_final = 4.0 / 2.02 ≈ 1.98 m/s.
- Example 5 — Using free-body diagram on an inclined plane (with friction coefficient): A 3 kg crate is on a 30° incline. μ = 0.15, g = 9.8 m/s². Will it slide? If so, find acceleration. Solution outline: Weight component down slope = mg sin30° = 3×9.8×0.5 = 14.7 N. Normal N = mg cos30° = 3×9.8×0.866 ≈ 25.5 N. Max static friction f_s(max) = μN ≈ 0.15×25.5 ≈ 3.83 N. Since driving component 14.7 N > 3.83 N, it slides. Kinetic friction approx same μ: Net force = 14.7 − 3.83 = 10.87 N. Acceleration a = 10.87 / 3 ≈ 3.62 m/s².
- Example 6 — Two forces on a mass (vector addition): A 4 kg object has two horizontal forces acting: 15 N to the right and 9 N to the left. Find acceleration. Solution: Net force = 15 − 9 = 6 N to the right. a = 6 / 4 = 1.5 m/s².
- Newton's second law: ΣF = ma (Force in newtons, mass in kg, acceleration in m/s²).
- Acceleration: a = F_net / m.
- Momentum: p = m v (kg·m/s).
- Impulse (relation between force and change in momentum): J = Δp = F_avg · Δt.
- Conservation of linear momentum (isolated system): m1 v1 + m2 v2 = (m1 + m2) v_final (for perfectly inelastic collision where objects stick together).
- Friction (when coefficient μ is given): frictional force f = μ N (N is the normal reaction).
Illustrative Experiments and Demonstrations
What these demonstrations show
Illustrative experiments in Class 9 (Force and Laws of Motion) are simple, classroom/bench experiments that make Newton's laws and related concepts (inertia, force, acceleration, friction, action–reaction, momentum) directly observable. Each demo has: aim, apparatus, procedure, observations and a short conclusion relating the observation to a law.
Common demonstrations and their essence
- Inertia (Newton's 1st law) – Coin and card: Place a card on a glass and a coin on the card. Flick the card horizontally. Observation: the coin falls into the glass. Conclusion: the coin remains at rest (due to inertia) while the card moves out from under it.
- Inertia (tablecloth trick) – Rapidly pull a tablecloth from under light objects and heavier objects stay almost undisturbed. Conclusion: an object resists sudden change of motion; larger mass ⇒ larger inertia.
- Newton's 2nd law (F = ma) — Trolley and hanging mass – A trolley on a low-friction track is connected over a pulley to a hanging mass. Vary the hanging mass (force) keeping trolley mass constant: acceleration increases with force. Or keep force constant and add mass to trolley: acceleration decreases. Conclusion: acceleration ∝ force and ∝ 1/mass; summarized by F = ma.
- Friction demonstration – Pull a block with a spring balance over different surfaces or increase weight: readings show limiting friction increases with normal force and depends on surface. Conclusion: friction opposes motion; limiting friction ≤ μ_s N; kinetic friction ≈ μ_k N.
- Newton's 3rd law (action–reaction) – Two spring balances connected and pulled, or two students push on each other on skateboards, or a balloon rocket: forces are equal in magnitude and opposite in direction and act on different bodies. Conclusion: forces come in equal-and-opposite pairs.
- Conservation of momentum (collisions) – Low-friction gliders collide on a track: total momentum before = total momentum after (if external impulses negligible). Observation: velocities change such that vector sum of momenta remains constant.
How to report an experiment
State aim, list apparatus, give a short stepwise procedure, record observations (table) and sketch graphs if needed, draw conclusions linking to the relevant law and mention main sources of error (friction, timing error, air resistance).
Tips for classroom clarity: use slow-motion video or repeatable small steps; use measured data for F and a and plot to verify linear relations; always indicate which body each force acts on when discussing action–reaction pairs.
- Coin on card: demonstrates inertia of rest — coin drops into glass when card is flicked away.
- Trolley & hanging mass: varying hanging mass (force) shows acceleration increases; adding trolley mass shows acceleration decreases — demonstration of F = ma.
- Tablecloth trick: objects resist sudden motion showing inertia; heavier objects show larger inertia.
- Balloon rocket: air expelled backwards produces forward motion — action and reaction (Newton's 3rd law).
- Block pulled on different surfaces: measure pull with spring balance to compare limiting and kinetic friction.
- Newton's 2nd law: F = m a (resultant force F acting on a body of mass m produces acceleration a)
- Proportional forms: a ∝ F (for fixed m), a ∝ 1/m (for fixed F)
- Atwood / connected masses example: two masses m1 and m2 over a pulley: a = (m1 - m2) g / (m1 + m2) (direction toward heavier mass)
- Weight: W = m g (g ≈ 9.8 m/s²)
- Friction (empirical): limiting friction F_l ≤ μ_s N ; kinetic friction F_k = μ_k N (N = normal reaction)
- Momentum: p = m v ; For an isolated system, total momentum before = total momentum after
Key Concepts
- Force
- A push or pull on an object that can change its state of rest or motion or shape. Measured in newtons (N).
- Mass
- Amount of matter in an object; a measure of its inertia. Mass is constant and measured in kilograms (kg).
- Weight
- Gravitational force acting on a mass; weight = mass × gravitational acceleration (W = mg). Measured in newtons (N).
- Gravitational acceleration (g)
- Acceleration due to Earth's gravity; approximately 9.8 m/s² near Earth's surface.
- Inertia
- Property of matter by which it resists any change in its state of rest or uniform motion in a straight line.
- Newton's First Law (Law of Inertia)
- An object remains at rest or in uniform motion in a straight line unless acted upon by an external unbalanced force.
- Newton's Second Law
- Acceleration produced in an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma).
- Newton's Third Law
- For every action there is an equal and opposite reaction; forces always occur in pairs acting on two different bodies.
- Momentum
- Product of mass and velocity of an object; a measure of how hard it is to stop the object (p = mv).
- Impulse
- Change in momentum of an object produced by a force acting for a time interval; impulse = force × time (J = FΔt).
- Conservation of Momentum
- In the absence of external forces, the total momentum of a system remains constant before and after a collision or interaction.
- Resultant Force
- Single force which has the same effect as the vector sum of all individual forces acting on a body.
- Balanced Force
- Forces acting on an object that are equal in magnitude and opposite in direction, producing no change in motion.
- Unbalanced Force
- Net force that is not zero, causing an object to accelerate or change its motion.
- Equilibrium
- State in which the net force on an object is zero, so the object remains at rest or moves with constant velocity.
- Friction
- Resistive force that acts opposite to the relative motion or tendency of motion between two surfaces in contact.
- Static Friction
- Frictional force that prevents relative motion between surfaces at rest with respect to each other; it adjusts up to a maximum value.
- Kinetic (Sliding) Friction
- Frictional force acting between surfaces that are sliding past each other; usually less than maximum static friction.
- Rolling Friction
- Frictional resistance experienced by an object rolling over a surface; typically much smaller than sliding friction.
- Newton (unit)
- SI unit of force; one newton is the force that gives a mass of 1 kg an acceleration of 1 m/s² (1 N = 1 kg·m/s²).
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
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A net force of 15 N acts on a 3 kg object. What is the acceleration? (a) 5 m/s² (b) 45 m/s² (c) 0.2 m/s² (d) 12 m/s² 3 kg द्रव्यमान की वस्तु पर 15 N का परिणामी बल लगता है। त्वरण क्या होगा? (a) 5 m/s² (b) 45 m/s² (c) 0.2 m/s² (d) 12 m/s²
Show answer
(a) 5 m/s² — a = F/m = 15/3 = 5 m/s² (Newton's second law). (a) 5 m/s² — a = F/m = 15/3 = 5 m/s² (न्यूटन का दूसरा नियम)।
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A bullet (0.02 kg) fired at 200 m/s from a gun (2 kg). The recoil velocity of the gun is: (a) 200 m/s (b) 2 m/s backward (c) 2 m/s forward (d) 0 m/s 0.02 kg की गोली 2 kg की बंदूक से 200 m/s पर दागी जाती है। बंदूक की पश्च-गति का वेग: (a) 200 m/s (b) 2 m/s पीछे (c) 2 m/s आगे (d) 0 m/s
Show answer
(b) 2 m/s backward — Conservation of momentum: 0 = 0.02×200 + 2×v, so v = −2 m/s (backward). (b) 2 m/s पीछे — संवेग संरक्षण: 0 = 0.02×200 + 2×v, अतः v = −2 m/s (पीछे)।
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Which of Newton's laws is called the 'Law of Inertia'? (a) First law (b) Second law (c) Third law (d) None न्यूटन का कौन-सा नियम 'जड़त्व का नियम' कहलाता है? (a) पहला नियम (b) दूसरा नियम (c) तीसरा नियम (d) कोई नहीं
Show answer
(a) First law — It states objects remain at rest or in uniform motion unless acted on by an external force. Inertia is this resistance to change. (a) पहला नियम — यह कहता है कि वस्तुएँ तब तक विराम या एकसमान गति में रहती हैं जब तक बाहरी बल न लगे। इस परिवर्तन-प्रतिरोध को जड़त्व कहते हैं।
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Momentum is defined as the product of ________ and ________. संवेग को ________ और ________ के गुणनफल के रूप में परिभाषित किया जाता है।
Show answer
Mass and velocity (p = mv). Momentum is a vector with SI unit kg·m/s. द्रव्यमान और वेग (p = mv)। संवेग एक सदिश राशि है जिसका SI मात्रक kg·m/s है।
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When passengers lean forward as a bus brakes suddenly, this demonstrates inertia of ________. बस के अचानक रुकने पर यात्रियों का आगे झुकना ________ के जड़त्व का उदाहरण है।
Show answer
Motion (inertia of motion). Passengers were moving with the bus and tend to continue forward even after the bus stops. गति का जड़त्व। यात्री बस के साथ गतिमान थे और बस रुकने के बाद भी आगे जाने की प्रवृत्ति रखते हैं।
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True or False: Action and reaction forces in Newton's Third Law act on the same body. सत्य या असत्य: न्यूटन के तीसरे नियम में क्रिया और प्रतिक्रिया बल एक ही वस्तु पर लगते हैं।
Show answer
False — They act on two different bodies, not the same body, so they cannot cancel each other. असत्य — ये दो अलग-अलग वस्तुओं पर लगते हैं, एक ही वस्तु पर नहीं, इसलिए वे एक-दूसरे को रद्द नहीं कर सकते।
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Explain why it is safer to fall on a sand bed than on a concrete floor. रेत पर गिरना कंक्रीट पर गिरने से सुरक्षित क्यों है? समझाइए।
Show answer
The change in momentum is the same in both cases. Sand increases the time of impact, which reduces the average force (F = Δp/Δt). Longer time → smaller force → less injury. दोनों मामलों में संवेग परिवर्तन समान है। रेत प्रभाव का समय बढ़ा देती है जिससे औसत बल कम होता है (F = Δp/Δt)। अधिक समय → कम बल → कम चोट।
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A 2 kg object at 3 m/s collides and sticks to a stationary 3 kg object. Find the common velocity after collision. 3 m/s से चलती 2 kg वस्तु एक स्थिर 3 kg वस्तु से टकराकर उससे चिपक जाती है। टक्कर के बाद साझा वेग ज्ञात कीजिए।
Show answer
1.2 m/s — m1v1 + m2v2 = (m1+m2)v → (2×3 + 0) = 5v → v = 6/5 = 1.2 m/s in the original direction. 1.2 m/s — m1v1 + m2v2 = (m1+m2)v → (2×3 + 0) = 5v → v = 6/5 = 1.2 m/s मूल दिशा में।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.