Chapter 9 — Force And Laws Of Motion

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).

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.

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).

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.

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.

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.

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.

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.

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.

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>