Laws of Motion: NEET notes.

01 · NCERT § "Aristotle's Fallacy" and § "The Law of Inertia"

Aristotle's fallacy and the law of inertia

• Aristotle held that a body needs a continuing external force just to keep moving. This is wrong; in daily life an applied force is needed only to overcome opposing forces such as friction and air drag.
• Galileo reasoned from motion on inclined planes: a body speeds up going down a slope and slows going up, so on a smooth horizontal plane it should neither speed up nor slow down.
• His double-inclined-plane argument: a ball rolled down one smooth slope climbs the other to nearly the same height; making the second slope flatter makes it travel farther, and with a horizontal second plane it would go on forever.
• Conclusion: a state of rest and a state of uniform straight-line motion are equivalent; no net force is required to maintain either.
• Inertia is the resistance of a body to a change in its state of rest or uniform motion. Mass is the quantitative measure of inertia.

02 · NCERT § "Newton's First Law of Motion"

Newton's first law of motion

• A body stays at rest, or keeps moving with constant velocity in a straight line, unless a net external force acts on it.
• Put the other way: zero net external force on a body means zero acceleration; several forces may act, but they must cancel.
• A book resting on a table has zero net force on it: its weight is balanced by the normal force from the table. These two are not an action-reaction pair.
• Everyday examples of inertia: passengers lurch forward when a bus stops suddenly and backward when it starts suddenly.
• The law defines the kind of frame (an inertial frame) in which Newton's laws hold as stated; a frame accelerating relative to such a frame is non-inertial.

03 · NCERT § "Newton's Second Law of Motion"

Newton's second law, momentum and impulse

• Linear momentum is p = mv, a vector along the velocity. Its SI unit is kg m s⁻¹.
• The second law: the net external force on a body equals the time rate of change of its momentum, F = dp/dt, and the momentum changes in the direction of that force. For constant mass this gives F = ma.
• The SI unit of force is the newton: 1 N = 1 kg m s⁻².
• The law is a vector equation and can be applied component by component; a force perpendicular to the velocity changes the direction of motion but not the speed.
• Acceleration at an instant depends on the force at that instant, not on the history of motion. A stone dropped from a moving train has, just after release, only the gravitational acceleration.
• Impulse is force multiplied by time for a force acting briefly, and it equals the change in momentum: J = FΔt = Δp. Its unit is N s, the same as momentum.
• The same change in momentum over a longer time needs a smaller average force; this is why a fielder draws the hands back while catching a fast ball.
• For a ball that rebounds, the change in velocity is the sum of the two speeds, because the direction reverses.

04 · NCERT § "Newton's Third Law of Motion"

Newton's third law of motion

• Forces always come in pairs: if body A exerts a force on body B, then B exerts an equal and opposite force on A, F_AB = −F_BA.
• The two forces of a pair act on different bodies, so they never cancel each other in the equation of motion of a single body.
• Action and reaction are simultaneous; neither is the cause of the other, so the labels action and reaction can be swapped.
• The pair is always of the same nature (both gravitational, both contact forces, and so on).
• The earth pulls a falling stone down and the stone pulls the earth up with an equal force; the earth's acceleration is tiny only because its mass is huge.
• Internal forces within a system come in third-law pairs and cancel in the total, so only external forces change the total momentum of a system.

05 · NCERT § "Conservation of Momentum"

Conservation of momentum

• For an isolated system of interacting particles, total momentum stays constant; this follows directly from the second and third laws.
• Recoil of a gun: gun and bullet start at rest, so after firing their momenta are equal and opposite; the heavier gun moves back much more slowly.
• In any collision the momentum lost by one body equals the momentum gained by the other, whatever happens to the kinetic energy.
• When one body throws another (for example a person on smooth ice throwing a ball), velocities must be measured relative to the ground before applying conservation.
• Momentum is conserved as a vector; in one-dimensional problems keep track of direction with signs.

06 · NCERT § "Equilibrium of a Particle"

Equilibrium of a particle

• A particle is in equilibrium when the net external force on it is zero; it is then at rest or in uniform motion.
• Two forces keep a particle in equilibrium only if they are equal in size and opposite in direction.
• Three concurrent forces in equilibrium add to zero and can be drawn head to tail as a closed triangle.
• In practice, resolve every force along two perpendicular axes and set the sum of components along each axis to zero.
• A mass hanging from two strings is the standard case: the vertical components of the tensions together balance the weight, and the horizontal components cancel.

07 · NCERT § "Common Forces in Mechanics"

Common forces in mechanics

• Gravity acts at a distance; all other everyday mechanical forces are contact forces that arise where bodies touch.
• The normal reaction is the component of a contact force perpendicular to the surfaces in contact; it adjusts to whatever value prevents the bodies from passing into each other.
• Tension is the pull transmitted by a string. For a light (massless) inextensible string over a frictionless pulley, the tension is the same all along it.
• A spring exerts a restoring force F = −kx for small extension or compression x, where k is the spring constant; the negative sign shows it opposes the deformation.
• In a lift accelerating upward at a, a person's apparent weight (the normal force from the floor or the scale reading) is m(g + a); accelerating downward it is m(g − a).
• A lift moving up but slowing down has downward acceleration, so the scale reads less than mg. The direction of acceleration matters, not the direction of motion.
• In free fall (a = g downward) the scale reads zero; the body appears weightless though gravity still acts on it.

08 · NCERT § "Friction"

Friction

• Friction is the part of the contact force that lies parallel to the surfaces; between the surfaces it opposes relative motion, or impending relative motion.
• Static friction is self-adjusting: it takes whatever value, up to a maximum, is needed to prevent sliding, f_s ≤ μ_s N.
• Its limiting value (f_s)max = μ_s N depends on the nature of the surfaces but not on the area of contact.
• If the applied force is less than the limiting value, static friction equals the applied force and the body stays at rest; it is not automatically μ_s N.
• Once sliding begins, kinetic friction acts: f_k = μ_k N, roughly independent of speed and area of contact. Usually μ_k < μ_s.
• Friction on a body can act along its direction of motion: the friction on a box on the floor of an accelerating truck points forward and is what accelerates the box.
• Rolling friction is much smaller than sliding friction, which is why wheels, ball bearings and lubricants are used to reduce energy loss.
• Friction is not purely a nuisance; walking, driving and braking all depend on it.

09 · NCERT § "Circular Motion"

Circular motion

• Moving at speed v in a circle of radius R, a body has an acceleration v²/R directed towards the centre, so it needs a net force mv²/R towards the centre.
• Centripetal force is not a new kind of force; it is supplied by an existing force such as tension, gravity, or friction.
• On a level (unbanked) road the centripetal force comes only from static friction between tyres and road, giving a maximum safe speed v_max = √(μ_s R g), independent of the car's mass.
• Static friction, not kinetic, is used here because the tyres do not slip sideways on the road.
• On a road banked at angle θ, a component of the normal force also points towards the centre, raising the maximum safe speed.
• At the optimum speed v₀ = √(R g tan θ) the normal force alone supplies the centripetal force and friction is not needed, which reduces tyre wear.
• Below the optimum speed friction acts up the slope; above it, friction acts down the slope.

10 · NCERT § "Solving Problems in Mechanics"

Solving problems in mechanics

• Pick the body (or system) and draw a free-body diagram showing every force acting on it, and only those forces.
• Choose axes, usually along and perpendicular to the direction of acceleration, and write the second law separately for each axis.
• For connected bodies, the string condition gives the same magnitude of acceleration for both bodies and, for a light string, the same tension at both ends.
• Treating connected bodies as one system gives the common acceleration quickly, because internal tensions cancel; a separate free-body diagram is then needed to find the tension.
• For two masses m₁ > m₂ hanging over a light frictionless pulley, a = (m₁ − m₂)g/(m₁ + m₂) and T = 2m₁m₂g/(m₁ + m₂).
• With friction on connected bodies, first check whether the driving force exceeds limiting static friction; if not, the system stays at rest and friction equals the driving force.
• When a force is applied at an angle to a floor, its vertical component changes the normal reaction and therefore the friction.

Revise

Must-know facts

1. 01Mass is the measure of inertia; heavier bodies are harder to start, stop or turn.
2. 02F = dp/dt; for constant mass F = ma. 1 N = 1 kg m s⁻².
3. 03Impulse = FΔt = Δp; unit N s, same as momentum.
4. 04Rebound: Δp = m(v₁ + v₂) in magnitude because the velocity reverses.
5. 05Action and reaction never cancel each other: the two act on different bodies.
6. 06Weight of a book and the normal force on it are balancing forces on one body, not a third-law pair.
7. 07Total momentum of an isolated system is conserved; recoil speed of a gun is (m_bullet/m_gun) × bullet speed.
8. 08Static friction is self-adjusting: f_s ≤ μ_s N; it equals μ_s N only at the point of slipping.
9. 09Kinetic friction f_k = μ_k N, and usually μ_k < μ_s.
10. 10Limiting friction does not depend on the area of contact.
11. 11Rolling friction is far smaller than sliding friction.
12. 12Apparent weight in a lift = m(g + a) for upward acceleration and m(g − a) for downward acceleration; zero in free fall.
13. 13Maximum speed on a level curve: v = √(μ_s R g), independent of mass.
14. 14Optimum speed on a banked curve (no friction needed): v₀ = √(R g tan θ).
15. 15Atwood machine: a = (m₁ − m₂)g/(m₁ + m₂), T = 2m₁m₂g/(m₁ + m₂).
16. 16Tension is the same throughout a light string over a frictionless pulley.
17. 17Acceleration at an instant depends only on the net force at that instant.

Where marks go

Common traps

Reference

Formulas

Linear momentum

p = mv

Vector along v; SI unit kg m s⁻¹.

Newton's second law

F = dp/dt = ma

F is the net external force; F = ma for constant mass.

Impulse

J = FΔt = Δp = m(v − u)

F is the average force during the short time Δt; unit N s.

Newton's third law

F_AB = −F_BA

The two forces act on different bodies.

Conservation of momentum (two bodies)

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Isolated system; velocities relative to the ground, with signs.

Recoil velocity

V = −(m/M)v

M, V for the gun; m, v for the bullet; starting from rest.

Equilibrium of a particle

ΣFx = 0, ΣFy = 0

Resolve forces along perpendicular axes.

Static friction

f_s ≤ μ_s N, (f_s)max = μ_s N

Self-adjusting up to the limiting value.

Kinetic friction

f_k = μ_k N

Usually μ_k < μ_s.

Spring force

F = −kx

Restoring force for small deformation x; k in N m⁻¹.

Apparent weight in a lift

R = m(g + a) upward acceleration; R = m(g − a) downward acceleration

R is the normal force (scale reading).

Centripetal force

F = mv²/R

Directed towards the centre of the circle.

Maximum speed on a level road

v_max = √(μ_s R g)

Independent of mass.

Maximum speed on a banked road

v_max = √[R g (μ_s + tan θ)/(1 − μ_s tan θ)]

θ is the banking angle.

Optimum speed on a banked road

v₀ = √(R g tan θ)

Friction not needed at this speed.

Two masses over a pulley (Atwood)

a = (m₁ − m₂)g/(m₁ + m₂), T = 2m₁m₂g/(m₁ + m₂)

Light inextensible string, frictionless light pulley, m₁ > m₂.

Glossary

Key terms

Inertia

The tendency of a body to keep its state of rest or uniform straight-line motion.

Inertial frame

A reference frame in which a body with no net force on it moves with constant velocity.

Linear momentum

Mass times velocity; the quantity whose rate of change equals the net force.

Impulse

The product of a large force and the short time it acts, equal to the change of momentum it produces.

Newton

The force that gives a 1 kg mass an acceleration of 1 m s⁻².

Normal reaction

The part of a contact force perpendicular to the surfaces in contact.

Tension

The pulling force a stretched string or rope exerts on whatever is attached to it.

Static friction

Friction that prevents sliding from starting; adjusts in size up to a limit.

Limiting friction

The largest static friction, μ_s N, reached just before sliding begins.

Kinetic friction

Friction that acts while surfaces slide over each other, μ_k N.

Rolling friction

The small resistance to a body rolling over a surface.

Centripetal force

The net inward force needed to keep a body moving in a circle.

Banking of roads

Raising the outer edge of a curved road so that the normal force helps provide the centripetal force.

Free-body diagram

A sketch of one body showing all the external forces acting on it.