Introduction
The laws of motion, formulated by Sir Isaac Newton in his work 'Principia Mathematica' (1687), form the foundation of classical mechanics.
Newtons First Law of Motion
'Every body continues in its state of rest or uniform motion in a straight line unless compelled by an external unbalanced force to change that state.'
Inertia
The tendency of a body to resist any change in its state of rest or motion is called inertia.
Types of Inertia:
- Inertia of rest: A passenger lurches backward when a bus suddenly starts.
- Inertia of motion: A passenger lurches forward when a bus suddenly stops.
- Inertia of direction: Mud flying off a rotating tyre.
Mass is a measure of inertia. Heavier objects have greater inertia.
Newtons Second Law of Motion
'The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force.'
Mathematically: vecF = d(vecp)/dt
For constant mass: vecF = m * veca
Important Points
- Force is required to change momentum, not to maintain motion (contradicts Aristotelian view).
- Second law is a vector equation. Component form:
F_x = ma_x,F_y = ma_y,F_z = ma_z. - SI unit of force: newton (N).
1 N = 1 kg m/s^2.
Momentum and Impulse
Linear momentum: vecp = m vecv
Impulse: I = int_(t_1)^(t_2) vecF dt = vecp_2 - vecp_1
Impulse equals change in momentum (Impulse-Momentum theorem).
Applications of Impulse
- Catching a cricket ball: Hands are pulled back to increase time, reducing force.
- Glassware wrapped in paper or straw: Increases collision time, reduces force.
- Karate blow: Short impact time means large force.
Newtons Third Law of Motion
'Every action has an equal and opposite reaction.'
- Action and reaction act on different bodies.
- Action and reaction are simultaneous.
- They are equal in magnitude and opposite in direction.
Examples
- Walking: Foot pushes ground backward (action), ground pushes foot forward (reaction).
- Rocket propulsion: Exhaust gases pushed downward (action), rocket pushed upward (reaction).
- Recoil of a gun: Bullet moves forward (action), gun moves backward (reaction).
Conservation of Linear Momentum
If no external force acts on a system, its total linear momentum remains constant.
m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2
Derivation from Newtons Laws
F_ext = dP/dt. If F_ext = 0, then dP/dt = 0, so P = text(constant).
Rocket propulsion: As mass decreases (fuel ejected), velocity increases. v = u + v_r ln(m_0/m).
Friction
The force that opposes relative motion between surfaces in contact.
Types of Friction
Static friction (f_s): Opposes initiation of motion. f_s <= mu_s N
Kinetic friction (f_k): Opposes ongoing motion. f_k = mu_k N
Rolling friction (f_r): Opposes rolling motion. f_r << f_k < f_s
Laws of Friction
- Friction is independent of area of contact.
- Friction is proportional to normal reaction.
- Kinetic friction is independent of relative speed (approximately).
- Static friction is greater than kinetic friction.
Angle of Friction
tan lambda = f_s/N = mu_s. Where lambda is the angle at which sliding begins.
Angle of Repose
The angle of an inclined plane at which a body just begins to slide: tan theta = mu_s.
Angle of repose = Angle of friction.
Circular Motion on Level and Banked Roads
On a Level Road
Centripetal force is provided by friction: mv^2/r = mu_s mg
Maximum safe speed: v_max = sqrt(mu_s rg)
On a Banked Road
For angle of banking theta:
tan theta = v^2/(rg)
For the design speed: No friction is needed. The horizontal component of normal reaction provides centripetal force.
Optimum speed: v_0 = sqrt(rg tan theta)
Worked Examples
Example 1: A 1000 kg car accelerates from 10 m/s to 20 m/s in 5 s. Find the force.
Solution: a = (20-10)/5 = 2 m/s^2. F = ma = 1000 * 2 = 2000 N.
Example 2: A 5 g bullet hits a wall with 200 m/s and stops in 0.01 s. Find force.
Solution: F = Delta p/Delta t = (0.005 * 200 - 0)/0.01 = 100 N.
Example 3: Find the minimum coefficient of friction for a car to safely turn a curve of radius 20 m at 10 m/s.
Solution: mu = v^2/(rg) = 100/(20*10) = 0.5.
Common Mistakes
- Action-reaction on same body: Action and reaction act on different bodies, not the same body.
- N2L and constant velocity: If velocity is constant, acceleration is zero, so net force is zero.
- Normal reaction equals weight: Only true for horizontal surfaces. On incline,
N = mg cos theta. - Static vs kinetic: Static friction adjusts up to its maximum; kinetic friction is constant.
ISC Exam Focus
- Theory (70%): Newtons laws, momentum, impulse, friction laws, banking of roads.
- Application (30%): Numerical problems using F=ma, conservation of momentum, friction.
- ISC frequently asks: "A body of mass m on an inclined plane ... find acceleration/force."
- 4-6 mark numericals on banked road and friction problems are common.
Self-Test Questions
Q1: State Newtons first law and explain inertia with an example. Answer: The law states a body resists change in its state. Example: when a car stops suddenly, passengers lurch forward (inertia of motion).
Q2: A 10 g bullet from a gun of mass 5 kg has muzzle velocity 400 m/s. Find recoil velocity.
Answer: m_b v_b + m_g v_g = 0 => 0.01*400 + 5*v_g = 0 => v_g = -0.8 m/s.
Q3: A 2 kg block on a rough horizontal surface (mu_k = 0.3) is pulled by 10 N. Find acceleration.
Answer: f_k = mu_k N = 0.3*2*10 = 6 N. Net force = 10 - 6 = 4 N. a = 4/2 = 2 m/s^2.
Q4: Derive the expression for optimum speed on a banked road.
Answer: tan theta = v^2/(rg) => v = sqrt(rg tan theta).
Q5: State the law of conservation of linear momentum. Answer: In the absence of external force, total linear momentum of a system remains constant.
Q6: Find impulse when a 0.5 kg ball hits a wall at 5 m/s and rebounds with 4 m/s.
Answer: Delta p = 0.5*(4 - (-5)) = 0.5*9 = 4.5 kg m/s. Impulse = 4.5 Ns.
