Comprehensive Physics Notes: Laws of Motion, Work, Energy, and Planar Motion

Chapter 5: Laws of Motion

Inertia

• Inertia is the property of an object that resists changes in its state of rest or uniform motion along a straight line.
• Inertia is a measure of mass; greater mass implies greater inertia, and vice versa.
• Types of Inertia:
  • Inertia of Rest: Demonstrated when passengers fall backward when a bus starts suddenly.
  • Inertia of Motion: Demonstrated when passengers jerk forward when a moving bus stops suddenly.
  • Inertia of Direction: Demonstrated by using an umbrella to protect from rain, as raindrops resist changing direction.

Force

• Force is a push or pull that changes or tries to change the state of rest, uniform motion, size, or shape of a body.
• SI unit is Newton (N); dimensional formula is [MLT^{-2}].
• Types of Forces:
  • Contact Forces: Examples include frictional force, tensional force, spring force, and normal force.
  • Action at a Distance Forces: Examples include electrostatic force, gravitational force, and magnetic force.

Impulsive Force

• A force acting on a body for a short time interval, producing a large change in momentum.

Linear Momentum

• The total amount of motion in a body.
• Linear momentum (p) is the product of mass (m) and velocity (u): p = mu.
• SI unit is kg-m/s; dimensional formula is [MLT^{-1}].
• It is a vector quantity with direction matching the body's velocity.

Impulse

• The product of impulsive force and time: Impulse = Force * Time = Change in momentum.
• SI unit is Newton-second or kg-m/s; dimension is [MLT^{-1}].
• It is a vector quantity in the direction of the force.

Newton’s Laws of Motion

1. Newton’s First Law of Motion

• A body remains in its state of rest or uniform motion unless an external force is applied.
• Also known as the law of inertia.
  • Examples:
    • Dust particles separating from a carpet when beaten with a stick.
    • Passengers bending outward when a moving vehicle suddenly stops.

2. Newton’s Second Law of Motion

• The rate of change of linear momentum is proportional to the applied force: F \propto dp/dt.
• Mathematically: F = k \frac{d}{dt}(mv), where k = 1 in SI and CGS systems.
• F = m \frac{dv}{dt} = ma
  • Examples:
    • It is easier for a strong adult to push a full shopping cart than a baby.
    • It is easier to push an empty shopping cart than a full one.

3. Newton’s Third Law of Motion

• For every action, there is an equal and opposite reaction, acting on different bodies: F{12} = -F{21}.
  • Examples:
    • Swimming is possible due to the third law.
    • Jumping from a boat onto the bank of a river.
    • Jerk produced in a gun when a bullet is fired.
    • Pulling of a cart by a horse.
• Newton’s second law is considered the real law of motion as the first and third laws can be derived from it.
• Modern Version of Newton’s Laws:
  • A body maintains its state of rest or uniform motion unless acted upon by an unbalanced external force.
  • Forces always occur in pairs; if body A exerts a force on body B, body B exerts an equal and opposite force on body A.

Law of Conservation of Linear Momentum

• If no external force acts on a system, the total linear momentum remains constant.
• Linear momentum depends on the frame of reference, but the law of conservation of linear momentum does not.
• Newton’s laws are valid only in inertial frames of reference.

Weight (w)

• The force with which a body is pulled towards the center of the Earth due to gravity: w = mg, where m is mass and g is acceleration due to gravity.

Apparent Weight in a Lift

• (i) At rest or constant speed: R = mg (actual weight).
• (ii) Accelerating upward: R_1 = m(g + a) (apparent weight is more than actual weight).
• (iii) Accelerating downward: R_2 = m(g - a) (apparent weight is less than actual weight).
• (iv) Falling freely: R_2 = m(g - g) = 0 (apparent weight is zero).
• (v) Accelerating downward with a > g: body will lift from the floor to the ceiling of the lift.

Rocket

• An example of variable mass system following the law of conservation of momentum.
• Thrust: F = -u \frac{dM}{dt}, where u is exhaust speed and \frac{dM}{dt} is the rate of fuel combustion.
• Velocity: u = v0 + u \loge(\frac{M0}{M}), where v0 is initial velocity, M_0 is initial mass, and M is present mass.
• Considering gravity: u = v0 + u \loge(\frac{M_0}{M}) - gt

Friction

• A force opposing relative motion at the contact point between objects, acting parallel to the contact surfaces.
• Caused by intermolecular interactions.

Types of Friction:

• 1. Static Friction:
  • The opposing force when one body tends to move over another, but actual motion hasn't started.
  • It is self-adjusting.
• 2. Limiting Friction:
  • The maximum value of static friction before motion begins: fs = \mus R, where \mu_s is the coefficient of limiting friction and R is the normal reaction.
  • Independent of the area of contact but depends on the nature (smoothness or roughness) of the surfaces.
  • If \theta is the angle of friction, then \mu_s = \tan \theta
• 3. Kinetic Friction:
  • The friction when the body is sliding: fk = \muk N, where \mu_k is the coefficient of kinetic friction and N is the normal force.
  • Types: sliding friction and rolling friction (rolling friction < sliding friction).

Angle of Repose or Angle of Sliding

• The minimum angle of inclination for a body to start sliding down an inclined plane.
• If \alpha is the angle of repose, then \mu_s = \tan \alpha

Motion on an Inclined Plane

• Normal reaction: R = mg \cos \theta
• Net force acting downward: F = mg \sin \theta - f
• Acceleration: a = g(\sin \theta - \mu \cos \theta)
• When the angle of inclination is less than the angle of repose:
  • Minimum force to move the body up: f_1 = mg(\sin \theta + \mu \cos \theta)
  • Minimum force to push the body down: f_2 = mg(\mu \cos \theta - \sin \theta)

Tension

• Tension force always pulls a body.
• It is a reactive force, not an active one.
• Tension remains constant across a massless, frictionless pulley.
• Rope becomes slack when the tension force is zero.

Motion of Bodies in Contact

(i) Two Bodies in Contact

• If force F is applied on mass m1, then acceleration: a = \frac{F}{m1 + m_2}
• Contact force on m1 = m1 a = \frac{m1 F}{m1 + m_2}
• Contact force on m2 = m2 a = \frac{m2 F}{m1 + m_2}

(ii) Three Bodies in Contact

• Acceleration: a = \frac{F}{m1 + m2 + m_3}
• Contact force between m1 and m2: F1 = \frac{(m2 + m3)F}{m1 + m2 + m3}
• Contact force between m2 and m3: F2 = \frac{m3 F}{m1 + m2 + m_3}

(iii) Motion of Two Bodies, One Resting on the Other

• (a) Force F applied on the lower body A, with friction coefficient \mu between A and B:
  • Common acceleration: a = \frac{F}{M + m}
  • Pseudo force on block B: f' = ma
  • Frictional force: f = \mu N = \mu mg
  • For equilibrium: ma \leq \mu mg or a \leq \mu g
• (b) Friction between ground and body A (coefficient \mu1) and between A and B (coefficient \mu2):
  • Net accelerating force: F - fA = F - \mu1 (M + m)g
  • Net acceleration: a = \frac{F - \mu1 (M + m)g}{M + m} = \frac{F}{M + m} - \mu1 g
  • Pseudo force on block B: f' = ma
  • Frictional force: f_B = \mu mg
  • For equilibrium: ma \leq \mu2 mg or a \leq \mu2 g
  • If a > \mu_2 g, the bodies will not move together.

(iv) Motion of Bodies Connected by Strings

• Acceleration of the system: a = \frac{F}{m1 + m2 + m_3}
• Tension in string T_1 = F
• Tension in string T2 = (m2 + m3)a = \frac{(m2 + m3)F}{m1 + m2 + m3}
• Tension in string T3 = m3 a = \frac{m3 F}{m1 + m2 + m3}

Pulley Mass System

(i) Unequal Masses Suspended from a Pulley (m1 > m2)

• Equations: m1 g - T = m1 a and T - m2 g = m2 a
• Acceleration: a = \frac{m1 - m2}{m1 + m2} g
• Tension: T = \frac{2 m1 m2}{m1 + m2} g

(ii) Mass m_2 on a Frictionless Horizontal Surface

• Acceleration: a = \frac{m1 g}{m1 + m_2}
• Tension: T = \frac{m1 m2 g}{m1 + m2}

(iii) Mass m_2 on a Rough Horizontal Surface

• Acceleration: a = \frac{m1 - \mu m2}{m1 + m2} g
• Tension: T = \frac{m1 m2 (1 + \mu)}{m1 + m2} g

(iv) Two Masses m1 and m2 Connected to a Single Mass M

• Equations:
  • m1 g - T1 = m_1 a
  • T2 - m2 g = m_2 a
  • T1 - T2 = Ma
• Acceleration: a = \frac{m1 - m2}{m1 + m2 + M} g
• Tension T1 = \frac{2 m2 + M}{m1 + m2 + M} m_1 g
• Tension T2 = \frac{2 m1 + M}{m1 + m2 + M} m_2 g

(v) Motion on a Smooth Inclined Plane

• Equations:
  • m1 g - T = m1 a
  • T - m2 g \sin \theta = m2 a
• Acceleration: a = \frac{m1 - m2 \sin \theta}{m1 + m2} g
• Tension: T = \frac{m1 m2 (1 + \sin \theta)}{m1 + m2} g

(vi) Two Bodies on Two Inclined Planes with Different Inclinations

• Acceleration: a = \frac{(m1 \sin \theta1 - m2 \sin \theta2) g}{m1 + m2}
• Tension: T = \frac{m1 m2}{m1 + m2} (\sin \theta1 - \sin \theta2) g

Chapter 6: Work, Energy, and Power

Work

• Work is done when a force causes displacement of an object in the direction of the force.
• W = F \cdot s = Fs \cos \theta, where \theta is the angle between force F and displacement s.
• Work is a scalar quantity with SI unit joule (J) and CGS unit erg (1 J = 10^7 erg).
• Dimensional formula is [ML^2T^{-2}].
• Work done is zero if:
  • No displacement (s = 0).
  • Displacement is perpendicular to the force (\theta = 90^\circ).
• Work is positive if \theta is acute and negative if \theta is obtuse.
• Work done by a constant force depends only on initial and final positions.

Work Done in Different Conditions

• (i) Variable force: W = \int F \cdot ds (equal to the area under the force-displacement graph).
• (ii) Multiple forces: Work done is equal to the work done by the resultant force.
• (iii) Equilibrium: Resultant force is zero, therefore, resultant work done is zero.
• (iv) Conservative vs. Non-conservative forces:
  • Conservative forces: Gravitational, electrostatic, magnetic forces.
  • Non-conservative forces: Frictional, viscous forces.
• (v) Work done by gravity: W = mgh, where h is the vertical displacement.
• (vi) Work done in compressing/stretching a spring: W = \frac{1}{2}kx^2, where k is the spring constant and x is the displacement from the mean position.
• (vii) Work done by a spring with initial and final positions x1 and x2: W = \frac{1}{2}k(x2^2 - x1^2)
• (viii) Work done by a couple (torque): W = i \cdot \theta, where i is the torque and \theta is the angular displacement.

Power

• The time rate of doing work: Power = Work done / Time taken.
• P = \frac{W}{t} = F \cdot \frac{s}{t} = F \cdot v, where v is velocity.
• P = Fv \cos \theta, where \theta is the angle between F and v.
• Power is a scalar quantity with SI unit watt (W) and dimensional formula [ML^2T^{-3}].
• Other units: 1 kilowatt = 1000 watt, 1 horsepower = 746 watt.

Energy

• Energy is a body's capacity to do work.
• It is a scalar quantity with SI unit joule (J) and dimensional formula [ML^2T^{-3}].
• Types: mechanical, chemical, light, heat, sound, nuclear, electric, etc.

Mechanical Energy

• Sum of kinetic and potential energies, remains constant throughout the motion (law of conservation of mechanical energy).
• Types:
  • 1. Kinetic Energy (K): Energy by virtue of motion: K = \frac{1}{2}mv^2 = \frac{p^2}{2m}, where m is mass, v is velocity, and p is momentum.
  • 2. Potential Energy (U): Energy by virtue of position or configuration.
    • (i) Gravitational: U = mgh, where h is height.
    • (ii) Elastic (spring): U = \frac{1}{2}kx^2.
    • (iii) Electric: U = \frac{1}{4\pi \epsilon0} \frac{q1 q2}{r}, where q1 and q_2 are charges separated by distance r.

Work-Energy Theorem

• Work done by a force is equal to the change in kinetic energy: W = Kf - Ki, where Ki is initial and Kf is final kinetic energy.
• If W_{net} is positive, kinetic energy increases and vice versa.
• Applies to non-inertial frames: Work done by all forces (including pseudo force) equals change in kinetic energy.

Mass-Energy Equivalence

• E = \Delta mc^2, where \Delta m is mass that disappears and c is the speed of light in vacuum.

Principle of Conservation of Energy

• The total energy in an isolated system remains constant.

Principle of Conservation of Mechanical Energy

• For conservative forces, the sum of kinetic and potential energies remains constant.

Collisions

• Interaction between two or more particles for a short time, involving strong forces.

Types of Collisions

• 1. Elastic Collision:
  • Both momentum and kinetic energy are conserved.
  • Involves conservative forces.
• 2. Inelastic Collision:
  • Only momentum is conserved.
  • Involves non-conservative forces.
  • Perfectly inelastic if bodies stick together after collision.

Coefficient of Restitution or Resilience (e)

• e = \frac{\text{relative velocity of separation}}{\text{relative velocity of approach}}
• Dependent on the material of the colliding bodies.
  • Elastic collision: e = 1
  • Inelastic collision: e = 0
  • Other collisions: 0 < e < 1

One-Dimensional (Head-on) Collision

• Velocities are along the same line.
  • Velocities after collision:
    • v1 = \frac{(m1 - m2)u1 + 2m2 u2}{m1 + m2}
    • v2 = \frac{(m2 - m1)u2 + 2m1 u1}{m1 + m2}
• Equal masses: Bodies exchange velocities (v1 = u2 and v2 = u1).
• If m1 = m2 and u2 = 0: v1 = 0 and v2 = u1.
• Light body m1 collides with heavy body m2 at rest: v1 = -u1 and v_2 = 0.
• Heavy body m1 collides with light body m2 at rest: v1 = u1 and v2 = 2u1.

Inelastic One-Dimensional Collision

• Loss of kinetic energy: \Delta E = \frac{m1 m2}{2(m1 + m2)} (u1 - u2)^2 (1 - e^2)

Perfectly Inelastic One-Dimensional Collision

• Velocity of separation = 0
• Loss of kinetic energy: \Delta E = \frac{m1 m2 (u1 - u2)^2}{2(m1 + m2)}
• Body dropped from height h0, rebounds to height h1:
  • e = \frac{vn}{v0} = \sqrt{\frac{hn}{h0}}

Two-Dimensional or Oblique Collision

• Velocities do not lie along the same line.
  • In horizontal direction: m1 u1 \cos \alpha1 + m2 u2 \cos \alpha2 = m1 v1 \cos \beta1 + m2 v2 \cos \beta2
  • In vertical direction: m1 u1 \sin \alpha1 - m2 u2 \sin \alpha2 = m1 u1 \sin \beta1 - m2 u2 \sin \beta2
• If m1 = m2 and \alpha1 + \alpha2 = 90^\circ then \beta1 + \beta2 = 90^\circ
• Particle A (mass m1) collides with stationary body B (mass m2):
  • Momentum conservation:
    • m1 u = m1 v1 \cos \alpha + m2 v_2 \cos \beta
    • 0 = m1 v1 \sin \alpha - m2 v2 \sin \beta

Chapter 4: Motion in a Plane

• Motion in a plane is two-dimensional (e.g., projectile motion, circular motion).

Scalar and Vector Quantities

Scalar Quantities

• Specified by magnitude alone (e.g., length, mass, density, speed, work).

Vector Quantities

• Characterized by magnitude and direction (e.g., velocity, displacement, acceleration, force, momentum, torque).

Characteristics of Vectors

• Possess magnitude and direction.
• Do not obey ordinary algebra.
• Change if magnitude, direction, or both change.
• Represented by bold-faced letters or letters with arrows.

Unit Vector

• Vector with unit magnitude, specifies direction only.
• Represented by a cap (^).

Equal Vectors

Zero Vector

Negative of a Vector

Parallel Vectors

Coplanar Vectors

• Lie in the same plane or are parallel to the same plane.

Displacement Vector

• Gives the position of a point with reference to another point (not the origin).

Parallelogram Law of Vector Addition

• If two vectors acting at a point are represented by adjacent sides of a parallelogram, then the resultant is represented by the diagonal.

Triangle Law of Vector Addition

• If two vectors are represented by two sides of a triangle in the same order, the resultant is the third side in the opposite order.

Polygon Law of Vector Addition

• If multiple vectors are represented by sides of a polygon in the same order, the resultant is the closing side in the opposite order.

• Resultant: \vec{R} = \vec{p} + \vec{q} + \vec{s} + \vec{t}

Properties of Vector Addition

• (i) Commutative Law: \vec{a} + \vec{b} = \vec{b} + \vec{a}
• (ii) Associative Law: \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}
• (iii) Distributive Property: \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}

Equilibriant Vector

• Balances two or more vectors; equal in magnitude and opposite in direction to the resultant vector.

• \vec{R'} = -\vec{R} = -(\vec{A} + \vec{B} + …)

• Multiplying a vector \vec{A} by a real number \lambda gives a vector \vec{B} with magnitude \lambda times that of \vec{A}, same or opposite direction depending on whether \lambda is positive or negative.

Subtraction of Vectors

• \vec{P} - \vec{Q} = \vec{P} + (-\vec{Q})

• Vector subtraction is non-commutative and non-associative:

  • \vec{A} - \vec{B} \neq \vec{B} - \vec{A}
  • \vec{A} - (\vec{B} - \vec{C}) \neq (\vec{A} - \vec{B}) - \vec{C}

Rectangular Components of a Vector

• Components that are perpendicular to each other.

Position Vector

Multiplication of Vectors

(i) Scalar Product (Dot Product)

• \vec{A} \cdot \vec{B} = |A||B| \cos \theta, where \theta is the angle between A and B.
• Scalar quantity.
• Geometrically: \vec{a} \cdot \vec{b} = (\text{Mod of a})(\text{Projection of b on a})

(ii) Vector Product (Cross Product)

• \vec{A} \times \vec{B} = |A||B| \sin \theta \hat{n}, where \theta is the angle between A and B, and \hat{n} is the unit vector perpendicular to the plane containing A and B.
• Geometrically: Area of parallelogram formed by the two vectors.

Properties of Scalar Product

• (i) Commutative Law: \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}
• (ii) Distributive Law: \vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}
• (iii) Zero for perpendicular vectors: \vec{A} \cdot \vec{B} = AB \cos 90^\circ = 0
• (iv) Maximum when parallel: \vec{A} \cdot \vec{B}_{max} = |A||B|
• (v) If a and b are unit vectors: \hat{a} \cdot \hat{b} = 1 \cdot 1 \cos 0 = 1
• (vi) Dot product of unit vectors: \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1 and \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0
• (vii) Square of a vector: \vec{a} \cdot \vec{a} = |a||a| \cos 0 = a^2
• (viii) In terms of rectangular components: If

\vec{A} = Ax \hat{i} + Ay \hat{j} + A_z \hat{k}

and

\vec{B} = Bx \hat{i} + By \hat{j} + B_z \hat{k}

then,

\vec{A} \cdot \vec{B} = Ax Bx + Ay By + Az Bz

Properties of Cross Product

• (i) Non-commutative: \vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}
• (ii) Non-associative: \vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}
• (iii) Distributive Law: \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
• (iv) Zero for collinear vectors: If \theta = 0 or \pi, then \vec{a} \times \vec{b} = 0
• (v) If \theta = 90^\circ, then \vec{a} \times \vec{b} = |a||b| \hat{n}
• (vi) Cross product of any vector with itself is zero:
• (vii) If \vec{a} \times \vec{b} = 0, then \vec{a} = 0 or \vec{b} = 0 or \vec{a} \parallel \vec{b}
• (viii) If a and b are unit vectors, then \vec{a} \times \vec{b} = \sin \theta \hat{n}
• (ix) Cross product of unit vectors:

\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0

\hat{i} \times \hat{j} = \hat{k} = -\hat{j} \times \hat{i}

\hat{j} \times \hat{k} = \hat{i} = -\hat{k} \times \hat{j}

\hat{k} \times \hat{i} = \hat{j} = -\hat{i} \times \hat{k}

Lami's Theorem: "If three forces acting at a point are in equilibrium, then each force has a constant ratio with the sine of the angle between the other two forces."

Projectile Motion

• The trajectory of an object with an initial inclined velocity acted on only by gravity and air resistance.

Angular Acceleration

• The rate of change of angular velocity in circular motion.

Angular Displacement

• Angle traced by the radius vector at the center of a circular path θ = arc/radius

Uniform Circular Motion

• Object moving in a circular path with constant speed.

Centripetal Acceleration

• Acceleration to maintain circular motion.