Physics Notes - Laws of Motion, Work, Energy, and Power

Laws of Motion

Inertia

Inertia is the property of an object that resists changes in its state of rest or uniform motion in a straight line.
It is a measure of mass; greater mass implies greater inertia.
Types of inertia:
- Inertia of Rest: Passengers fall backward when a bus starts suddenly.
- Inertia of Motion: Passengers jerk forward when a moving bus stops suddenly.
- Inertia of Direction: Using an umbrella to protect from rain due to raindrops resisting change in direction.

Force

Force is a push or pull that changes or tries to change the state of rest, motion, size, or shape of a body.
SI unit: Newton (N).
Dimensional formula: [MLT^{-2}]
Types of forces:
- Contact Forces: Frictional force, tensional force, spring force, normal force.
- Action at a Distance Forces: Electrostatic force, gravitational force, magnetic force.

Impulsive Force

A force acting for a short time, producing a large change in momentum.

Linear Momentum

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

Impulse

Impulse is the product of impulsive force and time.
Impulse = Force * Time = Change in momentum
SI unit: Newton-second (N-s) or kg-m/s.
Dimension: [MLT^{-1}]
It is a vector quantity, with direction matching 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 acted upon by an external force.
Also known as the law of inertia.
Examples:
- Dust particles separate from a carpet when beaten with a stick.
- Passengers bend outward when a moving vehicle stops suddenly.

2. Newton’s Second Law of Motion

The rate of change of linear momentum is proportional to the applied force.
Change in momentum occurs in the direction of the applied force.
Mathematically, F \propto dp/dt
F = k \frac{d}{dt}(mv), where k is a constant of proportionality (1 in SI and CGS).
F = m \frac{dv}{dt} = ma
Examples:
- It is easier for an adult to push a full shopping cart than for 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.
Mathematically, F{12} = -F{21}
Examples:
- Swimming is possible because of 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.
Note: Newton’s second law is the real law of motion because first and third laws can be derived from it.
Modern Version:
- A body remains in its initial state unless acted on by an unbalanced external force.
- Forces always occur in pairs; if A exerts a force on B, B exerts an equal and opposite force on A.

Law of Conservation of Linear Momentum

If no external force acts on a system, the total linear momentum remains conserved.
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

It is the force with which a body is pulled towards the center of the Earth due to gravity.
Magnitude: w = mg, where m is mass and g is the 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).
(iii) Accelerating downward: R_2 = m(g - a) (apparent weight is less).
(iv) Falling freely: R_2 = m(g - g) = 0 (apparent weight is zero).
(v) Accelerating downward with a > g: body lifts from the floor to the ceiling.

Rocket

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

Friction

A force opposing relative motion at the point of contact between objects.
Acts parallel to the contact surfaces.
Caused by intermolecular interactions between the molecules of the bodies in contact.
Types of Friction:

1. Static Friction

Opposing force that prevents motion when one body tends to move over another.
Self-adjusting force that increases with the applied force.

2. Limiting Friction

Maximum value of static friction just before the body starts moving.
fs = \mus R, where \mu_s is the coefficient of static 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 the angle of friction is \theta, then \mu_s = \tan \theta

3. Kinetic Friction

Frictional force when the body begins to slide.
Constant value: 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 (easier to roll than slide).

Angle of Repose or Angle of Sliding

Minimum angle of inclination of a plane for a body to just begin sliding down.
If the angle of repose is \alpha and the coefficient of limiting friction is \mus, then \mus = \tan \alpha

Motion on an Inclined Plane

Forces: normal reaction, friction.
Normal reaction: R = mg \cos \theta
Net force 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 \alpha:
- Minimum force to move the body up the plane: f_1 = mg(\sin \theta + \mu \cos \theta)
- Minimum force to push the body down the plane: f_2 = mg(\mu \cos \theta - \sin \theta)

Tension

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

Motion of Bodies in Contact

(i) Two Bodies in Contact

Acceleration: a = \frac{F}{m1 + m2}
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, \mu is the coefficient of friction 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 also present between ground and body A, \mu1 is the coefficient of friction between the ground and A, and \mu2 is that between A and B:
- 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: fB = \mu2 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: a = \frac{F}{m1 + m2 + m_3}
Tension in string T1: T1 = F
Tension in string T2: T2 = (m2 + m3) a = \frac{(m2 + m3) F}{m1 + m2 + m_3}
Tension in string T3: T3 = m3 a = \frac{m3 F}{m1 + m2 + m_3}

Pulley Mass System

(i) Unequal masses m1 and m2 suspended from a pulley (m1 > m2)

m1 g - T = m1 a and T - m2 g = m2 a
Solving, a = \frac{m1 - m2}{m1 + m2} g
T = \frac{2 m1 m2}{m1 + m2} g

(ii) Body of 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) Body of mass m_2 on a rough horizontal surface

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

(iv) Two masses m1 and m2 connected to a single mass M

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

(v) Motion on a smooth inclined plane

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) g}{m1 + m2}

(vi) Two bodies on two inclined planes with different angles of inclination

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

Work, Energy, and Power

Work

Work is done when a force causes an object to move in the direction of the force.
Work = Force * Displacement in the direction of the force.
W = F \cdot s = F s \cos \theta, where \theta is the angle between F and s.
Work is a scalar quantity.
SI unit: Joule (J).
CGS unit: erg.
1 Joule = 10^7 erg.
Dimensional formula: [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.
Work is negative if \theta is obtuse.
Work 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 = Work done by the resultant force.
(iii) Equilibrium: Resultant force is zero, so resultant work is zero.
(iv) Conservative vs. Non-conservative forces:
- If work done during a round trip is zero, the force is conservative.
- Examples: Gravitational, electrostatic, magnetic forces (central forces).
- Non-conservative forces: Frictional force, viscous force.
(v) Work done by gravity: W = mgh, where h is the height.
(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.
(vii) Work done with a block on a horizontal table from x = x1 to x = x2: W = \frac{1}{2} k(x2^2 - x1^2).
(viii) Work done by a couple: W = \tau \cdot \theta, where \tau is the torque and \theta is the angular displacement.

Power

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

Energy

Energy is the capacity of a body to do work.
It is a scalar quantity.
SI unit: Joule (J).
CGS unit: erg.
Dimensional formula: [ML^3T^{-3}].
Types of energies: Mechanical (kinetic and potential), chemical, light, heat, sound, nuclear, electric, etc.

Mechanical Energy

The sum of kinetic and potential energies remains constant.
Law of conservation of mechanical energy.
Types:

1. Kinetic Energy

Energy by virtue of motion.
K = \frac{1}{2} mv^2 = \frac{p^2}{2m}.

2. Potential Energy

Energy by virtue of position or configuration.
Types:
- Gravitational Potential Energy: U = mgh
- Elastic Potential Energy: U = \frac{1}{2} kx^2.
- Electric Potential Energy: U = \frac{1}{4\pi\epsilon0} \frac{q1 q_2}{r}.
Potential energy is defined only for conservative forces.
It depends on the frame of reference.

Work-Energy Theorem

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

Mass-Energy Equivalence

E = \Delta mc^2, where c is the speed of light.

Principle of Conservation of Energy

The sum of all kinds of energies in an isolated system remains constant.

Principle of Conservation of Mechanical Energy

For conservative forces, the sum of kinetic and potential energies remains constant.
Mass and energy are conserved as a single entity: mass-energy.

Collisions

Interaction for a short time where particles exert strong forces on each other.
Physical contact is not necessary.
Types:

1. Elastic Collision

Both momentum and kinetic energy are conserved.
Forces are conservative.
Total energy is conserved.

2. Inelastic Collision

Only momentum is conserved; kinetic energy is not.
Some forces are non-conservative.
Total energy is conserved.
Perfectly inelastic: bodies stick together after collision.

Coefficient of Restitution (e)

e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}}
0 \leq e \leq 1
Perfectly elastic: e = 1
Perfectly inelastic: e = 0

One Dimensional or Head-on Collision

Velocities lie along the same line.

Inelastic One-Dimensional Collision

v1 = \frac{(m1 - m2)u1 + 2m2 u2}{m1 + m2}
v2 = \frac{(m2 - m1)u2 + 2m1 u1}{m1 + m2}
If m1 = m2, bodies exchange velocities: v1 = u2, v2 = u1.
If m1 = m2 and the second body is at rest, after the collision, the first body stops, and the second body moves with the initial velocity of the first body. v1 = 0, v2 = u_1
If a light body collides with a heavy body at rest, the light body rebounds with its own velocity v1 = -u1 and the heavy body remains at rest v_2 = 0
If a heavy body collides with a light body at rest, the heavy body continues with its initial velocity v1 = u1 and the light body moves with twice the velocity of the heavy body 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 after collision = 0.
Loss of kinetic energy: \Delta E = \frac{m1 m2 (u1 - u2)^2}{2(m1 + m2)}
Height and Velocity after n collisions:
- e^n = \frac{vn}{v0} = \sqrt{\frac{hn}{h0}}

Two Dimensional or Oblique Collision

Velocities do not lie along the same line.
Horizontal: m1 u1 \cos \alpha1 + m2 u2 \cos \alpha2 = m1 v1 \cos \beta1 + m2 v2 \cos \beta2
Vertical: 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
If a particle A of mass m1 moving along the z-axis with speed u collides elastically with a stationary body B of mass m2:
- m1 u = m1 v1 \cos \alpha + m2 v_2 \cos \beta
- O = m1 v1 \sin \alpha - m2 v2 \sin \beta

Motion in a Plane

Motion in two dimensions (e.g., projectile motion, circular motion).
Reference is made to an origin and two coordinate axes X and Y.

Scalar and Vector Quantities

Scalar Quantities: Specified by magnitude alone (e.g., length, mass, density, speed, work).
Vector Quantities: Characterized by both magnitude and direction (e.g., velocity, displacement, acceleration, force, momentum, torque).

Characteristics of Vectors

Possess magnitude and direction.
Do not obey ordinary laws of algebra.
Change if either magnitude or direction or both change.
Represented by bold-faced letters or letters with an arrow over them.

Unit Vector

A vector of unit magnitude in a particular direction.
Used to specify direction only.
Denoted by a cap (^).

Equal Vectors

Zero Vector

Negative of a Vector

Parallel Vectors

Coplanar Vectors

Displacement Vector

A vector giving the position of a point with reference to another point (not the origin).

Parallelogram Law of Vector Addition

If two vectors acting simultaneously at a point are represented in magnitude and direction by two adjacent sides of a parallelogram, the resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

Triangle Law of Vector Addition

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

Polygon Law of Vector Addition

If a number of vectors are represented in magnitude and direction by the sides of a polygon taken in the same order, the resultant vector is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.
Resultant \vec{R} = \vec{p} + \vec{q} + \vec{r} + \vec{s} + \vec{t}

Properties of Vector Addition

Commutative law: \vec{a} + \vec{b} = \vec{b} + \vec{a}
Associative law: \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}
Distributive property: \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}

Equilibriant Vector

A vector that balances two or more vectors acting simultaneously at a point.
Equal in magnitude and opposite in direction to the resultant vector.
\vec{R'} = -\vec{R} = -(\vec{A} + \vec{B} + …)
If vector \vec{A} is multiplied by a real number\lambda , it gives a vector \vec{B} whose magnitude is \lambda times the magnitude of \vec{A}, and whose direction is the same or opposite 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}
If the components of a given vector are perpendicular, they are called rectangular components.
Position Vector

Multiplication of Vectors

(i) Scalar product (Dot product)

The product of the magnitudes of two vectors with the cosine of the smaller angle between them.
It is always a scalar: \vec{A} \cdot \vec{B} = |A||B|\cos\theta
Geometrically: \vec{a} \cdot \vec{b} = (\text{Mod of } a)(\text{Projection of } b \text{ on } a)

(ii) Vector product (Cross product)

\vec{A} \times \vec{B} = |A||B|\sin\theta \hat{n}, where \theta is the angle between \vec{A} and \vec{B} (anti-clockwise) and \hat{n} is a unit vector perpendicular to the plane containing \vec{A} and \vec{B}.
Geometrically, \vec{A} \times \vec{B} is a vector whose modulus is the area of the parallelogram formed by \vec{A} and \vec{B} as adjacent sides, and direction is perpendicular to both \vec{A} and \vec{B}.

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) Scalar (Dot) product of two mutually perpendicular vectors is zero: (\vec{A} \cdot \vec{B}) = AB \cos 90^\circ = 0
(iv) Scalar (Dot) product is maximum when \theta = 0^\circ: (\vec{A} \cdot \vec{B})_{\text{max}} = |A||B|
(v) If\vec{a} and \vec{b} are unit vectors, then \vec{a} \cdot \vec{b} = 1 \cdot 1 \cos 0 = 1
(vi) Dot product of unit vectors \hat{i}, \hat{j}, \hat{k}:
- \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1
- \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) If the components of two vectors \vec{A} and \vec{B} are given by:
- \vec{A} = Ax \hat{i} + Ay \hat{j} + A_z \hat{k}
- \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) Cross product is not commutative: \vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}
(ii) Cross product is not associative: \vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}
(iii) Cross product obeys the distributive law: \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
(iv) If \theta = 0 or \pi: \vec{a} \times \vec{b} = \vec{0}
- Conversely, if \vec{a} \times \vec{b} = \vec{0}, then \vec{a} and \vec{b} are parallel, provided \vec{a} and \vec{b} are non-zero vectors.
(v) If \theta = 90^\circ: \vec{a} \times \vec{b} = |a||b| \sin 90^\circ \hat{n} = |a||b| \hat{n}
(vi) The vector product of a vector with itself is \vec{0}.
(vii) If \vec{a} \times \vec{b} = 0 then, \vec{a} = 0 or \vec{b} = 0 or \vec{a} \parallel \vec{b}
(viii) If \vec{a} and \vec{b} are unit vectors, then \vec{a} \times \vec{b} = 1 \cdot 1 \sin \theta \hat{n} = \sin \theta \hat{n}
(ix) Cross product of unit vectors \hat{i}, \hat{j} and \hat{k}:
- \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}
  Equilibrium Condition
“If three concurrent forces are in equilibrium, then each force has a constant ratio with the sine of the angle between the other two forces.”

Projectile Motion

The projectile is an object given an initial