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, $F<em>{12} = -F</em>{21}$
• 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: $u = v<em>0 + u \log</em>e(\frac{M<em>0}{M})$, where $v</em>0$ is the initial velocity, $M_0$ is the initial mass, and $M$ is the present mass.
• With gravity: $u = v<em>0 + u \log</em>e(\frac{M_0}{M}) - gt$

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.
• $f<em>s = \mu</em>s R$, 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: $f<em>k = \mu</em>k N$, 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 $\mu<em>s$, then $\mu</em>s = \tan \alpha$

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: $a = \frac{F}{m<em>1 + m</em>2}$
• Contact force on $m<em>1$: $m</em>1 a = \frac{m<em>1 F}{m</em>1 + m_2}$
• Contact force on $m<em>2$: $m</em>2 a = \frac{m<em>2 F}{m</em>1 + m_2}$

(ii) Three Bodies in Contact

• Acceleration: $a = \frac{F}{m<em>1 + m</em>2 + m_3}$
• Contact force between $m<em>1$ and $m</em>2$: $F<em>1 = \frac{(m</em>2 + m<em>3)F}{m</em>1 + m<em>2 + m</em>3}$
• Contact force between $m<em>2$ and $m</em>3$: $F<em>2 = \frac{m</em>3 F}{m<em>1 + m</em>2 + m_3}$

(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, $\mu<em>1$ is the coefficient of friction between the ground and A, and $\mu</em>2$ is that between A and B:
  • Net accelerating force: $F - f<em>A = F - \mu</em>1 (M + m)g$
  • Net acceleration: $a = \frac{F - \mu<em>1 (M + m)g}{M + m} = \frac{F}{M + m} - \mu</em>1 g$
  • Pseudo force on block B: $f' = ma$
  • Frictional force: $f<em>B = \mu</em>2 mg$
  • For equilibrium: $ma \leq \mu<em>2 mg$ or $a \leq \mu</em>2 g$. If a > \mu_2 g, the bodies will not move together.

(iv) Motion of Bodies Connected by Strings

• Acceleration: $a = \frac{F}{m<em>1 + m</em>2 + m_3}$
• Tension in string $T<em>1$: $T</em>1 = F$
• Tension in string $T<em>2$: $T</em>2 = (m<em>2 + m</em>3) a = \frac{(m<em>2 + m</em>3) F}{m<em>1 + m</em>2 + m_3}$
• Tension in string $T<em>3$: $T</em>3 = m<em>3 a = \frac{m</em>3 F}{m<em>1 + m</em>2 + m_3}$

Pulley Mass System

(i) Unequal masses $m<em>1$ and $m</em>2$ suspended from a pulley ($m<em>1 > m</em>2$)

• $m<em>1 g - T = m</em>1 a$ and $T - m<em>2 g = m</em>2 a$
• Solving, $a = \frac{m<em>1 - m</em>2}{m<em>1 + m</em>2} g$
• $T = \frac{2 m<em>1 m</em>2}{m<em>1 + m</em>2} g$

(ii) Body of mass $m_2$ on a frictionless horizontal surface

• Acceleration: $a = \frac{m<em>1 g}{m</em>1 + m_2}$
• Tension: $T = \frac{m<em>1 m</em>2 g}{m<em>1 + m</em>2}$

(iii) Body of mass $m_2$ on a rough horizontal surface

• Acceleration: $a = \frac{m<em>1 - \mu m</em>2}{m<em>1 + m</em>2} g$
• Tension: $T = \frac{m<em>1 m</em>2 (1 + \mu) g}{m<em>1 + m</em>2}$

(iv) Two masses $m<em>1$ and $m</em>2$ connected to a single mass $M$

• $m<em>1 g - T</em>1 = m_1 a$
• $T<em>2 - m</em>2 g = m_2 a$
• $T<em>1 - T</em>2 = M a$
• Acceleration: $a = \frac{m<em>1 - m</em>2}{m<em>1 + m</em>2 + M} g$
• Tension $T<em>1$: $T</em>1 = \frac{2 m<em>2 + M}{m</em>1 + m<em>2 + M} m</em>1 g$
• Tension $T<em>2$: $T</em>2 = \frac{2 m<em>1 + M}{m</em>1 + m<em>2 + M} m</em>2 g$

(v) Motion on a smooth inclined plane

• $m<em>1 g - T = m</em>1 a$
• $T - m<em>2 g \sin \theta = m</em>2 a$
• Acceleration: $a = \frac{m<em>1 - m</em>2 \sin \theta}{m<em>1 + m</em>2} g$
• Tension: $T = \frac{m<em>1 m</em>2 (1 + \sin \theta) g}{m<em>1 + m</em>2}$

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

• Acceleration: $a = \frac{(m<em>1 \sin \theta</em>1 - m<em>2 \sin \theta</em>2) g}{m<em>1 + m</em>2}$
• Tension: $T = \frac{m<em>1 m</em>2}{m<em>1 + m</em>2} (\sin \theta<em>1 - \sin \theta</em>2) g$

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 $x = x<em>1$ to $x = x</em>2$: $W = \frac{1}{2} k(x<em>2^2 - x</em>1^2)$.
• (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: $U = \frac{1}{4\pi\epsilon<em>0} \frac{q</em>1 q_2}{r}$.
• Potential energy is defined only for conservative forces.
• It depends on the frame of reference.

Work-Energy Theorem

• Work done = Change in kinetic energy: $W = K<em>f - K</em>i$.
• 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

• $v<em>1 = \frac{(m</em>1 - m<em>2)u</em>1 + 2m<em>2 u</em>2}{m<em>1 + m</em>2}$
• $v<em>2 = \frac{(m</em>2 - m<em>1)u</em>2 + 2m<em>1 u</em>1}{m<em>1 + m</em>2}$
• If $m<em>1 = m</em>2$, bodies exchange velocities: $v<em>1 = u</em>2$, $v<em>2 = u</em>1$.
• If $m<em>1 = m</em>2$ 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. $v<em>1 = 0$, $v</em>2 = u_1$
• If a light body collides with a heavy body at rest, the light body rebounds with its own velocity $v<em>1 = -u</em>1$ 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 $v<em>1 = u</em>1$ and the light body moves with twice the velocity of the heavy body $v<em>2 = 2u</em>1$

Inelastic One-Dimensional Collision

• Loss of kinetic energy: $\Delta E = \frac{m<em>1 m</em>2}{2(m<em>1 + m</em>2)} (u<em>1 - u</em>2)^2 (1 - e^2)$

Perfectly Inelastic One-Dimensional Collision

• Velocity of separation after collision = 0.
• Loss of kinetic energy: $\Delta E = \frac{m<em>1 m</em>2 (u<em>1 - u</em>2)^2}{2(m<em>1 + m</em>2)}$
• Height and Velocity after n collisions:
  • $e^n = \frac{v<em>n}{v</em>0} = \sqrt{\frac{h<em>n}{h</em>0}}$

Two Dimensional or Oblique Collision

• Velocities do not lie along the same line.
• Horizontal: $m<em>1 u</em>1 \cos \alpha<em>1 + m</em>2 u<em>2 \cos \alpha</em>2 = m<em>1 v</em>1 \cos \beta<em>1 + m</em>2 v<em>2 \cos \beta</em>2$
• Vertical: $m<em>1 u</em>1 \sin \alpha<em>1 - m</em>2 u<em>2 \sin \alpha</em>2 = m<em>1 u</em>1 \sin \beta<em>1 - m</em>2 u<em>2 \sin \beta</em>2$
• If $m<em>1 = m</em>2$ and $\alpha<em>1 + \alpha</em>2 = 90^\circ$, then $\beta<em>1 + \beta</em>2 = 90^\circ$
• If a particle A of mass $m<em>1$ moving along the z-axis with speed $u$ collides elastically with a stationary body B of mass $m</em>2$:
  • $m<em>1 u = m</em>1 v<em>1 \cos \alpha + m</em>2 v_2 \cos \beta$
  • $O = m<em>1 v</em>1 \sin \alpha - m<em>2 v</em>2 \sin \beta$

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:
  • $\vec{A} = A<em>x \hat{i} + A</em>y \hat{j} + A_z \hat{k}$
  • $\vec{B} = B<em>x \hat{i} + B</em>y \hat{j} + B_z \hat{k}$ then,
  • $\vec{A} \cdot \vec{B} = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$

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