Class 11 – Chapter 0: All Important Maths Formulas

1) Algebraic Identities

\[ (a+b)^2 = a^2 + 2ab + b^2 \] \[ (a-b)^2 = a^2 - 2ab + b^2 \] \[ (a+b)(a-b) = a^2 - b^2 \] \[ (x+a)(x+b) = x^2 + (a+b)x + ab \] \[ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca) \] \[ (a+b)^3 = a^3 + b^3 + 3ab(a+b) \] \[ (a-b)^3 = a^3 - b^3 - 3ab(a-b) \] \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \] \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \] \[ a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]

2) Factorisation / Quadratic

\[ ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{\,b^2 - 4ac\,}}{2a} \] \[ x^2 + (p+q)x + pq = (x+p)(x+q) \] \[ b^2 - 4ac = \Delta \text{ (discriminant)} \]

3) Indices (Exponents)

\[ a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n} \] \[ (a^m)^n = a^{mn}, \quad (ab)^n = a^n b^n \] \[ a^0 = 1 \ (a\neq0), \quad a^{-n} = \frac{1}{a^n} \]

4) Surds

\[ \sqrt{ab} = \sqrt a\,\sqrt b \ (a,b \ge 0), \quad \sqrt{\tfrac{a}{b}} = \frac{\sqrt a}{\sqrt b} \ (b>0) \] \[ \sqrt a \cdot \sqrt a = a, \quad (\sqrt a + \sqrt b)^2 = a+b+2\sqrt{ab} \]

5) Logarithms

\[ \log_a (mn) = \log_a m + \log_a n \] \[ \log_a \!\left(\frac{m}{n}\right) = \log_a m - \log_a n \] \[ \log_a (m^k) = k\,\log_a m \] \[ \log_a 1 = 0,\quad \log_a a = 1 \] \[ \log_a b = \frac{\log_c b}{\log_c a} \ (\text{change of base}) \] \[ a^{\log_a m} = m \]

6) Series: AP & GP

\[ \text{AP: } a,\,a+d,\,a+2d,\dots \] \[ n\text{th term: } a_n = a + (n-1)d \] \[ \text{Sum of }n\text{ terms: } S_n = \frac{n}{2}\,[2a + (n-1)d] \] \[ \text{GP: } a,\,ar,\,ar^2,\dots \] \[ n\text{th term: } a_n = a\,r^{\,n-1} \] \[ \text{Sum of }n\text{ terms }(r\neq1):\; S_n = a\,\frac{1-r^{\,n}}{1-r} \] \[ \text{Infinite GP }(|r|<1):\; S_\infty = \frac{a}{1-r} \]

7) Binomial Basics

\[ (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{\,n-k} y^{\,k} \] \[ \binom{n}{k} = \frac{n!}{k!\,(n-k)!}, \quad \binom{n}{k} = \binom{n}{n-k} \]

8) Trigonometry – Pythagorean & Identities

\[ \sin^2\theta + \cos^2\theta = 1 \] \[ 1 + \tan^2\theta = \sec^2\theta,\quad 1 + \cot^2\theta = \csc^2\theta \] \[ \sin(2\theta) = 2\sin\theta\cos\theta \] \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta = 1-2\sin^2\theta = 2\cos^2\theta-1 \] \[ \tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta} \] \[ \sin(A\pm B)=\sin A\cos B \pm \cos A\sin B \] \[ \cos(A\pm B)=\cos A\cos B \mp \sin A\sin B \] \[ \tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \]

Trigonometric Ratios (Standard Angles)

Values of sin θ, cos θ, tan θ, cosec θ, sec θ, and cot θ at standard angles:

\(\theta\) \(\sin\theta\) \(\cos\theta\) \(\tan\theta\) \(\csc\theta\) \(\sec\theta\) \(\cot\theta\)
\(0^\circ\) 0 1 0 Undefined 1 Undefined
\(30^\circ\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{\sqrt{3}}\) 2 \(\frac{2}{\sqrt{3}}\) \(\sqrt{3}\)
\(45^\circ\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) 1 \(\sqrt{2}\) \(\sqrt{2}\) 1
\(60^\circ\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\) \(\frac{2}{\sqrt{3}}\) 2 \(\frac{1}{\sqrt{3}}\)
\(90^\circ\) 1 0 Undefined 1 Undefined 0
\(120^\circ\) \(\frac{\sqrt{3}}{2}\) \(-\frac{1}{2}\) \(-\sqrt{3}\) \(\frac{2}{\sqrt{3}}\) -2 \(-\frac{1}{\sqrt{3}}\)
\(135^\circ\) \(\frac{\sqrt{2}}{2}\) \(-\frac{\sqrt{2}}{2}\) -1 \(\sqrt{2}\) \(-\sqrt{2}\) -1
\(150^\circ\) \(\frac{1}{2}\) \(-\frac{\sqrt{3}}{2}\) \(-\frac{1}{\sqrt{3}}\) 2 \(-\frac{2}{\sqrt{3}}\) \(-\sqrt{3}\)
\(180^\circ\) 0 -1 0 Undefined -1 Undefined
\(270^\circ\) -1 0 Undefined -1 Undefined 0
\(360^\circ\) 0 1 0 Undefined 1 Undefined

9) Coordinate Geometry

\[ \text{Distance: } d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \] \[ \text{Midpoint: } M\!\left(\frac{x_1+x_2}{2},\,\frac{y_1+y_2}{2}\right) \] \[ \text{Section (internal }m:n\text{): } \left(\frac{mx_2+nx_1}{m+n},\,\frac{my_2+ny_1}{m+n}\right) \] \[ \text{Slope: } m = \frac{y_2-y_1}{x_2-x_1} \] \[ \text{Point–slope: } y-y_1 = m(x-x_1) \] \[ \text{Slope–intercept: } y = mx + c \] \[ \text{Two-point: } \frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1} \] \[ \text{Intercept form: } \frac{x}{a} + \frac{y}{b} = 1 \] \[ \text{General form: } Ax + By + C = 0 \]

10) Circle (Basics)

\[ (x - h)^2 + (y - k)^2 = r^2 \] \[ x^2 + y^2 + 2gx + 2fy + c = 0 \ \Rightarrow \ \text{center }(-g,-f),\ r=\sqrt{g^2+f^2-c} \]

11) Triangle Geometry (2D)

\[ \text{Area: } \Delta = \frac{1}{2}\,|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)| \] \[ \text{Heron: } \Delta = \sqrt{s(s-a)(s-b)(s-c)},\ s=\frac{a+b+c}{2} \] \[ \text{Centroid: } \left(\frac{x_1+x_2+x_3}{3},\,\frac{y_1+y_2+y_3}{3}\right) \]

12) Mensuration (Quick)

\[ \text{Area(circle)}=\pi r^2,\ \text{Circumference}=2\pi r \] \[ \text{Surface Area(sphere)}=4\pi r^2,\ \text{Volume(sphere)}=\frac{4}{3}\pi r^3 \] \[ \text{Volume(cylinder)}=\pi r^2 h,\ \text{Volume(cone)}=\frac{1}{3}\pi r^2 h \]

13) Probability (Basics)

\[ P(E) = \frac{\text{favourable outcomes}}{\text{total outcomes}} \] \[ P(\bar E) = 1 - P(E) \] \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

14) Inequalities (Basics)

\[ |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases} \] \[ |x| < a \quad (a > 0) \quad \Rightarrow \quad -a < x < a \] \[ |x| > a \quad (a > 0) \quad \Rightarrow \quad x < -a \ \text{or} \ x > a \]

Differentiation Formulas

1. \(\frac{d}{dx}(c) = 0\)

2. \(\frac{d}{dx}(x) = 1\)

3. \(\frac{d}{dx}(x^n) = n x^{n-1}\)

4. \(\frac{d}{dx}(e^x) = e^x\)

5. \(\frac{d}{dx}(a^x) = a^x \ln(a)\)

6. \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)

7. \(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)

8. \(\frac{d}{dx}(\sin x) = \cos x\)

9. \(\frac{d}{dx}(\cos x) = -\sin x\)

10. \(\frac{d}{dx}(\tan x) = \sec^2 x\)

11. \(\frac{d}{dx}(\cot x) = -\csc^2 x\)

12. \(\frac{d}{dx}(\sec x) = \sec x \tan x\)

13. \(\frac{d}{dx}(\csc x) = -\csc x \cot x\)

14. \(\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}\)

15. \(\frac{d}{dx}(\cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}\)

16. \(\frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}\)

17. \(\frac{d}{dx}(\cot^{-1}x) = -\frac{1}{1+x^2}\)

18. \(\frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}\)

19. \(\frac{d}{dx}(\csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}}\)

Integration Formulas

1. \(\int 0 \, dx = C\)

2. \(\int 1 \, dx = x + C\)

3. \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \; n \neq -1\)

4. \(\int e^x \, dx = e^x + C\)

5. \(\int a^x \, dx = \frac{a^x}{\ln a} + C\)

6. \(\int \frac{1}{x} \, dx = \ln|x| + C\)

7. \(\int \sin x \, dx = -\cos x + C\)

8. \(\int \cos x \, dx = \sin x + C\)

9. \(\int \sec^2 x \, dx = \tan x + C\)

10. \(\int \csc^2 x \, dx = -\cot x + C\)

11. \(\int \sec x \tan x \, dx = \sec x + C\)

12. \(\int \csc x \cot x \, dx = -\csc x + C\)

13. \(\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1}x + C\)

14. \(\int \frac{-1}{\sqrt{1-x^2}} \, dx = \cos^{-1}x + C\)

15. \(\int \frac{1}{1+x^2} \, dx = \tan^{-1}x + C\)

16. \(\int \frac{-1}{1+x^2} \, dx = \cot^{-1}x + C\)

17. \(\int \frac{1}{|x|\sqrt{x^2-1}} \, dx = \sec^{-1}x + C\)

18. \(\int \frac{-1}{|x|\sqrt{x^2-1}} \, dx = \csc^{-1}x + C\)

Units and Measurements - Class 11 Physics Notes

Physics is a science based on the measurement of physical quantities. Every observation and experiment requires measurement in a well-defined unit. To make communication universal, the International System of Units (SI system) is used worldwide.

1. Physical Quantities and Measurement

Physical Quantity: Any quantity that can be measured and expressed in terms of a unit.
Examples: Length, Mass, Time, Temperature, Electric Current, etc.

Measurement: Comparison of a physical quantity with a standard reference unit.

Example: If the length of a rod is 2 m, it means the rod is 2 times the chosen unit of length (1 m).

2. Systems of Units

Historically, different unit systems were used:

SystemLengthMassTime
CGScmgs
FPSfootpounds
MKSmetrekgs
SImetrekgs

SI System (International System of Units)

Introduced in 1960, the SI system is the most widely accepted. It defines 7 fundamental quantities:

Base QuantitySI UnitSymbol
Lengthmetrem
Masskilogramkg
Timeseconds
Electric currentampereA
TemperaturekelvinK
Luminous intensitycandelacd
Amount of substancemolemol

👉 All other quantities like velocity, force, energy, pressure are derived quantities, expressed using these base units.

3. Significant Figures

Significant figures represent the precision and reliability of a measurement.

Rules for Counting Significant Figures

  • All non-zero digits are significant.
    Example: 456 → 3 significant figures.
  • Zeros between non-zero digits are significant.
    Example: 1002 → 4 significant figures.
  • Leading zeros are not significant.
    Example: 0.0045 → 2 significant figures.
  • Trailing zeros after a decimal are significant.
    Example: 2.300 → 4 significant figures.

Importance: They show the reliability of experimental data.

4. Accuracy, Precision, and Errors

Accuracy: Closeness of a measured value to the true value.
Precision: Closeness of repeated measurements with each other.

Examples:
– If a clock shows 2:59 when the actual time is 3:00 → Accurate.
– If multiple readings are 2:59, 2:59, 2:59 → Precise.

Types of Errors

  • Systematic Errors: Due to faulty instruments, poor calibration, or observer bias.
    Example: Meter scale with worn-out edges.
  • Random Errors: Due to unpredictable variations.
    Example: Fluctuation in temperature readings.
  • Least Count Error: Due to limitation of instrument resolution.

Formulas for Errors

\[ \text{Absolute Error} = | \text{Measured Value – True Value} | \]
\[ \text{Mean Absolute Error} = \frac{\text{Sum of absolute errors}}{n} \]
\[ \text{Relative Error} = \frac{\text{Mean Absolute Error}}{\text{True Value}} \]
\[ \text{Percentage Error} = \text{Relative Error} \times 100 \]

5. Dimensional Analysis

Dimensions express a physical quantity in terms of fundamental quantities (M, L, T).

Examples:
Velocity = distance / time = \([M^0L^1T^{-1}]\)
Force = mass × acceleration = \([M^1L^1T^{-2}]\)

Uses of Dimensional Analysis

  • Conversion of units from one system to another.
  • Checking correctness of physical equations.
  • Deriving relations between physical quantities.

Limitations

  • Cannot find dimensionless constants (π, e).
  • Cannot be applied to empirical formulas.
  • Cannot give information about additive or multiplicative constants.

Summary - Key Points

  • Measurement is comparison with a standard unit.
  • SI system has 7 base quantities.
  • Significant figures show precision of data.
  • Accuracy ≠ Precision.
  • Errors are unavoidable but can be minimized.
  • Dimensional analysis is useful for unit conversion and equation verification.

Class 11 Physics Notes – Chapter 2: Motion in a Straight Line

Introduction

The study of motion without considering its cause is called Kinematics. Motion along a straight line is the simplest type, also called one-dimensional motion.

Points to Remember

  • Scalar: only magnitude (distance, speed).
  • Vector: magnitude + direction (displacement, velocity, acceleration).

Position, Path Length and Displacement

  • Position: Location of a particle on a straight line w.r.t origin.
  • Path length (Distance): Actual length of the path travelled; a scalar.
  • Displacement: Change in position of the particle; a vector.

Formula: \(\Delta x = x_2 - x_1\)

Points to Remember

  • Distance ≥ Displacement (always).
  • Displacement can be zero even if distance is not zero.

Speed and Its Types

Speed = Rate of change of distance (scalar).

  1. Uniform Speed: Equal distances in equal intervals of time.
  2. Non-Uniform Speed: Unequal distances in equal intervals of time.
  3. Average Speed: \(\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}\)
  4. Instantaneous Speed: Speed at a given instant (speedometer reading).

Velocity and Its Types

Velocity = Rate of change of displacement (vector).

  1. Uniform Velocity: Equal displacements in equal intervals of time in the same direction.
  2. Non-Uniform Velocity: Unequal displacements OR change in direction.
  3. Average Velocity: \(v_{avg} = \frac{\Delta x}{\Delta t}\)
  4. Instantaneous Velocity: \(v = \frac{dx}{dt}\)
  5. Relative Velocity: \(v_{AB} = v_A - v_B\)

Points to Remember

  • Speed ≥ |Velocity| (in magnitude).
  • Velocity can be positive, negative or zero; speed cannot be negative.
  • Uniform speed ≠ Uniform velocity (velocity depends on direction).

Acceleration

Acceleration = Rate of change of velocity.

Formulas: \(a_{avg} = \frac{\Delta v}{\Delta t}, \quad a = \frac{dv}{dt}\)

Points to Remember

  • If velocity increases → acceleration positive.
  • If velocity decreases → acceleration negative (retardation).
  • Acceleration can be zero even if velocity is not zero (uniform velocity motion).

Equations of Motion (for uniform acceleration)

  1. \( v = u + at \)
  2. \( s = ut + \frac{1}{2}at^2 \)
  3. \( v^2 - u^2 = 2as \)

Where u = initial velocity, v = final velocity, a = acceleration, s = displacement, t = time

Points to Remember

  • Valid only for constant acceleration.
  • Derivable using calculus or graphs.

Graphs in Straight Line Motion

  1. Position–Time Graph (x–t): slope = velocity. Straight line → uniform velocity, Curve → variable velocity.
  2. Velocity–Time Graph (v–t): slope = acceleration, area under graph = displacement.
  3. Acceleration–Time Graph (a–t): area under graph = change in velocity.

Points to Remember

  • Steeper slope = higher velocity/acceleration.
  • v–t graph is most important for exam problems.

Relative Velocity in One Dimension

Formula: \(v_{AB} = v_A - v_B\)

  • Same direction → difference of speeds.
  • Opposite direction → sum of speeds.

Summary / Quick Revision Box

  • Distance = scalar, Displacement = vector.
  • Speed = scalar, Velocity = vector.
  • Instantaneous velocity = slope of x–t graph.
  • Acceleration = slope of v–t graph.
  • Displacement = area under v–t graph.
  • Change in velocity = area under a–t graph.
  • Three equations of motion valid only for uniform acceleration.
  • Relative velocity (1D): \( v_{AB} = v_A - v_B \).

Chapter 3: Motion in a Plane (Vectors and 2D Motion)

1. Scalars and Vectors

Scalar: Physical quantity having only magnitude (e.g., mass, time, speed, temperature).

Vector: Physical quantity having both magnitude and direction (e.g., displacement, velocity, force).

2. Types of Vectors

  • Zero Vector: Magnitude = 0, direction arbitrary.
  • Unit Vector: Vector with magnitude = 1, represents direction.
  • Equal Vectors: Same magnitude and direction.
  • Like Vectors: Same direction.
  • Unlike Vectors: Opposite direction.
  • Position Vector: Denotes the position of a point relative to origin.
  • Co-initial Vectors: Vectors starting from the same point.
  • Collinear Vectors: Parallel vectors.
  • Coplanar Vectors: Vectors lying in the same plane.

3. Vector Operations

(a) Addition of Vectors

Triangle Law: If two vectors are represented as two sides of a triangle taken in order, then their sum is represented by the third side taken in opposite order.

Parallelogram Law: If two vectors are represented by adjacent sides of a parallelogram, their sum is represented by the diagonal.

\[ R = \sqrt{A^2 + B^2 + 2AB\cos\theta} \] \[ \tan\phi = \frac{B\sin\theta}{A + B\cos\theta} \]

(b) Subtraction of Vectors

\[\vec{A} - \vec{B} = \vec{A} + (-\vec{B})\]

(c) Resolution of a Vector

If a vector \(\vec{A}\) makes angle \(\theta\) with x-axis:

\[ A_x = A\cos\theta, \quad A_y = A\sin\theta \] \[ |\vec{A}| = \sqrt{A_x^2 + A_y^2}, \quad \tan\theta = \frac{A_y}{A_x} \]

4. Scalar Product (Dot Product)

\[\vec{A} \cdot \vec{B} = AB\cos\theta\]

Work done:

\[ W = \vec{F} \cdot \vec{d} = Fd\cos\theta \]

Properties:

  • If vectors are perpendicular → \(\vec{A} \cdot \vec{B} = 0\)
  • If vectors are parallel → \(\vec{A} \cdot \vec{B} = AB\)

5. Vector Product (Cross Product)

\[\vec{A} \times \vec{B} = AB\sin\theta \, \hat{n}\]

Here, \(\hat{n}\) is a unit vector perpendicular to the plane of \(\vec{A}, \vec{B}\). Direction is given by the Right-Hand Rule.

Magnitude:

\[ |\vec{A} \times \vec{B}| = AB\sin\theta \]

Applications:

  • Torque: \(\vec{\tau} = \vec{r} \times \vec{F}\)
  • Angular momentum: \(\vec{L} = \vec{r} \times \vec{p}\)

6. Geometrical Applications of Vectors

(a) Area of a Triangle:

\[ \text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| \]

(b) Area of a Parallelogram:

\[ \text{Area} = |\vec{a} \times \vec{b}| \]

(c) Area of a Rectangle:

\[ \text{Area} = l \times b \]

(d) Area of a Rhombus:

\[ \text{Area} = \frac{1}{2} d_1 d_2 \]

where \(d_1, d_2\) are diagonals.

7. Direction Cosines and Direction Ratios

If a vector \(\vec{A}\) makes angles \(\alpha, \beta, \gamma\) with x, y, z axes:

\[ l = \cos\alpha, \quad m = \cos\beta, \quad n = \cos\gamma \]

Relation:

\[ l^2 + m^2 + n^2 = 1 \]

Direction Ratios (DRs): Any numbers proportional to \((l, m, n)\).

8. Projectile Motion

(a) Horizontal Projectile

When a body is projected horizontally with velocity \(u\) from a height \(h\):

  • Time of flight: \(T = \sqrt{\frac{2h}{g}}\)
  • Horizontal range: \(R = uT = u \sqrt{\frac{2h}{g}}\)
  • Path equation: \(y = \frac{1}{2} g \left(\frac{x}{u}\right)^2\)

(b) Oblique Projectile

When a body is projected with velocity \(u\) at an angle \(\theta\) with the horizontal:

  • Time of flight: \(T = \frac{2u \sin \theta}{g}\)
  • Maximum height: \(H = \frac{u^2 \sin^2 \theta}{2g}\)
  • Horizontal range: \(R = \frac{u^2 \sin 2\theta}{g}\)
  • Equation of trajectory: \(y = x \tan \theta - \frac{gx^2}{2u^2 \cos^2 \theta}\)
  • Path: Parabola

Laws of Motion & Dynamics – Class 11 Physics Notes

1. Intuitive Concept of Force

Force is any influence that can change the state of rest or motion of a body. Represented by \(\vec{F}\), measured in Newton (N).

Types of Force:

  • Contact forces: Push, pull, tension
  • Non-contact forces: Gravity, magnetic, electrostatic

Effect of Force: Changes velocity (causes acceleration) and can deform a body.

2. Inertia & Newton’s First Law of Motion

Inertia: Tendency of a body to resist changes in motion. Depends on mass: larger mass → larger inertia.

Newton’s First Law: A body remains at rest or in uniform motion unless acted upon by an external force.

Examples:

  • Passenger falls forward when a bus stops suddenly
  • Book remains on a table until pushed

Applications: Seat belts, vehicle brakes, safety measures in transport

3. Momentum & Newton’s Second Law

Momentum:
\(\vec{p} = m \vec{v}\)

Newton’s Second Law:
\(\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a} \quad (\text{if } m \text{ is constant})\)

Impulse:
\(\vec{J} = \vec{F} \Delta t = \Delta \vec{p}\)

Applications: Catching a ball, vehicle collisions, airbags

4. Newton’s Third Law of Motion

Statement: For every action, there is an equal and opposite reaction. Forces act on different bodies.

Examples: Recoil of a gun, rocket propulsion, swimming

5. Law of Conservation of Linear Momentum

Statement: Total momentum of a system remains constant if no external force acts:
\(\vec{p}_\text{initial} = \vec{p}_\text{final}\)

Applications:

  • Elastic and inelastic collisions
  • Recoil of gun
  • Rocket motion

6. Equilibrium of Concurrent Forces

Particle is in equilibrium if:
\(\sum \vec{F} = 0\)

Resolve forces along axes:
\(\sum F_x = 0, \quad \sum F_y = 0\)

Example: Hanging lamp supported by two ropes

7. Friction

Definition: Force opposing relative motion between surfaces. Originates from microscopic roughness and interlocking of surfaces.

Types:

  • Static friction (\(f_s\)) – prevents motion at rest
  • Kinetic friction (\(f_k\)) – opposes sliding motion
  • Rolling friction – smaller than sliding friction

Laws of Friction:

  1. Friction ∝ Normal force: \( f \propto N \)
  2. Independent of contact area
  3. Static friction > Kinetic friction

Limiting friction: \( f_\text{max} = \mu N \)

Angle of Friction (\(\theta\)): \(\tan \theta = \mu\)

Angle of Repose (\(\alpha\)): \(\tan \alpha = \mu\)

Lubrication: Reduces friction using oil or grease

Applications: Vehicles, machinery, ramps

8. Dynamics of Uniform Circular Motion

Centripetal Force: Force towards center keeping body in circular motion:
\[ F_c = \frac{mv^2}{r} \]

Centripetal acceleration:
\[ a_c = \frac{v^2}{r} \]

Examples:

  • Vehicle on a level circular road: friction provides centripetal force
  • Vehicle on a banked road (frictionless): \(\tan \theta = \frac{v^2}{rg}\)

Applications: Banking of roads, roller coasters, satellites, planets

9. Pulley System (Two-Mass)

Two masses \( m_1 \) and \( m_2 \) connected by a light string over frictionless pulley (\( m_1 > m_2 \))

Forces:

  • \( m_1 \) downward: \( m_1 g - T = m_1 a \)
  • \( m_2 \) upward: \( T - m_2 g = m_2 a \)

Acceleration: \( a = \frac{(m_1 - m_2) g}{m_1 + m_2} \)

Tension: \( T = \frac{2 m_1 m_2 g}{m_1 + m_2} \)

Applications: Lifting weights, cranes, elevators

10. Important Formulas

ConceptFormula
Momentum\(\vec{p} = m \vec{v}\)
Force\(\vec{F} = m\vec{a}\)
Impulse\(\vec{J} = \vec{F} \Delta t = \Delta \vec{p}\)
Friction\(f \leq \mu N\)
Limiting friction\(f_\text{max} = \mu N\)
Circular motion\(F_c = \frac{mv^2}{r}, a_c = \frac{v^2}{r}\)
Banked road\(\tan \theta = \frac{v^2}{rg}\)
Pulley acceleration\(a = \frac{(m_1 - m_2) g}{m_1 + m_2}\)
Pulley tension\(T = \frac{2 m_1 m_2 g}{m_1 + m_2}\)
Weight\(W = mg\)
Normal reaction (inclined plane)\(N = W \cos \theta\)

Unit IV: Work, Energy, and Power – Class 11 Physics Notes

1. Work Done by a Force

Definition: Work is done when a force displaces a body in the direction of the force.

Formula (constant force):
\[ W = \vec{F} \cdot \vec{d} = Fd \cos \theta \]

\(\theta\) = angle between force and displacement. Work is positive if along displacement, negative if opposite.

Variable force:
\[ W = \int_{x_1}^{x_2} F(x) \, dx \]

Unit: Joule (1 J = 1 N·m)

Example: Pushing a block, lifting a weight, stretching a spring.

2. Kinetic Energy and Work-Energy Theorem

Kinetic Energy (KE):
\[ KE = \frac{1}{2} m v^2 \]

Work-Energy Theorem: Work done by net force = change in kinetic energy:
\[ W_\text{net} = KE_f - KE_i = \Delta KE \]

Example: A car accelerating or braking.

3. Power

Definition: Rate of doing work:
\[ P = \frac{dW}{dt} \]

Average Power:
\[ P_\text{avg} = \frac{W}{\Delta t} \]

Unit: Watt (1 W = 1 J/s)

Example: A motor lifting a weight, running engine power.

4. Potential Energy (PE)

Gravitational PE:
\[ PE = m g h \]

Elastic potential energy of a spring:
\[ PE_\text{spring} = \frac{1}{2} k x^2 \]

\(k\) = spring constant, \(x\) = displacement from equilibrium

5. Conservative and Non-Conservative Forces

5.1 Conservative Forces

Work done is independent of path:
\[ \oint \vec{F} \cdot d\vec{r} = 0 \]

Can be expressed as:
\[ \vec{F} = -\nabla U \]

Examples: Gravity, spring force, electrostatic force

Mechanical Energy Conservation:
\[ KE + PE = \text{constant} \]

5.2 Non-Conservative Forces

Work depends on the path; energy dissipates as heat or sound.

Examples: Friction, air resistance, viscous forces

Work-energy relation:
\[ W_\text{nc} = \Delta KE + \Delta PE \]

Applications: Braking vehicles, sliding blocks, dissipative systems

6. Motion in a Vertical Circle

Particle of mass \(m\) in vertical circle of radius \(r\):

  • Minimum speed at top: \(v_\text{top} = \sqrt{r g}\)
  • Tension at top:
    \[ T + mg = \frac{mv^2}{r} \]

Applications: Pendulum, roller coaster loops, satellites.

7. Collisions

7.1 Elastic Collision

Both momentum and kinetic energy conserved.

1D Elastic Collision Formulas:
\[ v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2 m_2}{m_1 + m_2} v_2 \]
\[ v_2' = \frac{2 m_1}{m_1 + m_2} v_1 - \frac{m_1 - m_2}{m_1 + m_2} v_2 \]

7.2 Inelastic Collision

Momentum conserved, kinetic energy not conserved.

2D collision momentum conservation along axes:
\[ m_1 \vec{v}_1 + m_2 \vec{v}_2 = m_1 \vec{v}_1' + m_2 \vec{v}_2' \]

8. Important Formulas

ConceptFormula
Work (constant force)\( W = F d \cos \theta \)
Work (variable force)\( W = \int F(x) dx \)
Kinetic Energy\( KE = \frac{1}{2} m v^2 \)
Work-Energy Theorem\( W_\text{net} = \Delta KE \)
Power\( P = \frac{dW}{dt}, \quad P_\text{avg} = \frac{W}{\Delta t} \)
Gravitational PE\( PE = m g h \)
Elastic PE\( PE_\text{spring} = \frac{1}{2} k x^2 \)
Mechanical energy conservation\( KE + PE = \text{constant} \)
Circular motion (tension at top)\( T + mg = \frac{mv^2}{r} \)
1D Elastic collision\( v_1', v_2' \) formulas above
Momentum (2D collision)\( m_1 \vec{v}_1 + m_2 \vec{v}_2 = m_1 \vec{v}_1' + m_2 \vec{v}_2' \)

Chapter 6: System of Particles and Rotational Motion

1. Centre of Mass of a Two-Particle System

Definition: The point where the total mass of the system is considered concentrated for translational motion.

\( \vec{r_{cm}} = \frac{m_1 \vec{r_1} + m_2 \vec{r_2}}{m_1 + m_2} \)
\( x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \) (1D)
  • Equal masses: COM at midpoint.
  • COM closer to heavier mass.
  • Exam Tip: Used in collision and external force problems.

2. Momentum Conservation and Centre of Mass Motion

\( \vec{P} = m_1 \vec{v_1} + m_2 \vec{v_2} = M \vec{v_{cm}} \), where \( M = m_1 + m_2 \)
\( M \vec{a_{cm}} = \vec{F_{ext}} \)

Conservation: If no external force, \( \vec{P}_{initial} = \vec{P}_{final} \).

  • Isolated system: COM velocity constant.
  • Applications: Rocket propulsion, explosions (COM at rest if initial momentum zero).
  • Exam Tip: Solve collision velocities in COM frame.

3. Centre of Mass of a Rigid Body

\( \vec{r_{cm}} = \frac{1}{M} \int \vec{r} \, dm \)

Uniform Rod: COM at midpoint,

\( x_{cm} = \frac{L}{2} \)

  • Symmetric bodies: COM at centre of symmetry.
  • Exam Tip: Use integration for non-uniform density; symmetry for uniform bodies.

4. Moment of a Force and Torque

\( \vec{\tau} = \vec{r} \times \vec{F} \), magnitude \( \tau = r F \sin \theta \)

Unit: N·m.

  • Causes rotational motion.
  • Direction: Right-hand rule.
  • Exam Tip: Zero torque if force through axis or parallel to \( \vec{r} \).

5. Angular Momentum

\( \vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v} \), magnitude \( L = m v r \sin \theta \)
\( \vec{L} = \sum \vec{L_i} \) (System)
\( \vec{\tau} = \frac{d\vec{L}}{dt} \)
  • Unit: kg·m²/s.
  • Exam Tip: Use perpendicular components in calculations.

6. Law of Conservation of Angular Momentum

Law: If no external torque,

\( \vec{L}_{initial} = \vec{L}_{final} \)

Applications:

  • Ice skater: Pulling arms in increases \( \omega \).
  • Diving: Tucking body increases rotation speed.
  • Planetary motion, boomerang.
  • Exam Tip: Use
    \( I_1 \omega_1 = I_2 \omega_2 \)

7. Equilibrium of Rigid Bodies

\( \sum \vec{F} = 0 \)
\( \sum \vec{\tau} = 0 \)

Types: Stable, unstable, neutral.

  • Exam Tip: Solve ladder problems by balancing torques about pivot.

8. Rigid Body Rotation and Equations of Rotational Motion

Angular Quantities: Displacement \( \theta \), velocity

\( \omega = \frac{d\theta}{dt} \)
, acceleration
\( \alpha = \frac{d\omega}{dt} \)

Equations (Constant \( \alpha \)):

  • \( \omega = \omega_0 + \alpha t \)
  • \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \)
  • \( \omega^2 = \omega_0^2 + 2 \alpha \theta \)
\( \tau = I \alpha \)
  • Exam Tip: Analogous to \( F = ma \).

9. Comparison of Linear and Rotational Motions

LinearRotational
Displacement: \( x \)Angular displacement: \( \theta \)
Velocity:
\( v = \frac{dx}{dt} \)
Angular velocity:
\( \omega = \frac{d\theta}{dt} \)
Acceleration:
\( a = \frac{dv}{dt} \)
Angular acceleration:
\( \alpha = \frac{d\omega}{dt} \)
Mass: \( m \)Moment of inertia: \( I \)
Force:
\( F = ma \)
Torque:
\( \tau = I \alpha \)
Momentum:
\( p = mv \)
Angular momentum:
\( L = I \omega \)
Kinetic Energy:
\( \frac{1}{2} m v^2 \)
Rotational KE:
\( \frac{1}{2} I \omega^2 \)
Work:
\( F \cdot s \)
Work:
\( \tau \cdot \theta \)
  • Exam Tip: Use analogies for problem-solving.

10. Moment of Inertia and Radius of Gyration

\( I = \sum m_i r_i^2 \) or \( I = \int r^2 dm \)
\( k = \sqrt{\frac{I}{M}} \), where \( I = M k^2 \)

Values for Objects:

  • Thin rod (centre, perpendicular):
    \( I = \frac{1}{12} M L^2 \)
  • Thin rod (end):
    \( I = \frac{1}{3} M L^2 \)
  • Rectangular lamina (centre):
    \( I = \frac{1}{12} M (a^2 + b^2) \)
  • Thin ring (diameter):
    \( I = \frac{1}{2} M R^2 \)
  • Thin ring (axis):
    \( I = M R^2 \)
  • Disc (diameter):
    \( I = \frac{1}{4} M R^2 \)
  • Disc (axis):
    \( I = \frac{1}{2} M R^2 \)
  • Solid sphere (diameter):
    \( I = \frac{2}{5} M R^2 \)
  • Hollow sphere (diameter):
    \( I = \frac{2}{3} M R^2 \)
  • Theorems: Parallel axis:
    \( I = I_{cm} + M d^2 \)
    ; Perpendicular axis:
    \( I_z = I_x + I_y \)
    .
  • Exam Tip: Choose axis carefully; apply theorems to shift axes.

Summary / Quick Revision Box

  • COM: Weighted average position;
    \( M \vec{a_{cm}} = \vec{F_{ext}} \)
    .
  • Momentum: Conserved if no external force.
  • Torque:
    \( \vec{\tau} = \vec{r} \times \vec{F} = I \alpha \)
    .
  • Angular Momentum:
    \( \vec{L} = I \omega \)
    , conserved if no external torque.
  • Equilibrium:
    \( \sum \vec{F} = 0 \)
    ,
    \( \sum \vec{\tau} = 0 \)
    .
  • Rotational Equations: Analogous to linear; use for constant \( \alpha \).
  • Linear vs Rotational: Analogous quantities (e.g., \( m \leftrightarrow I \)).
  • Moment of Inertia: Depends on mass distribution; use standard values.
  • Radius of Gyration:
    \( k = \sqrt{\frac{I}{M}} \)
    .

Chapter 7: Gravitation

1. Kepler’s Laws of Planetary Motion

First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus.

Second Law (Law of Equal Areas): The line joining a planet to the Sun sweeps equal areas in equal time intervals.

\( \frac{dA}{dt} = \frac{L}{2m} \) (constant, where \( L \) is angular momentum, \( m \) is planet’s mass)

Third Law (Law of Periods): The square of the orbital period is proportional to the cube of the semi-major axis.

\( T^2 \propto r^3 \) or \( T^2 = \frac{4\pi^2}{GM} r^3 \)
  • Exam Tip: Use third law to find orbital period or radius; applies to satellites too.

2. Universal Law of Gravitation

Newton’s Law: Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

\( F = \frac{G m_1 m_2}{r^2} \)

G: Gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2 \).

  • Vector form:
    \( \vec{F} = -\frac{G m_1 m_2}{r^2} \hat{r} \)
  • Exam Tip: Use for calculating force between masses; direction is along the line joining them.

3. Acceleration Due to Gravity

Definition: Acceleration of an object due to Earth’s gravitational pull, \( g = 9.8 \, \text{m/s}^2 \) (surface value).

\( g = \frac{G M}{R^2} \), where \( M \) is Earth’s mass, \( R \) is Earth’s radius.

Variation with Altitude (h):

\( g_h = \frac{G M}{(R + h)^2} = g \left( \frac{R}{R + h} \right)^2 \approx g \left( 1 - \frac{2h}{R} \right) \) (for \( h \ll R \))

Variation with Depth (d):

\( g_d = g \left( 1 - \frac{d}{R} \right) \)
  • At Earth’s centre: \( g_d = 0 \) (d = R).
  • Exam Tip: Use approximate formulas for small \( h \) or \( d \); note \( g \) decreases with altitude and depth.

4. Gravitational Potential Energy

Definition: Energy due to the position of a mass in a gravitational field.

\( U = -\frac{G M m}{r} \)
  • Negative sign indicates bound system; zero at infinity.
  • Exam Tip: Use to calculate work done in moving masses; change in \( U = -\text{Work} \).

5. Gravitational Potential

Definition: Potential energy per unit mass at a point in the gravitational field.

\( V = -\frac{G M}{r} \)
  • Unit: J/kg.
  • Relation: \( U = m V \).
  • Exam Tip: Use to find potential at a point; useful for energy conservation problems.

6. Escape Speed

Definition: Minimum speed required for an object to escape Earth’s gravitational field.

\( v_e = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R} \)
  • For Earth: \( v_e \approx 11.2 \, \text{km/s} \).
  • Exam Tip: Independent of object’s mass; use conservation of energy to derive.

7. Orbital Velocity of a Satellite

Definition: Speed required for a satellite to maintain a circular orbit.

\( v_o = \sqrt{\frac{G M}{r}} = \sqrt{\frac{g R^2}{r}} \), where \( r = R + h \).
  • For near-Earth orbit: \( v_o \approx 7.9 \, \text{km/s} \).
  • Exam Tip: Derived from balancing gravitational force and centripetal force.

8. Energy of an Orbiting Satellite

Kinetic Energy:

\( KE = \frac{1}{2} m v_o^2 = \frac{G M m}{2 r} \)

Potential Energy:

\( U = -\frac{G M m}{r} \)

Total Energy:

\( E = KE + U = -\frac{G M m}{2 r} \)
  • Negative total energy indicates bound orbit.
  • Exam Tip: Use total energy to compare orbits; smaller \( r \) means lower (more negative) energy.

Chapter 8: Mechanical Properties of Solids

1. Elasticity

Definition: Property of a material to regain its original shape and size after removal of deforming force.

  • Elastic Limit: Maximum stress a material can withstand without permanent deformation.
  • Types: Perfectly elastic (returns to original shape completely, e.g., quartz), partially elastic (partial recovery, e.g., rubber).
  • Exam Tip: Elasticity is key for materials in springs, bridges, and structures.

2. Stress-Strain Relationship

Stress: Internal force per unit area resisting deformation.

\( \text{Stress} = \frac{F}{A} \)

Types of Stress:

  • Normal Stress: Perpendicular to surface (tensile or compressive).
  • Shear Stress: Parallel to surface.

Strain: Relative deformation due to stress.

\( \text{Strain} = \frac{\text{Change in dimension}}{\text{Original dimension}} \)

Types of Strain:

  • Longitudinal Strain:
    \( \frac{\Delta L}{L} \)
  • Shear Strain: Angular deformation (qualitative).
  • Volumetric Strain:
    \( \frac{\Delta V}{V} \)
  • Stress-Strain Curve: Shows elastic region (linear), yield point, and plastic region.
  • Exam Tip: Linear portion obeys Hooke’s law; beyond yield point, permanent deformation occurs.

3. Hooke’s Law

Statement: Within elastic limit, stress is directly proportional to strain.

\( \text{Stress} = k \cdot \text{Strain} \)
  • \( k \): Modulus of elasticity (depends on type of stress/strain).
  • Exam Tip: Applies only in elastic region; used to calculate moduli.

4. Young’s Modulus

Definition: Measure of stiffness under tensile or compressive stress.

\( Y = \frac{\text{Tensile stress}}{\text{Longitudinal strain}} = \frac{F / A}{\Delta L / L} = \frac{F L}{A \Delta L} \)
  • Unit: N/m² or Pa.
  • Higher \( Y \): Stiffer material (e.g., steel > rubber).
  • Exam Tip: Use for problems involving stretching/compressing wires or rods.

5. Bulk Modulus

Definition: Measure of resistance to uniform compression (volume change).

\( K = \frac{\text{Normal stress}}{\text{Volumetric strain}} = \frac{P}{\Delta V / V} = -\frac{P V}{\Delta V} \)
  • Unit: N/m² or Pa.
  • Negative sign: Volume decreases with pressure.
  • Exam Tip: Liquids have high \( K \); gases have low \( K \).

6. Shear Modulus of Rigidity (Qualitative Idea)

Definition: Measure of resistance to shear deformation.

\( G = \frac{\text{Shear stress}}{\text{Shear strain}} \)
  • Unit: N/m² or Pa.
  • Qualitative: Describes how materials resist shape change without volume change (e.g., twisting of a shaft).
  • Exam Tip: Focus on concept; numerical problems less common.

7. Poisson’s Ratio

Definition: Ratio of transverse strain to longitudinal strain under tensile stress.

\( \sigma = -\frac{\text{Lateral strain}}{\text{Longitudinal strain}} = -\frac{\Delta D / D}{\Delta L / L} \)
  • Negative sign: Lateral contraction with longitudinal extension.
  • Range: Typically 0 to 0.5 for most materials.
  • Exam Tip: Understand concept; rarely used in numericals but important for material properties.

8. Elastic Energy

Definition: Potential energy stored in a deformed elastic body.

\( U = \frac{1}{2} \text{Stress} \times \text{Strain} \times \text{Volume} \)
For Young’s modulus: \( U = \frac{1}{2} Y (\text{Strain})^2 \times \text{Volume} \)
  • Analogous to spring energy:
    \( U = \frac{1}{2} k x^2 \)
    .
  • Exam Tip: Calculate for stretched wires or compressed materials; use in energy conservation problems.

9. Application of Elastic Behavior of Materials (Qualitative Idea)

  • Construction: High Young’s modulus materials (e.g., steel) used in bridges, buildings for rigidity.
  • Springs: Elastic energy storage for mechanical devices.
  • Shock Absorbers: Materials with controlled elasticity to absorb vibrations.
  • Medical: Elastic properties in prosthetics or implants to mimic tissues.
  • Exam Tip: Focus on examples like beams bending or wires stretching; relate to moduli.

Summary / Quick Revision Box

  • Elasticity: Ability to regain shape; elastic limit key for deformation.
  • Stress:
    \( \frac{F}{A} \)
    , types: normal, shear.
  • Strain:
    \( \frac{\text{Change}}{\text{Original}} \)
    , types: longitudinal, shear, volumetric.
  • Hooke’s Law:
    \( \text{Stress} \propto \text{Strain} \)
    .
  • Young’s Modulus:
    \( Y = \frac{F L}{A \Delta L} \)
    .
  • Bulk Modulus:
    \( K = -\frac{P V}{\Delta V} \)
    .\( G = \frac{\text{Shear stress}}{\text{Shear strain}} \)