Tatva Practice Book: Physics Class 11 JEE Summary Notes

PHYSICAL QUANTITIES AND UNITS

• Physical Quantity: Any quantity that can be measured is classified as a physical quantity. Examples include length, weight, and time.
• Fundamental Physical Quantities: These are quantities that are independent of others. The seven base SI units are:
  • Length: Meter ($m$)
  • Mass: Kilogram ($kg$)
  • Time: Second ($s$)
  • Electric Current: Ampere ($A$)
  • Temperature: Kelvin ($K$)
  • Amount of Substance: Mole ($mol$)
  • Luminous Intensity: Candela ($Cd$)
• Derived Physical Quantities: These depend on fundamental quantities. Examples include:
  • Acceleration: $\frac{\text{length}}{\text{time}^2}$
  • Density: $\frac{\text{mass}}{\text{length}^3}$
  • Volume: $\text{length}^3$
  • Force: $\frac{\text{mass} \times \text{length}}{\text{time}^2}$
  • Momentum: $\frac{\text{mass} \times \text{length}}{\text{time}}$
  • Pressure: $\frac{\text{mass}}{\text{length} \times \text{time}^2}$
• Measurement and Units: Measurement is expressed as a magnitude multiplied by a unit ($Q = nu$). If a quantity is measured in two different units, the relationship is $n_1u_1 = n_2u_2$.
• Supplementary Units:
  • Plane Angle: Radian ($rad$); defined as $\theta = \frac{s}{r}$.
  • Solid Angle: Steradian ($sr$); defined as $\Omega = \frac{A}{r^2}$.

DIMENSIONAL ANALYSIS

• Definition: Dimensions are the powers to which fundamental units must be raised to represent a physical quantity.
• Behavior in Mathematical Formulae:
  • Rule 1: Terms that are added or subtracted must have the same dimensions.
  • Rule 2: Dimensions obey the rules of algebraic multiplication and division.
• Applications:
  • Unit Conversion: Using the formula $n_1[M_1^a L_1^b T_1^c] = n_2[M_2^a L_2^b T_2^c]$. For example, converting $1\,J$ to erg yields $1\,J = 10^7\,erg$.
  • Checking Consistency: The Principle of Homogeneity states that dimensions on the left and right sides of an equation must match (e.g., in $s = ut + \frac{1}{2}at^2$, all terms have dimension $[L]$).
  • Deducing Relations: Finding relationships if the dependencies are known (e.g., the time period of a simple pendulum).
• Limitations:
  • Cannot determine dimensionless constants of proportionality.
  • Not applicable to trigonometric, logarithmic, or exponential functions.
  • Difficult when a quantity depends on more than three variables.
  • Ineffective if equations involve the addition or subtraction of distinct physical quantities.

SIGNIFICANT FIGURES AND ERROR ANALYSIS

• Significant Figures: Digits known reliably plus one uncertain digit.
  • Rule 1: Every non-zero digit is significant.
  • Rule 2: Zeros between non-zeros are significant.
  • Rule 3: Leftmost zeros are not significant.
  • Rule 4: Trailing zeros are significant only if they are to the right of a decimal point.
• Rounding Off Rules:
  • If the digit to be dropped is $< 5$, drop it.
  • If the digit to be dropped is $> 5$, increase the preceding digit by 1.
  • If the digit is exactly 5: if the preceding digit is even, drop it; if odd, increase the preceding digit by 1.
• Types of Errors:
  • Random Errors: Unpredictable; minimized by taking the mean of multiple readings.
  • Systematic Errors: Consistent shifts due to faulty instruments (calibration), faulty technique (parallax), or personal habits.
• Error Calculation:
  • Absolute Error: $|a_{mean} - a_i|$, always positive.
  • Relative Error: $\frac{\text{Mean Absolute Error}}{\text{Mean Value}}$.
  • Percentage Error: $\text{Relative Error} \times 100\%$.
• Propagation of Errors:
  • Sum/Difference ($Z = A \pm B$): $\Delta Z = \Delta A + \Delta B$.
  • Product/Division ($Z = AB$ or $A/B$): $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$.
  • Power ($Z = A^k$): $\frac{\Delta Z}{Z} = k \frac{\Delta A}{A}$.

BASIC MATHEMATICS FOR PHYSICS

• Quadratic Equations: Standard form $ax^2 + bx + c = 0$.
  • Discriminant ($D$): $D = b^2 - 4ac$.
    • $D > 0$: Two distinct real roots.
    • $D = 0$: Real and equal roots.
    • $D < 0$: No real roots (imaginary).
  • Root Formulas: $\text{Sum} = -\frac{b}{a}$, $\text{Product} = \frac{c}{a}$.
• Binomial Expansion: $(1 + x)^n \approx 1 + nx$ if $x \ll 1$.
• Graphs:
  • Straight Line: $y = mx + c$; slope $m = \tan(\theta)$.
  • Parabola: $y^2 = kx$ or $x^2 = ky$.
  • Hyperbola: $xy = \text{constant}$.
  • Circle: $x^2 + y^2 = a^2$.
  • Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
  • Trigonometric Graphs: Sine and Cosine have an amplitude of 1 and period of $2\pi$. Tangent has period $\pi$ and undefined amplitude.

VECTORS

• Scalars vs. Vectors: Scalars have magnitude only; Vectors have magnitude and direction and follow vector addition laws.
• Unit Vector: $|\hat{x}| = 1$. Calculated as $\hat{x} = \frac{\vec{x}}{|\vec{x}|}$.
• Vector Addition Laws:
  • Parallelogram/Triangle Laws: $R = \sqrt{A^2 + B^2 + 2AB\cos(\theta)}$.
  • Direction: $\tan(\alpha) = \frac{B\sin(\theta)}{A + B\cos(\theta)}$.
• Dot Product (Scalar Product): $\vec{a} \cdot \vec{b} = ab\cos(\theta)$.
  • Distributive and Commutative.
  • Applications: Work ($W = \vec{F} \cdot \vec{s}$), Power ($P = \vec{F} \cdot \vec{v}$).
• Cross Product (Vector Product): $\vec{a} \times \vec{b} = ab\sin(\theta) \hat{n}$.
  • Anti-commutative ($\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$).
  • Direction determined by Right Hand Thumb Rule.
  • Applications: Torque ($\vec{\tau} = \vec{r} \times \vec{F}$), Angular Momentum ($\vec{L} = \vec{r} \times \vec{p}$).

MOTION IN A STRAIGHT LINE

• Mechanics Branches: Statics (rest), Kinematics (motion without causes), Dynamics (motion with causes).
• Rest and Motion: These are relative to the observer.
• Kinematic Definitions:
  • Distance: Actual path length (scalar).
  • Displacement: Shortest distance between initial and final points (vector).
  • Average Velocity: $\frac{\Delta x}{\Delta t}$.
  • Average Speed: $\frac{\text{Total path}}{\text{Total time}}$.
  • Instantaneous Velocity: $v = \frac{dx}{dt}$.
  • Acceleration: Average is $\frac{\Delta v}{\Delta t}$, Instantaneous is $a = \frac{dv}{dt}$.
• Equations of Uniform Acceleration:
  • $v = u + at$
  • $s = ut + \frac{1}{2}at^2$
  • $v^2 = u^2 + 2as$
  • $S_{nth\,second} = u + \frac{a}{2}(2n - 1)$
• Vertical Motion under Gravity:
  • Free fall ($u = 0$): $v = gt$, $h = \frac{1}{2}gt^2$, $v = \sqrt{2gh}$.
  • Thrown up: Replace $g$ with $-g$. Time to max height $T = \frac{u}{g}$. Max height $h = \frac{u^2}{2g}$.
• Graphs:
  • Slope of $x-t$ graph equals velocity.
  • Slope of $v-t$ graph equals acceleration.
  • Area under $v-t$ graph equals displacement.
  • Area under $a-t$ graph equals change in velocity.

MOTION IN A PLANE AND RELATIVE MOTION

• Projectile Motion Parameters:
  • Time of Flight ($T$): $\frac{2u\sin(\theta)}{g}$.
  • Horizontal Range ($R$): $\frac{u^2\sin(2\theta)}{g}$. Maximum range at $\theta = 45^\circ$.
  • Maximum Height ($H$): $\frac{u^2\sin^2(\theta)}{2g}$.
  • Equation of Trajectory: $y = x\tan(\theta) - \frac{gx^2}{2u^2\cos^2(\theta)}$ or $y = x\tan(\theta) \left[ 1 - \frac{x}{R} \right]$.
• Projectile from a Height: Horizontal velocity $u$ is constant; Vertical motion is free fall ($u_y = 0$). Time of flight $T = \sqrt{\frac{2h}{g}}$. Range $R = u\sqrt{\frac{2h}{g}}$.
• Relative Motion:
  • Relative velocity of B with respect to A: $\vec{v}_{BA} = \vec{v}_B - \vec{v}_A$.
• River-Boat Problems:
  • Shortest Time: Crossed at $\theta = 0^\circ$ (stearing perpendicular to current). Time $t = \frac{d}{v_{br}}$.
  • Shortest Path (Zero Drift): Crossed at angle $\sin(\theta) = \frac{v_r}{v_{br}}$ upstream from the bank to reach directly opposite point.
• Rain-Man Problems: To protect from rain, hold umbrella at $\tan(\theta) = \frac{v_{man}}{v_{rain}}$ where $\theta$ is relative to vertical.

LAWS OF MOTION AND FRICTION

• Types of Forces: Weight ($mg$), Contact (Normal and Friction), Tension, and Spring Force ($F = -kx$).
• Newton’s Laws:
  1. First Law (Inertia): Objects remain at rest or in uniform motion unless acted upon by a net force.
  2. Second Law: $\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a}$.
  3. Third Law: Action and reaction are equal and opposite, acting on different bodies ($\vec{F}_{AB} = -\vec{F}_{BA}$).
• Free Body Diagram (FBD): Isolating an object to show all external forces acting on it.
• Pseudo Force: Essential in non-inertial (accelerated) frames, applied in the direction opposite to frame acceleration ($\vec{F}_{pseudo} = -m\vec{a}_{frame}$).
• Apparent Weight in Lift:
  • Accelerating up: $R = m(g + a)$.
  • Accelerating down: $R = m(g - a)$.
  • Free fall: $R = 0$.
• Friction:
  • Static Friction ($f_s$): Adjusts to applied force up to $\text{limiting friction } f_{max} = \mu_s N$.
  • Kinetic Friction ($f_k$): Opposes actual motion, $f_k = \mu_k N$. Note $\mu_k < \mu_s$.
  • Rolling Friction: Opposes a wheel rolling; usually much smaller than kinetic friction.
  • Angle of friction ($\theta$): Resultant of friction and normal forces; $\mu = \tan(\theta)$.
  • Angle of repose ($\alpha$): Maximum inclination for an object to remain at rest; $\alpha = \theta$.