Physics Half Yearly Revision - Class 11 JKBOSE Physics

Units, Measurements, and Core Mechanics

Physical Quantities and Their Types

Physical quantities are those which can be measured and expressed numerically with a unit. They are the foundation of physics, as all measurable aspects of nature are described using physical quantities. Non-physical quantities, such as emotions, cannot be measured and are not studied in physics.

• Examples of Physical Quantities: length, mass, time, area, volume, force, work, energy, kinetic energy, potential energy.
• Non-Physical Quantities: love, hate, emotions (cannot be measured).
• Physics: The study of physical quantities.

Types of Physical Quantities:

• Fundamental (Base) Quantities: Seven in…

Units, Measurements, and Core Mechanics

Physical Quantities and Their Types

Physical quantities are those which can be measured and expressed numerically with a unit. They are the foundation of physics, as all measurable aspects of nature are described using physical quantities. Non-physical quantities, such as emotions, cannot be measured and are not studied in physics.

• Examples of Physical Quantities: length, mass, time, area, volume, force, work, energy, kinetic energy, potential energy.
• Non-Physical Quantities: love, hate, emotions (cannot be measured).
• Physics: The study of physical quantities.

Types of Physical Quantities:

• Fundamental (Base) Quantities: Seven in number—mass, length, time, temperature, electric current, amount of substance, luminous intensity.
• Derived Quantities: Obtained from base quantities (e.g., speed, velocity, acceleration, force, area, volume, density, work, power, pressure).
• Supplementary Quantities: Angle and solid angle; have units but no dimensions.

Units and Systems of Units

A unit is a fixed quantity used as a standard of measurement for physical quantities. The choice of unit must be universal to avoid confusion.

• Examples: Length (meter), mass (kilogram), time (second).
• Historical Units: Different regions used different units (e.g., hand-span, foot), leading to inconsistency.
• Standardization: The SI (International System of Units) was adopted for universal consistency.

Major Systems of Units:

• FPS: Foot, Pound, Second.
• CGS: Centimeter, Gram, Second.
• MKS (SI): Meter, Kilogram, Second (most widely used).

Dimensions and Dimensional Analysis

Dimensions are the powers to which the base quantities are raised in order to represent a physical quantity. Dimensional analysis is used to check the correctness of equations, derive formulas, and convert units.

• Base Quantity Symbols: Mass ($M$), Length ($L$), Time ($T$), Temperature ($K$), Current ($A$), Luminous Intensity ($cd$), Amount of Substance ($mol$).
• Dimensional Formula: Expresses a derived quantity in terms of base quantities (e.g., speed: $[L^1 T^{-1}]$).
• Dimensional Equation: The equation showing the dimensional formula.

Examples:

• Speed: $[L^1 T^{-1}]$
• Acceleration: $[L^1 T^{-2}]$
• Force: $[M^1 L^1 T^{-2}]$
• Work/Energy/Torque: $[M^1 L^2 T^{-2}]$
• Pressure: $[M^1 L^{-1} T^{-2}]$

Applications and Limitations of Dimensional Analysis

Dimensional analysis is applied to verify equations, derive relationships, and convert units between systems. However, it cannot determine dimensionless constants or handle equations involving trigonometric or exponential functions.

Applications:

• Checking Correctness: Law of homogeneity—only quantities with the same dimensions can be added, subtracted, or equated.
• Deriving Formulas: By expressing dependence on base quantities and comparing dimensions.
• Unit Conversion: Using the formula:$$n_2 = n_1 \left(\frac{m_1}{m_2}\right)^a \left(\frac{l_1}{l_2}\right)^b \left(\frac{t_1}{t_2}\right)^c$$where $a, b, c$ are the powers from the dimensional formula.

Limitations:

• Cannot determine numerical constants (e.g., $2\pi$ in pendulum formula).
• Cannot handle equations with addition/subtraction of unlike dimensions.
• Not unique: Different quantities can have the same dimensional formula (e.g., work, energy, torque).

Significant Figures

Significant figures indicate the number of important digits in a measured or calculated quantity. They reflect the precision of measurement.

• Rules:
  • All non-zero digits are significant.
  • Zeros between non-zero digits are significant.
  • Leading zeros are not significant.
  • Trailing zeros are significant only if there is a decimal point.
  • In scientific notation, only the digits in the coefficient are significant.

Examples:

• 42345: 5 significant figures.
• 42005: 5 significant figures.
• 42000: 2 significant figures (if no decimal).
• 420.00: 5 significant figures.
• 0.0042: 2 significant figures.

Motion in a Straight Line: Equations and Graphical Method

Motion in a straight line is described using three equations of motion, which can be derived graphically using the velocity-time graph.

• First Equation: $v = u + at$
• Second Equation: $s = ut + \frac{1}{2}at^2$
• Third Equation: $v^2 = u^2 + 2as$

Method:

• Draw a $v$-$t$ graph with initial velocity $u$, final velocity $v$, and time $t$.
• Area under the graph gives displacement.
• Use the equations to relate $v$, $u$, $a$, $t$, and $s$.

Vectors and Their Operations

Vectors have both magnitude and direction. Vector addition follows the parallelogram law, and vectors can be resolved into rectangular components.

• Parallelogram Law: If two vectors $A$ and $B$ are adjacent sides of a parallelogram, the diagonal represents the resultant $R$.$$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$$$$\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$$
• Resolution: Any vector $R$ at angle $\theta$ can be resolved as $R_x = R\cos\theta$, $R_y = R\sin\theta$.

Vector Multiplication:

• Dot Product: $A \cdot B = AB\cos\theta$ (scalar, commutative).
• Cross Product: $A \times B = AB\sin\theta\,\hat{n}$ (vector, anti-commutative).

Projectile Motion

Projectile motion is the motion of an object thrown into the air under gravity. The path is parabolic.

• Time of Flight: $T = \frac{2u\sin\theta}{g}$
• Maximum Height: $H = \frac{u^2\sin^2\theta}{2g}$
• Range: $R = \frac{u^2\sin2\theta}{g}$
• Equation of Trajectory: $y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$

Laws of Motion and Friction

Newton’s laws describe the relationship between force and motion. Friction opposes relative motion and depends on the nature of surfaces and normal reaction.

• Newton’s Second Law: Force is the rate of change of momentum; $F = ma$.
• Impulse: Force acting for a short time; $Impulse = F \times \Delta t = \Delta p$.
• Friction: $f = \mu N$, where $\mu$ is the coefficient of friction.
• Types of Friction: Static (at rest), limiting (maximum static), kinetic (in motion), rolling (for rolling bodies).
• Reducing Friction: Use of lubricants, polishing, or powder.

Circular Motion and Banking of Roads

Circular motion requires a centripetal force directed towards the center. Banking of roads prevents vehicles from skidding at turns.

• Centripetal Acceleration: $a_c = \frac{v^2}{r}$
• Centripetal Force: $F_c = m\frac{v^2}{r}$
• Banking Formula:$$\frac{v^2}{rg} = \frac{\tan\theta + \mu}{1 - \mu\tan\theta}$$$$v = \sqrt{rg\frac{\tan\theta + \mu}{1 - \mu\tan\theta}}$$

Work, Energy, Power, and Collisions

Work is the product of force and displacement. The work-energy theorem relates work done to the change in kinetic energy. Collisions are classified as elastic or inelastic.

• Work-Energy Theorem: $W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$
• Elastic Collision: Both momentum and kinetic energy are conserved.
• Inelastic Collision: Only momentum is conserved.
• Final Velocity (Inelastic): $v = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$

Rotational Motion and Moment of Inertia

Rotational motion involves quantities analogous to linear motion. Moment of inertia measures resistance to rotational acceleration.

• Torque: $\tau = r \times F = rF\sin\theta$; unit: Nm; dimension: $[ML^2T^{-2}]$
• Angular Momentum: $L = r \times p$; unit: $kg\,m^2/s$; dimension: $[ML^2T^{-1}]$
• Moment of Inertia: $I = mr^2$; unit: $kg\,m^2$; dimension: $[ML^2]$
• Conservation of Angular Momentum: If external torque is zero, initial angular momentum equals final angular momentum; $I_1\omega_1 = I_2\omega_2$.

Gravitation and Kepler’s Laws

Every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Universal Law of Gravitation: $F = G\frac{m_1m_2}{r^2}$
• Gravitational Constant: $G = 6.67 \times 10^{-11}$ (unit: $Nm^2/kg^2$)
• Kepler’s Laws:
  • Law of Orbits: Planets move in ellipses with the sun at one focus.
  • Law of Areas: Equal areas are swept in equal times.
  • Law of Periods: Square of period is proportional to cube of semi-major axis.

Unit Conversion Example:

• $1\,\text{Newton} = 10^5\,\text{dyne}$
• $1\,\text{Joule} = 10^7\,\text{erg}$

Special Units:

• Light Year: Distance light travels in one year ($9.147 \times 10^{15}$ m).
• Parsec: $3.1 \times 10^{16}$ m.
• Calorie: $1\,\text{calorie} = 4.2\,\text{Joule}$

Homework Questions (as per transcript):

• What is the unit of coefficient of friction ($\mu$)?
• Which force is called intelligent force?
• Write the dimensional formula of Planck’s constant.