Definitions & Formulas: Solutions

Ans: Measurement: It is the process of comparing an unknown physical quantity with a known standard quantity of the same nature.
Unit: The internationally accepted basic reference standard used for measurement is called a unit.
Characteristics of a standard unit: 1) It should be well-defined. 2) It should be of suitable size. 3) It should be easily reproducible. 4) It should not change with time or physical conditions (like temp, pressure). 5) It should be internationally accepted.

Ans: Fundamental Quantities: Physical quantities which are independent of each other and cannot be derived from other quantities. Example: Mass (SI: kg, Dim: \([M]\)), Length (SI: m, Dim: \([L]\)).
Derived Quantities: Physical quantities which are derived from fundamental quantities using mathematical operations. Example: Velocity (SI: m/s, Dim: \([LT^{-1}]\)), Force (SI: Newton, Dim: \([MLT^{-2}]\)).

Ans: System of Units: A complete set of fundamental and derived units for all kinds of physical quantities.
7 Fundamental Units: 1) Length (meter, m), 2) Mass (kilogram, kg), 3) Time (second, s), 4) Electric Current (ampere, A), 5) Thermodynamic Temperature (kelvin, K), 6) Luminous Intensity (candela, cd), 7) Amount of Substance (mole, mol).
2 Supplementary Units: 1) Plane Angle (radian, rad), 2) Solid Angle (steradian, sr).

Ans:
a) Astronomical Unit (AU): The average distance between the center of the Earth and the center of the Sun. \( 1 \text{ AU} = 1.496 \times 10^{11} \text{ m} \).
b) Light Year (ly): The distance traveled by light in a vacuum in one Earth year. \( 1 \text{ ly} = 9.46 \times 10^{15} \text{ m} \).
c) Parsec (Parallactic Second): The distance at which an arc of 1 AU subtends an angle of 1 arc-second. \( 1 \text{ Parsec} = 3.08 \times 10^{16} \text{ m} = 3.26 \text{ ly} \).

Ans: Accuracy: Refers to how close a measured value is to the true or accepted value of the quantity.
Precision: Refers to the resolution or the limit to which the quantity is measured (how close multiple measurements are to each other).
Example: If true length is 3.678 cm. Measurement A: 3.5 cm (less precise, less accurate). Measurement B: 3.385 cm (more precise, less accurate). Measurement C: 3.68 cm (more accurate).

Ans: Error: The uncertainty or difference between the measured value and the true value.
a) Systematic Error: Errors that tend to be in one direction (either positive or negative) due to known causes (e.g., zero error in vernier calipers).
b) Gross Error: Errors caused by human carelessness, like misreading an instrument.
c) Least Count Error: Error associated with the resolution of the instrument. Minimized by using instruments with higher resolution (smaller least count).

Ans: Rest: An object is at rest if it does not change its position with respect to its surroundings with time.
Motion: An object is in motion if it changes its position with respect to its surroundings with time.
Frame of Reference: A set of coordinate axes (X, Y, Z) attached to an observer having a clock, with respect to which the observer describes the state of rest or motion of an object.

Ans: Distance: The actual total length of the path covered by a moving body. It is a scalar quantity. Can never be zero or negative for a moving body.
Displacement: The shortest straight-line distance between the initial and final positions, directed from initial to final. It is a vector quantity. Can be positive, negative, or zero.
SI Unit: meter (m). Dimensional Formula: \([L]\).

Ans: Average Speed: Total distance traveled divided by total time taken. Formula: \( v_{avg} = \frac{\Delta x}{\Delta t} \). (Scalar)
Average Velocity: Total displacement divided by the total time taken. Formula: \( \vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t} \). (Vector)
SI Unit: m/s. Dimensional Formula: \([LT^{-1}]\).

Ans: It is the velocity of an object at a specific instant of time or at a specific point in its path. Mathematically, it is the limit of average velocity as the time interval \( \Delta t \) approaches zero.
Formula: \( \vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt} \)

Ans: The equations are:
1) \( v = u + at \) (Velocity-time relation)
2) \( s = ut + \frac{1}{2}at^2 \) (Position-time relation)
3) \( v^2 - u^2 = 2as \) (Position-velocity relation)
Condition: These equations are strictly valid only when the acceleration (\(a\)) of the body is constant (uniform).

Ans: Scalar: A physical quantity having only magnitude and no direction. Follows simple algebraic addition. Ex: Mass (kg), Work (Joules).
Vector: A physical quantity having both magnitude and direction, and which obeys the laws of vector addition. Ex: Velocity (m/s), Force (N).

Ans: A vector whose magnitude is exactly 1 (unity) and points in a particular direction is called a unit vector. It is used exclusively to specify direction.
Notation & Formula: Unit vector of \( \vec{A} \) is denoted by \( \hat{A} \) (read as A cap).
\( \hat{A} = \frac{\vec{A}}{|\vec{A}|} \)

Ans: The scalar product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them.
Formula: \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta \).
The result is a purely scalar quantity.

Ans: Projectile Motion: When a body is thrown with an initial velocity at an angle to the horizontal, it moves under the sole effect of gravity along a curved path called a trajectory (Parabola).
Trajectory Equation: \( y = x \tan\theta - \frac{gx^2}{2u^2 \cos^2\theta} \)
a) Time of Flight (T): \( T = \frac{2u \sin\theta}{g} \)
b) Horizontal Range (R): \( R = \frac{u^2 \sin(2\theta)}{g} \)
c) Max Height (H): \( H = \frac{u^2 \sin^2\theta}{2g} \)

Ans: The acceleration acting on a body undergoing circular motion, directed continuously towards the center of the circular path.
Formulas: \( a_c = \frac{v^2}{r} \) or \( a_c = r\omega^2 \)
SI Unit: \( \text{m/s}^2 \). Dim: \([LT^{-2}]\).

Ans: A body continues to be in its state of rest or of uniform motion in a straight line unless it is compelled by some external unbalanced force to change that state. (Also known as Law of Inertia).

Ans: The rate of change of linear momentum of a body is directly proportional to the applied external force and takes place in the direction of the force.
Formula: \( \vec{F} = \frac{d\vec{p}}{dt} \). Since \( \vec{p} = m\vec{v} \), if mass is constant, \( \vec{F} = m\vec{a} \).

Ans: Impulse: When a large force acts on a body for a very short time, the product of force and time is called Impulse. \( J = F \times \Delta t \). SI Unit: Ns or kg m/s.
Impulse-Momentum Theorem: The impulse applied to an object is equal to the change in its linear momentum. \( \vec{J} = \vec{p}_f - \vec{p}_i = \Delta \vec{p} \).

Ans: Friction: The opposing force that comes into play when one body moves or tends to move over the surface of another body.
a) Static: Opposes impending motion (body hasn't moved yet).
b) Limiting: The maximum value of static friction before motion begins.
c) Kinetic: Friction when the body is in actual relative motion.
d) Rolling: Friction when a body (like a cylinder/sphere) rolls without slipping over a surface.

Ans: Work is said to be done by a force when the body is displaced in the direction of the applied force.
Dot Product Formula: \( W = \vec{F} \cdot \vec{S} = F S \cos\theta \).
SI Unit: Joule (J). Dim: \([ML^2T^{-2}]\). It is a scalar quantity.

Ans: It states that the net work done by all the forces acting on a body is equal to the change in its kinetic energy.
Equation: \( W_{net} = K_f - K_i = \Delta K \).

Ans: Conservative: Work done depends only on initial and final positions, not on the path taken. Work done in a closed loop is zero. Ex: Gravitational force, Electrostatic force.
Non-Conservative: Work done depends on the actual path taken. Work in a closed loop is not zero. Ex: Frictional force, Viscous drag.

Ans: Coefficient of Restitution (e): It is the ratio of relative velocity of separation after collision to the relative velocity of approach before collision. \( e = \frac{v_2 - v_1}{u_1 - u_2} \).
For Perfectly Elastic: \( e = 1 \).
For Inelastic: \( 0 < e < 1 \).
For Perfectly Inelastic: \( e = 0 \).

Ans: A Rigid Body is a body with a perfectly definite and unchanging shape.
Condition: The distance between any two given points of a rigid body remains absolutely constant in time regardless of the external forces acting on it.

Ans: Centre of Mass (COM): A theoretical point where the entire mass of a system is assumed to be concentrated to describe its translational motion. It can lie inside or outside the physical body.
Centre of Gravity (COG): The point through which the total weight (gravitational force) of the body acts. COM and COG coincide in a uniform gravitational field.

Ans: Torque (\(\tau\)): The turning effect of a force. \(\vec{\tau} = \vec{r} \times \vec{F}\). SI: Nm. Vector quantity.
Angular Momentum (L): The moment of linear momentum. \(\vec{L} = \vec{r} \times \vec{p}\). SI: kg·m²/s. Vector quantity.
Relation: Torque is the rate of change of angular momentum: \( \vec{\tau} = \frac{d\vec{L}}{dt} \).

Ans: The property of a rotating body to resist any change in its state of rotational motion. It is the rotational analog of mass.
Formula: \( I = \sum m_i r_i^2 \).
SI Unit: kg·m². Dim: \([ML^2]\).

Ans: Perpendicular Axes: For a planar body, \( I_z = I_x + I_y \). (The moment of inertia about a perpendicular axis equals the sum of moments of inertia about two mutually perpendicular concurrent axes in the plane).
Parallel Axes: \( I = I_{cm} + Md^2 \). (The moment of inertia about any axis equals the moment of inertia about a parallel axis through the COM plus the product of mass and the square of the distance between them).

Ans: 1) Law of Orbits: All planets move in elliptical orbits with the Sun at one of the foci.
2) Law of Areas: The line joining the planet to the Sun sweeps out equal areas in equal intervals of time (Areal velocity is constant).
3) Law of Periods: The square of the time period of revolution is proportional to the cube of the semi-major axis: \( T^2 \propto a^3 \).

Ans: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Formula: \( F = G\frac{m_1 m_2}{r^2} \). (It is a central, attractive, and conservative force).

Ans: a) Altitude (\(h\)): Decreases with height. \( g_h \approx g(1 - \frac{2h}{R}) \).
b) Depth (\(d\)): Decreases linearly with depth. \( g_d = g(1 - \frac{d}{R}) \). Zero at the center of Earth.
c) Shape: Earth is an oblate spheroid. \(g\) is maximum at the poles and minimum at the equator.
d) Rotation: Decreases at the equator due to centrifugal effect. \( g' = g - R\omega^2 \cos^2\lambda \).

Ans: Escape Speed: Minimum initial speed required for an object to permanently overcome the gravitational pull of a planet. \( v_e = \sqrt{2gR} \) (approx 11.2 km/s on Earth).
Orbital Velocity: Velocity required to keep a satellite in its circular orbit around a planet. \( v_o = \sqrt{GM/r} \).

Ans: It is a situation where the apparent weight of a body becomes zero. In an orbiting satellite, both the astronaut and the satellite fall toward the Earth with the exact same acceleration (free fall), making the normal reaction exerted by the floor zero.

Ans: Stress: The internal restoring force developed per unit cross-sectional area of a deformed body. \(\sigma = F/A\). SI: N/m².
Strain: The ratio of change in dimension to the original dimension. It is a dimensionless ratio.
Hooke's Law: Within the elastic limit, the stress developed in a body is directly proportional to the strain produced.

Ans: 1) Young's Modulus (\(Y\)): Ratio of longitudinal stress to longitudinal strain. \( Y = \frac{FL}{A\Delta L} \).
2) Bulk Modulus (\(B\)): Ratio of volumetric stress to volumetric strain. \( B = -\frac{PV}{\Delta V} \). (Compressibility = \(1/B\)).
3) Shear Modulus (\(G\)): Ratio of shearing stress to shearing strain.

Ans: The pressure exerted anywhere on a confined, incompressible fluid is transmitted equally and undiminished in all directions throughout the fluid and onto the walls of the container.

Ans: For the streamline flow of an ideal (non-viscous, incompressible) fluid, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant at all points along its path.
Equation: \( P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \).

Ans: Stokes' Law: The backward viscous drag force acting on a small sphere falling through a viscous fluid is \( F = 6\pi\eta rv \).
Terminal Velocity: The maximum constant velocity acquired by a body falling freely through a viscous fluid, achieved when the viscous drag and buoyant force exactly balance the gravitational weight.

Ans: Surface Tension (\(S\)): The property of a liquid surface at rest to shrink to the minimum possible surface area. Mathematically, it is the force per unit length drawn on the liquid surface (\(S = F/L\)).
Capillarity: The phenomenon of the spontaneous rise or fall of a liquid inside a fine capillary tube. Formula for rise: \( h = \frac{2S\cos\theta}{r\rho g} \).

Ans: \( PV = nRT \). Where P is pressure, V is volume, n is number of moles, T is absolute temperature, and \( R \) is the Universal Gas Constant (\(8.314 \text{ J/mol·K}\)).

Ans: Specific Heat (\(c\)): The amount of heat required to raise the temperature of unit mass (1 kg) of a substance by 1°C or 1 K. \( c = \frac{Q}{m\Delta T} \).
Molar Specific Heat (\(C\)): The amount of heat required to raise the temperature of 1 mole of a substance by 1°C.

Ans: First Law: Energy conservation applied to heat. Heat supplied is used partially to increase internal energy and partially to do external work: \( \Delta Q = \Delta U + \Delta W \).
Second Law (Kelvin-Planck): It is completely impossible to construct a heat engine that absorbs heat from a hot reservoir and converts 100% of it into work without rejecting some heat to a cold sink.