Comprehensive Physics Study Guide: Mechanics, Kinematics, and Dynamics

Newton's Second Law of Motion: * The acceleration ( $a$ ) of an object is directly proportional to the net force acting upon it and inversely proportional to its mass ( $m$ ). This relationship is expressed as: $\text{a} = \frac{\text{Net Force}}{\text{mass}}$ * If a net force is applied to a body, the resulting acceleration occurs in the same direction as that net force.
Newton's First Law of Motion: * A body that is not acted upon by any net force ( $\sum F = 0$ ) will maintain a constant velocity. * In such a state, the body experiences zero acceleration ( $a = 0\,m/s^2$ ).

Vector Addition (Q14): * Given Vector $\mathbf{A} = 2i - 3j - k$ and Vector $\mathbf{B} = i + 4j - 2k$ . * To find the sum $\mathbf{A} + \mathbf{B}$ , add the components individually: * $x$ -components: $2i + 1i = 3i$ * $y$ -components: $-3j + 4j = j$ * $z$ -components: $-k - 2k = -3k$ * Resulting Vector: $3i + j - 3k$ .

Acceleration Calculation (Q15): * Given data: * Mass ( $m$ ) = $32.5\,kg$ * Initial state: at rest ( $u = 0\,m/s$ ) * Net horizontal force ( $F$ ) = $140\,N$ * Calculation: $a = \frac{F}{m} = \frac{140\,N}{32.5\,kg} \approx 4.31\,m/s^2$ .
Force and Acceleration Proportionality (Q28): * A force produces an acceleration of $0.5\,m/s^2$ in a body of mass $3.0\,kg$ . * The magnitude of the force is $F = m \times a = 3.0\,kg \times 0.5\,m/s^2 = 1.5\,N$ . * If the same force ( $1.5\,N$ ) acts on a body of mass $1.5\,kg$ , the new acceleration ( $a'$ ) is: * $a' = \frac{1.5\,N}{1.5\,kg} = 1.0\,m/s^2$ .
Friction (Q24): * The coefficient of kinetic friction ( $\mu_k$ ) is the ratio of the kinetic frictional force to the normal force. * Given parameters: * Weight ( $W$ ), which equals the Normal Force ( $N$ ) on a level surface = $200\,N$ * Force to initiate motion (Static Friction limit) = $80\,N$ * Force to maintain constant velocity (Kinetic Friction, $F_k$ ) = $40\,N$ * Calculation: $\mu_k = \frac{F_k}{N} = \frac{40\,N}{200\,N} = 0.2$ .

Kepler's Laws (Q16): * Kepler's First Law (The Law of Orbits): The planets move in elliptical orbits with the Sun at one of the two foci.
Universal Law of Gravitation (Q18): * The gravitational force ( $F$ ) between two masses ( $m_1$ and $m_2$ ) separated by distance ( $r$ ) is given by: $F = \frac{G m_1 m_2}{r^2}$ * Dimensions of Gravitational Constant ( $G$ ): * Rearranging the formula: $G = \frac{F r^2}{m_1 m_2}$ * Dimensions of Force: $[M L T^{-2}]$ * Dimensions of $r^2$ : $[L^2]$ * Dimensions of mass product: $[M^2]$ * Resulting Dimensions of $G$ : $[M^{-1} L^3 T^{-2}]$ .

Dimension of Pressure (Q17): * Pressure is defined as Force per unit Area ( $P = F/A$ ). * Dimensions of Force: $[M L T^{-2}]$ * Dimensions of Area: $[L^2]$ * Dimensional Formula for Pressure: $[M L^{-1} T^{-2}]$ .
Comparison of Quantities (Q20): * Quantities with equal dimensions usually involve energy or rotational force: * Torque: $[M L^2 T^{-2}]$ * Work: $[M L^2 T^{-2}]$ * Energy: $[M L^2 T^{-2}]$ * Non-matching quantity: Momentum ( $p = mv$ ), which has dimensions of $[M L T^{-1}]$ .

Projectile Motion Formulas (Q19): * The Range ( $R$ ) of a projectile launched with initial velocity ( $u$ ) at an angle ( $\theta$ ) is calculated using: $R = \frac{u^2 \sin(2\theta)}{g}$
Graphical Analysis (Q22, Q25, Q26): * Velocity-Time ( $v-t$ ) Graph Slope: The slope of a velocity-time graph represents the acceleration of the object. * Velocity-Time ( $v-t$ ) Graph Area: The area under the curve of a velocity-time graph represents the displacement of the object.
Motion Terminology (Q21): * The path traced by a particle during the course of its motion is known as its Trajectory.
Horizontal Projection Problem (Q23): * Scenario: Object projected horizontally at $v_x = 80\,m/s$ from a height $h = 160\,m$ . * Components of final velocity at impact: * Horizontal component ( $v_x$ ): Remains constant at $80\,m/s$ . * Vertical component ( $v_y$ ): Calculated using $v_y^2 = u_y^2 + 2gh$ . * $v_y^2 = 0 + 2 \times 9.8 \times 160 = 3136 \Rightarrow v_y = 56\,m/s$ . * Final velocity magnitude ( $v$ ): $v = \sqrt{v_x^2 + v_y^2} = \sqrt{80^2 + 56^2} = \sqrt{6400 + 3136} = \sqrt{9536} \approx 97.6\,m/s$ .

Collision Types (Q27): * Inelastic Collision: A collision where momentum is conserved, but kinetic energy is not conserved.
Fluid Statics (Q29): * Archimedes' Principle: When an object is completely or partially immersed in a fluid, it experiences an upward force (upthrust) equal to the weight of the fluid displaced by the object.

Power Calculation (Q30): * Power ( $P$ ) is defined as the rate of doing work. * Calculated as $P = \frac{\text{Work}}{\text{time}} = \frac{\text{Force} \times \text{distance}}{\text{time}}$ . * Given data: * Force ( $F$ ) = $2000\,N$ * Distance ( $d$ ) = $10\,m$ * Time ( $t$ ) = $50\,s$ * Calculation: $P = \frac{2000\,N \times 10\,m}{50\,s} = \frac{20000\,J}{50\,s} = 400\,W$ .