Centre of Mass and Rotational Motion

Centre of Mass

• Centre of Mass (CoM) is the point where the total mass of a system is considered concentrated.
• For discrete particles, CoM is calculated as where is the mass of each particle and is the position vector.
• For continuous bodies, CoM is given by , where is the density and is the position vector.
• The CoM moves as if all external forces act on it.
• The CoM of a uniform rod is at its midpoint.
• Applications include balance and system motion analysis.

Rotational Motion

• Angular quantities include angle (θ), angular velocity (ω), and angular acceleration (α).
• Key equations relate angular quantities to linear motion quantities.
• Moment of Inertia (I) depends on mass distribution and the axis of rotation.
• Torque (τ) is a measure of the twisting force.
• Conservation of Angular Momentum states that angular momentum (L) remains constant if no external torque acts.

Fluid Statics

• Pressure (P) in a fluid increases with depth.
• Pascal's Law: Pressure applied at one point in a confined fluid is transmitted undiminished to all points.
• Buoyancy (upthrust) = weight of displaced fluid (ρVg).
• Archimedes' Principle: An immersed body experiences an upward force equal to the weight of the displaced fluid.

Thermodynamics

• Thermodynamics involves the relationship between heat, work, and temperature.
• Zeroth Law establishes the concept of thermal equilibrium.
• First Law of Thermodynamics describes the conservation of energy: ΔU = Q - W.
• Second Law states that entropy always tends to increase in a closed system.
• Third Law states that absolute zero temperature is unattainable.
• Thermodynamic processes include isothermal, adiabatic, isobaric, and isochoric processes.

Thermochemistry

• Enthalpy (H) is the heat change at constant pressure.
• Endothermic processes absorb heat (ΔH > 0), while exothermic processes release heat (ΔH < 0).
• Hess's Law: The total enthalpy change of a reaction is equal to the sum of the enthalpy changes along different paths.
• Bond energy is the energy needed to break one mole of bonds.

Equilibrium

• Dynamic equilibrium occurs when forward and reverse reaction rates are equal.
• Law of Mass Action describes the relationship between reactant and product concentrations.
• Le Chatelier's Principle describes how a system in equilibrium responds to changes in conditions.

Straight Lines

• General equation of a straight line is Ax + By + C = 0.
• Slope-intercept form of a straight line is y = mx + b.
• Slope (m) = (y₂ - y₁) / (x₂ - x₁).
• Distance of a point (x₀, y₀) from a line Ax + By + C = 0 is given by |Ax₀ + By₀ + C| / √(A² + B²).

Permutations & Combinations (P&C)

• Permutations (P(n,r)) arrange objects from a group of n.
• Combinations (C(n,r)) select objects from a group of n.
• Factorial (n!) = n × (n-1) × ... × 2 × 1.

Binomial Theorem

• Binomial expansion of is. (x + y)^n
• General term in the expansion of (x + y)^n is .
• Middle term in the expansion of (x + y)^n is if n is even.

Parabola [Basics]

• Standard equation of a parabola opening horizontally is x = ay² or y = ax².
• Parabolas' vertices are at the origin (0, 0).
• Focus depends on the direction (e.g., for x = ay² the focus is (1/4a,0)).
• Parabolas have a directrix (e.g., for x = ay² the directrix is x = -1/4a).

Circle

• Standard equation of a circle is (x-h)² + (y-k)² = r².
• (h, k) is the center and r is the radius.
• General equation is x² + y² + 2gx + 2fy + c = 0.
• Center is (-g, -f) and radius is √(g² + f² - c).