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).

Explore the concepts of Centre of Mass, Rotational Motion, and Fluid Statics through this quiz. From understanding how the CoM is calculated to the intricacies of angular quantities and fluid pressure, this quiz covers essential topics in physics. Test your knowledge and application skills in these key areas of mechanics.