Physics Study Notes: Rotational Motion, Equilibrium & Elasticity, Momentum, Energy & Work, Using Energy, and Fluids

Rotational Motion

Introduction to Rotational Motion

Rotational motion involves the movement of objects around a fixed axis. It is characterized by angular quantities analogous to linear motion.

• Angular Displacement (\(\theta\)): The angle through which an object rotates, measured in radians.

• Angular Velocity (\(\omega\)): The rate of change of angular displacement, \(\omega = \frac{d\theta}{dt}\).

• Angular Acceleration (\(\alpha\)): The rate of change of angular velocity, \(\alpha = \frac{d\omega}{dt}\).

• Moment of Inertia (\(I\)): The rotational equivalent of mass, representing an object's resistance to angular acceleration.

Key Equations:

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega = \omega_0 + \alpha t$

• $\tau = I \alpha$ (Torque and angular acceleration)

Example: A solid disk of mass \(M\) and radius \(R\) rotating about its center has moment of inertia \(I = \frac{1}{2}MR^2\).

Equilibrium & Elasticity

Conditions for Equilibrium

Equilibrium occurs when the net force and net torque on a system are zero, resulting in no linear or rotational acceleration.

• Translational Equilibrium: $\sum \vec{F} = 0$

• Rotational Equilibrium: $\sum \tau = 0$

Elasticity

Elasticity describes how materials deform and return to their original shape when forces are applied and removed.

• Stress: Force per unit area, $\text{Stress} = \frac{F}{A}$

• Strain: Relative deformation, $\text{Strain} = \frac{\Delta L}{L_0}$

• Young's Modulus (Y): $Y = \frac{\text{Stress}}{\text{Strain}}$

Example: A steel wire stretches by 1 mm under a load; its Young's modulus quantifies its stiffness.

Momentum

Linear Momentum and Impulse

Momentum is a measure of an object's motion, and impulse is the change in momentum due to a force applied over time.

• Momentum (\(\vec{p}\)): $\vec{p} = m\vec{v}$

• Impulse (\(\vec{J}\)): $\vec{J} = \Delta \vec{p} = F_{\text{avg}} \Delta t$

Conservation of Momentum

In a closed system, total momentum remains constant if no external forces act.

• Elastic Collisions: Both momentum and kinetic energy are conserved.

• Inelastic Collisions: Momentum is conserved, but kinetic energy is not.

Example: Two ice skaters push off each other and move in opposite directions; their combined momentum before and after is the same.

Energy & Work

Work and Kinetic Energy

Work is done when a force causes displacement. Energy is the capacity to do work.

• Work (W): $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$

• Work-Energy Theorem: $W_{\text{net}} = \Delta KE$

Potential Energy

• Gravitational Potential Energy: $U_g = mgh$

• Elastic Potential Energy: $U_s = \frac{1}{2}kx^2$

Example: Lifting a box increases its gravitational potential energy.

Using Energy

Conservation of Energy

Energy cannot be created or destroyed, only transformed from one form to another.

• Mechanical Energy: $E_{\text{mech}} = KE + PE$

• Conservation Principle: $E_{\text{mech, initial}} = E_{\text{mech, final}}$ (if no non-conservative forces)

Power

• Power (P): $P = \frac{W}{t}$

• Instantaneous Power: $P = \vec{F} \cdot \vec{v}$

Example: An electric motor lifting a weight does work and consumes energy at a certain rate (power).

Fluids

Properties of Fluids

Fluids are substances that flow and take the shape of their container, including liquids and gases.

• Density (\(\rho\)): $\rho = \frac{m}{V}$

• Pressure (P): $P = \frac{F}{A}$

Fluid Statics

• Hydrostatic Pressure: $P = P_0 + \rho gh$

• Buoyant Force: $F_b = \rho_{\text{fluid}} V_{\text{disp}} g$

Fluid Dynamics

• Continuity Equation: $A_1 v_1 = A_2 v_2$

• Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

Example: A floating object experiences an upward buoyant force equal to the weight of the fluid displaced (Archimedes' principle).