Physics Chapter on Motion and Forces

Acceleration and Velocity

• Acceleration is defined by the formula ( a = \frac{Δv}{t} ).
• Instantaneous velocity is expressed as ( v_x = v_0 + at ).

Displacement and Motion

• Displacement can be calculated using the equation ( Δd = v_0t + \frac{1}{2}at^2 ).
• For velocity without time, use ( v_x^2 = v_0^2 + 2aΔd ).

Forces and Friction

• Newton’s second law states ( a = \frac{F}{m} ), relating force, mass, and acceleration.
• Friction is given by ( f = μN ), where ( μ ) is the coefficient of friction and ( N ) is the normal force.

Momentum and Energy

• Momentum is defined as ( p = mv ).
• Impulse, representing the change in momentum, can be calculated by ( Δp = Ft ) or ( Δp = mΔv ).
• The linear kinetic energy formula is ( K = \frac{1}{2}mv^2 ).

Work and Power

• The work-energy theorem states ( ΔE = W ), highlighting the relationship between work and energy.
• Work can be calculated as ( W = Fd \cos(θ) ).
• General power is expressed as ( P = \frac{ΔE}{t} ) and mechanical power as ( P = \frac{W}{t} ).

Rotational Motion

• Rotational displacement is given by ( θ = θ_0 + ω_0t + \frac{1}{2}αt^2 ).
• The angular acceleration formula is ( α = \frac{Στ}{I} ) and net torque is ( Στ = Iα ).
• Torque in terms of force is calculated by ( τ = rF \sin(θ) ).
• Angular momentum is defined as ( L = Iω ) and its change as ( ΔL = τt ).
• Rotational kinetic energy is given by ( K = \frac{1}{2}Iω^2 ).

Energy in Elastic Systems

• Hooke's Law describes the restoring force in springs as ( F_s = kx ).
• The potential energy stored in a spring is calculated using ( U_s = \frac{1}{2}kx^2 ).

Density and Gravitational Concepts

• Density is defined as ( ρ = \frac{m}{V} ).
• Gravitational potential energy is calculated as ( ΔU_g = mgΔy ).

Periodic Motion

• The period in terms of angular velocity is ( T = \frac{2π}{ω} ).
• The period with frequency is ( T = \frac{1}{f} ).
• The period of a spring is given by ( T_s = 2π\sqrt{\frac{m}{k}} ) and for a pendulum, ( T_p = 2π\sqrt{\frac{l}{g}} ).

Universal Gravitation

• Newton's law of universal gravitation is expressed as ( F_g = \frac{Gm_1m_2}{r^2} ).
• Gravitational acceleration on any planet is calculated by ( g = \frac{GM}{r^2} ).

Electrical Concepts

• The electrical force between charges is determined by ( F_e = \frac{kq_1q_2}{r^2} ).
• Current in terms of charge is defined as ( I = \frac{Δq}{Δt} ).
• Resistivity relates to resistance through ( R = \frac{ρl}{A} ).
• Ohm’s Law states ( I = \frac{V}{R} ) relating current, voltage, and resistance.

Electrical Power and Circuits

• Electrical power can be calculated using ( P = IV ).
• Resistance in series is given by ( R_s = R_1 + R_2 + ... + R_n ).
• In parallel circuits, resistance is determined by ( \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} ).

Wave Mechanics

• Wavespeed is calculated using ( λ = \frac{v}{f} ).

Free Fall and Velocity Components

• The time to fall from a height is ( t = \sqrt{\frac{2h}{g}} ).
• The x-component of velocity is ( v_x = v_0 \cos(θ) ) and y-component is ( v_{oy} = v_0 \sin(θ) ).

Forces in Elevators

• The net force on an elevator is expressed as ( ΣF = F_s - mg ).

Critical Velocity

• The critical velocity at the top of a circle is ( v = \sqrt{gr} ), while the maximum velocity on a road curve is ( v = \sqrt{rμg} ).
• Satellite velocity is calculated using ( v = \sqrt{\frac{GM}{r}} ).