Circular motion and SHM

• For something moving in a circle, there must be a force continuously acting at 90 degrees to the object's velocity or direction of motion
• Velocity and direction of travel constantly change, but the object still travels at a constant speed
• Velocity is a vector, so it's still accelerating because the direction of the velocity is changing, even though the speed is constant
• Centripetal acceleration is equal to V^2 / R, where R is the radius of the circle
• Centripetal force is equal to mv^2 / R
• Angular velocity (angular speed or angular frequency) is represented by the symbol Omega (Ω), which is equal to 2π / T (radians per second)
• V = Omega * R
• Simple harmonic motion (SHM) describes any object oscillating around a point like a pendulum or a mass on a spring
• Acceleration in SHM is proportional to the object's displacement from equilibrium and in the opposite direction to the displacement
• Restoring force in SHM tries to return the object to equilibrium
• Equation for SHM: a = -Omega^2 * x
• Maximum acceleration occurs at the maximum displacement (amplitude)
• Equation for time period of a pendulum: T = 2π * sqrt(L / g), where L is the length of the string and g is gravitational field strength
• Equation for time period of a mass on a spring: T = 2π * sqrt(m / k), where k is the spring constant
• For a pendulum, the frequency is inversely proportional to the length
• Damping force opposes the motion of the object and removes energy from the system
• Light damping gradually decreases the amplitude of oscillations
• Heavy damping quickly reduces the displacement of the object
• Critical damping stops the object at equilibrium as quickly as possible
• Resonance occurs when an external force drives the oscillations, increasing the amplitude and adding energy into the system
• Maximum resonance occurs when the driving force frequency matches the natural frequency of the system
• Driving force and restoring force are 90 degrees out of phase with each other during resonance
• Resonance is most effective when the system is lightly damped
• Resonance graph shows the highest amplitude when the driving frequency equals the natural frequency
• Adding more damping decreases the height of the resonance peak
• Peak width does not change with increased damping
• Energy is proportional to the square of the amplitude in a damped system
• Total energy remains constant in a closed system with no energy loss to the surroundings