Periodic motion

• An object moving in a circular path at constant speed has a constantly changing velocity as velocity is a vector, and so the object must also be accelerating.
• An object moving in a circular path experiences a centripetal force acting towards the centre of the circle (perpendicular to velocity, and in the same direction as acceleration).
• Angular velocity, w, is the angle an object moves through per unit time, measured in radians. It can be found by dividing linear speed by radius or by multiplying frequency by two pi.
• Centripetal acceleration can be found by dividing linear speed squared by radius or by multiplying angular velocity by radius.
• Centripetal force can be derived from F = m a to equal (mass x linear velocity squared)/ radius or mass x (angular velocity squared) x radius.
• An object is experiencing simple harmonic motion when its acceleration is directly proportional, but in the opposite direction, to displacement. This can be shown by the equation a = w ^2 x.
• For a pendulum, amplitude can be used to form two more equations: x = A cos (w t) and v = w root(A^2 - x^2).
• The maximum speed of a pendulum is angular velocity multiplied by amplitude, and the maximum acceleration is angular velocity squared multiplied by amplitude.
• When a mass m hangs from a pendulum length l, the timer period of oscillation is T = 2 pi root(l / g).
• When a mass m hangs from a spring, the time period of oscillation is T = 2 pi root(m / k).
• For any simple harmonic system, kinetic energy is transferred to potential energy and back as the system oscillates, with total energy staying constant.
• Damping is where the energy in an oscillating system is lost to the environment, leading to reduced amplitude of oscillations.
• Light damping, also known as underdamping, is where the amplitude slowly decreases with each oscillation.
• Critical damping is where the amplitude is reduced to zero quickly without any oscillation (within 1/4 of a cycle).
• Heavy damping, also known as overdamping, is where the amplitude reduces slowly but without any oscillations.
• Free oscillations occur when no external force is continuously acting on the system, and therefore the system will oscillate at its natural frequency.
• Forced oscillations are when a system experiences an external driving force which forces it to oscillate at the driving frequency.
• If the driving frequency is the same as the natural frequency of a system, resonance occurs, which is when the amplitude of oscillations increase dramatically to a maximum due to gaining energy from the driving force.
• As the degree of damping increases, the resonant frequency decreases and appears over a wider range of frequencies, and maximum amplitude decreases.