Simple Harmonic Motion

Simple Harmonic Motion Periodic Motion Periodic motion is any motion that repeats itself. Examples of periodic motion include: Rise and fall of tides Heart beat Pendulum Horizontal & vertical springs Simple Harmonic Motion For an object to be doing SHM the following two (three) conditions must apply to the periodic motion: Acceleration is proportional to displacement Acceleration directed towards the equilibrium position There must exist a restorative force

acceleration is  to negative displacement For an object undergoing SHM: The velocity is NOT constant and will vary between Vmax and zero. Since V is always changing there must be acceleration occurring The acceleration is NOT constant and will vary between amax and zero. x=-A x=0 x=A Vmax a =0 V = 0 amax V =0 amax acceleration is  to negative displacement

V = 0, a = max V = max, a = 0 V = 0, a = max

a a a a Simple Pendulum Equil. position

Essential features for Horizontal & Vertical Springs x = 0 F m A = xmax F Equilibrium Position x m No difference in SHM calculations between horizontal & vertical springs mg (k) is the spring constant Displacement (x) is measured from the equilibrium point Amplitude (A) is the maximum displacement A cycle is a full up-and-down or left-to-right motion Period (T) is the time required to complete one cycle Frequency (f) is the number of cycles completed per second Similar features for a pendulum…

Periodic and Simple Harmonic Motion (SHM) For these events discuss: if they are periodic; and if they approximate SHM. Piston Baby swing Ski jumping Heart beat Bungy jumping

Periodic and approximates SHM Periodic and Simple Harmonic Motion (SHM) For these events discuss: if they are periodic; and if they approximate SHM. Baby swing Periodic and approximates SHM The movement repeats over and over again in regular cycles There is a restoring force which acts towards the equilibrium position (resting position) The greater the displacement the greater the restoring force The displacement and the restoring force are continually in opposite directions Acceleration is a - displacement Baby swing

Periodic and Simple Harmonic Motion (SHM) For these events discuss: if they are periodic; and if they approximate SHM. Bungy jumping In free fall the motion is not periodic and hence cannot be SHM because After maximum displacement of the bungy cord the motion is periodic and approximates SHM because: the movement repeats over and over again in regular cycles even though the amplitude decreases; there is a restoring force which acts towards the equilibrium position. (final resting position); the greater the displacement the greater the restoring force; there are no unbalanced forces acting in the equilibrium position; the displacement and the restoring force are continually in opposite directions; acceleration a – displacement; and force = - kx where the –ve stands for direction, the constant k is +ve the one force – gravity – acts downwards; the force is constant; as the displacement increases the force remains constant in the same direction; and acceleration is constant ie -9.81 ms-1

Periodic and approximates SHM Periodic and Simple Harmonic Motion (SHM) For these events discuss: if they are periodic; and if they approximate SHM. Periodic and approximates SHM The piston is periodic and approximates SHM at constant r.p.m. The piston is driven i.e., given an impulse periodically to keep it going The motion repeats over and over again in regular cycles The crankshaft has constant angular velocity at constant r.p.m. Piston Crank shaft

Periodic but not SHM Periodic and Simple Harmonic Motion (SHM) For these events discuss: if they are periodic; and if they approximate SHM. Heart beat Periodic but not SHM A pulse beats at regular intervals. Has no other SHM properties. There is one main force – gravity – that acts downwards The force is constant As the displacement increases the force remains constant in the same direction Acceleration is constant Ski jumping In free fall the motion is not periodic and hence cannot be SHM