Simple Harmonic Motion

• There are four different kinds of motion in Physics: Linear (in a straight line), Circular (going round in a circle), Rotational (spinning on an axis), and Oscillations (going backwards and forwards in a to-and-fro movement).
• Anything that swings or bounces or vibrates in a regular to-and-fro motion is said to oscillate, examples include a swinging pendulum or a spring bouncing up and down.
• The regularity of a swinging object was first described by a teenage Galileo who watched a chandelier swinging during a church service in Pisa.
• Simple Harmonic Motion (SHM) describes the way that oscillating objects move, consider a spring with a mass going from side to side.
• A mass is mounted on a small railway truck, which is free to move from side to side, and there is negligible friction in the truck.
• The rest or equilibrium position at O is where the spring would hold the mass when it is not bouncing.
• A is the position where the spring is stretched the most, and B is where the spring is squashed most.
• At A there is a large restoring force because that is where the spring is stretched most, as a result of this the mass is accelerated.
• The acceleration of a body is towards the equilibrium position or in the opposite direction of the displacement.
• SHM can be linked to circular motion.
• Some useful relationships for SHM are: the period, which is the time taken to make a complete oscillation or cycle, the frequency, which is the reciprocal of the period, and acceleration which can be linked to displacement by a = - (2pf )2 x.
• If x = 0, v has a maximum value; if x = A, v = 0.
• The velocity is 0 at each extreme of the oscillation.
• The displacement, x, is given by: x = ± A cos 2pf t.
• The plus and minus sign here tells us that the motion is forwards and backwards.
• The displacement, velocity, and acceleration are p 1800 radians out of phase.
• The extension of a spring is directly proportional to the force (Hooke’s Law).
• Consider a mass, m, put onto a spring of spring constant k so that so that it stretches by an extension l.
• The force on the spring = mg, and the stretching tension = k l.
• mg = k l.
• If the spring is pulled down by a distance x below the rest position, the stretching force becomes k(l + x).
• The restoring force, Fup = k(l + x) – mg.
• Since mg is the weight, which always acts downwards, the restoring force is Fup = k(l + x) – mg.
• If k l = mg, we can write: Fup = k l + kx – k l.
• We can apply Newton II to write: -kx = ma.
• The condition for SHM is satisfied in this system, as long as Hooke’s Law is obeyed.
• The period of a simple harmonic oscillator is given by T = 1/f.
• The period of a simple harmonic oscillator is independent of amplitude or mass of the bob.
• The velocity at any point in the oscillation is given by: v2 = (2pf )2(A2 – x2), where A is the amplitude and x is the displacement from the equilibrium position.
• The gradient of the graph of T2 against L is 4p2/g.
• To measure g, divide 4p2 by the gradient.
• A relatively accurate determination of g can be obtained by counting at least 100 swings, using a swing angle of less than 10o, measuring L to the centre of the bob and counting the oscillations as the bob passes the equilibrium position.
• Potential energy is highest when the oscillator is at the maximum amplitude;
• Kinetic energy is highest when the oscillator
• An oscillation is any to-and-fro movement;
• It can arise from a swinging pendulum, a mass bouncing on a spring, or a vibrating system.
• A cycle is a complete to-and-fro movement;
• The period is the time taken for a complete to-and-fro movement.
• The frequency is how many cycles there are in a second;
• The Physics symbol for frequency is f and it is measured in Hertz (Hz).