Simple Harmonic Motion

A mass on a spring A vibrating tuning fork Examples A mass on a spring A vibrating tuning fork A swinging pendulum

The Definition Simple harmonic motion is the motion of a particle about a fixed point such that its acceleration towards that point is proportional to its displacement from the point. The acceleration is directed toward the point

In Mathematical Terms a = acceleration w2 = constant s = displacement

Hookes Law The extension of a spring is in directly proportional to the force applied to it as long as this force does not exceed the elastic limit. F = Force k = Spring constant s = Extension

A body obeying Hooke's Law exhibits simple harmonic motion Hooke's Law F = - ks ma = - ks a = - s = a constant So acceleration is proportional to the displacement and in the opposite direction. This is simple harmonic motion.

For a spring By comparison

Periodic Time The periodic time T is the time taken for one complete oscillation. For an object in simple harmonic motion For a pendulum swinging through an angle less that 50 L = length of pendulum g = acceleration due to gravity

State Hooke’s law. (6) A stretched spring obeys Hooke’s law. When a small sphere of mass 300 g is attached to a spring of length 200 mm, its length increases to 285 mm. Calculate its spring constant. (9) The sphere is pulled down until the length of the spring is 310 mm. The sphere is then released and oscillates about a fixed point. Derive the relationship between the acceleration of the sphere and its displacement from the fixed point. Why does the sphere oscillate with simple harmonic motion? (18) Calculate: (i) the period of oscillation of the sphere (ii) the maximum acceleration of the sphere (iii) the length of the spring when the acceleration of the sphere is zero. 2007 Q6

State Newton’s second law of motion. (6) The equation F = – ks, where k is a constant, is an expression for a law that governs the motion of a body. Name this law and give a statement of it. (9) Give the name for this type of motion and describe the motion. (9) A mass at the end of a spring is an example of a system that obeys this law. Give two other examples of systems that obey this law. (6) The springs of a mountain bike are compressed vertically by 5 mm when a cyclist of mass 60 kg sits on it. When the cyclist rides the bike over a bump on a track, the frame of the bike and the cyclist oscillate up and down. Using the formula F = – ks, calculate the value of k, the constant for the springs of the bike. (6) The total mass of the frame of the bike and the cyclist is 80 kg. Calculate (i) the period of oscillation of the cyclist, (ii) the number of oscillations of the cyclist per second. (20) 2002 Q6