Year 12 Physics week beginning 18 th May

Year 12 Physics – week beginning 18 th May 2020 Simple Harmonic Motion (SHM) - session 1

Introduction There are four different kinds of motion that we can encounter in Physics: • Linear (in a straight line) • Circular (going round in a circle) • Rotational (spinning on an axis) • 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. It is said that 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.

Example of a simple harmonic oscillator 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 motion of the mass on the truck is described as SHM

Displacement x T +A time -A Amplitude ( A ): The maximum distance that an object moves from its rest position. x = A and x = - A. Period ( T ): The time that it takes to execute one complete cycle of its motion. Units seconds, Frequency ( f ): The number or oscillations the object completes per unit time. Units Hz = s-1. Angular Frequency ( ω ): The frequency in radians per second, 2π per cycle.

The rest or equilibrium position at O is where the spring would hold the mass when it is not bouncing. Going back to the example 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 (maximum acceleration to the left) because that is where the spring is stretched most. (note: velocity = 0) • As a result of this the mass is accelerated. It accelerates towards the rest position. • Its velocity to the left increases. (reaches a maximum at the equilibrium position). • As the trolley approaches the O (equilibrium position) the acceleration decreases, since the restoring force decreases (spring is becoming less stretched). • The acceleration is zero at the equilibrium position (restoring force = zero) What is the acceleration and velocity at B? The acceleration is a maximum, but this time to the right. Velocity is zero since its at maximum displacement (amplitude)

From this example we can note two important points: 1. The acceleration depends on the displacement, for e. g. at maximum displacement the acceleration is a maximum and as the displacement decreases so does the acceleration. (note the displacement is the distance from the equilibrium position and it is a vector so it can be positive or negative depending on if its on the right or left of the equilibrium position). 2. The acceleration is always in the direction of the equilibrium position regardless of which way the object is moving, this is because the restoring force is always towards the equilibrium position as it is trying to restore the object to its equilibrium position. If any of this is unclear or you require further clarification please let me know. Definition of SHM (a system to performing SHM the following conditions must be satisfied) 1. its acceleration is directly proportional to its displacement from a fixed point (the equilibrium position) 2. its acceleration always acts towards the equilibrium position.

Study the following animation and make sure you understand the definition for SHM

Mathematical Definition for SHM a = acceleration x = displacement a µ -x a = -kx a=- 2 (2 f) x a=- 2 ( ) x The minus sign is important as it tells us that the acceleration is towards the equilibrium position. It can be shown that k = (2 f)2 (not required for the Remember = 2 f (fromcourse) circular motion) Important relationship

Graphical relationship between acceleration and displacement acceleration a µ -x displacement 0 a a Equil. position a a