# Simple harmonic motion relationship between acceleration and displacement

### Simple harmonic motion

When an object is in simple harmonic motion, the rate at which it oscillates the acceleration the second derivative of the displacement function with respect to time. . Notice that in equation 11, the relationship between T and m is not linear. Simple harmonic motion occurs when the acceleration is proportional to displacement but they are in opposite directions. Position is simple harmonic motion is [math]x=Acos(ωt+φ)[/math]. between acceleration and displacement for a simple harmonic motion?.

The acceleration can in fact be written as: All of the equations above, for displacement, velocity, and acceleration as a function of time, apply to any system undergoing simple harmonic motion.

### Introduction to simple harmonic motion review (article) | Khan Academy

What distinguishes one system from another is what determines the frequency of the motion. We'll look at that for two systems, a mass on a spring, and a pendulum. The frequency of the motion for a mass on a spring For SHM, the oscillation frequency depends on the restoring force.

This is the net force acting, so it equals ma: This gives a relationship between the angular velocity, the spring constant, and the mass: The simple pendulum A simple pendulum is a pendulum with all the mass the same distance from the support point, like a ball on the end of a string.

Gravity provides the restoring force a component of the weight of the pendulum. Summing torques, the restoring torque being the only one, gives: For small angular displacements: So, the torque equation becomes: Whenever the acceleration is proportional to, and in the opposite direction as, the displacement, the motion is simple harmonic.

## Introduction to simple harmonic motion review

For a simple pendulum, with all the mass the same distance from the suspension point, the moment of inertia is: The equation relating the angular acceleration to the angular displacement for a simple pendulum thus becomes: This gives the angular frequency of the simple harmonic motion of the simple pendulum, because: Note that the frequency is independent of the mass of the pendulum. If we consider just the y-component of the motion the path with time we can see that it traces out a wave as shown in Flash 2.

Simple Harmonic Motion, is also a component of circular motion. We can also see that the period of the motion is equal to the time it takes for one rotation. The frequency is the number of oscilations per second.

Consider the particle undergoing simple harmonic motion in Flash 3. The displacement with time takes the form of a sinusoidial wave. The velocity of the particle can be calculated by differentiating the displacement.

### Simple Harmonic Motion (SHM)

The result is also a wave but the maximum amplitude is delayed, so that when the displacement is at a maximum the velocity is at a minimum and when the displacement is zero the velocity has its greatest value. Simple Harmonic Motion is characterised by the acceleration a being oppositely proportional to the displacement, y. Displacement, velocity and acceleration vectors of a particle undergoing simple harmonic motion We set this out mathematically, using a differential equation as in equation 4.

We specify the equation in terms of the forces acting on the object. The acceleration is the second derivative of the position with respect to time and this is proportional to the position with respect to time. The minus sign indicates that the position is in the opposite direction to the acceleration.

Intuitively, we can imagine that this velocity occurs when the oscillating system has reached the equilibrium position and is about to overshoot.

The minus sign indicates the direction of travel is in the opposite direction. The maximum acceleration occurs where the argument of cosine in eqn 7 is also -1 or 1. If the mass is displaced from its equilibrium position by a distance x a force F will act in the opposite direction to the displacement. However, if there is no friction the inertia of the mass will cause it to overshoot the equilibrium position and the force will act in the opposite direction slowing it down and pulling it back.

The action of this force on the mass keeps it oscillating backwards and forwards. Spring Mass system undergoing SHM.

The spring mass system consists of a spring with a spring constant of k attached to a mass, m. The mass is displaced a distance x from its equilibrium position work is done and potential energy is stored in the spring. The force on the spring is assumed to obey Hooke's law, therefore, the restoring force is proportional to the extension. For a particle undergoing simple harmonic motion, the displacement x is given by equation 1.

**SHM: Velocity and Acceleration Equations**

The potential energy is gradually transferred to kinetic energy. The velocity is given by equation 6. The blue line shows the potential energy.

- Simple Harmonic Motion (SHM)
- Simple Harmonic Motion