Simple Harmonic Motion

Chapter 12 Section 1

Periodic Motion

A repeated motion is what describes Periodic Motion

Examples: Swinging on a playground swing Pendulum of a clock Wrecking ball Springs and oscillators Etc…

Mass-Spring System

x = 0

Equilibrium

Maximum Displacement

Maximum Displacement

-x +x

Felastic

Felastic

Felastic = 0

Velocity In Periodic Motion

At the equilibrium position, the mass in periodic motion reaches its maximum velocity.

At the maximum displacement away from the equilibrium position, the velocity will be zero.

Acceleration In Periodic Motion The acceleration of a mass is greatest at the

maximum displacement away from the equilibrium position.

The acceleration of a mass is zero at the equilibrium position.

Graphing Position, Velocity & Acceleration as a Function of Time

Time (sec)

Pos

ition

(m) /

Vel

ocity

(m/s

) / A

ccel

(m/s

2 )

Forces In Periodic Motion

According to Newton’s Laws of Motion if there is a net force acting on an object, the object will acceleration in the direction of the net force.

The acceleration of the mass is greatest when the force is greatest. This is when the mass reaches the maximum

distance away from the equilibrium position. The direction of the spring force will always

point towards the equilibrium position.

Ideal Periodic Motion

In an ideal system the mass in a periodic motion will oscillate indefinitely.

But, there is always friction present in the physics world which slows down the motion. This slowing down motion is called damping.

Damping is minimal during short periods of time, so it can be considered an ideal system.

Restoring Forces

The forces acting on the object will always pull it towards the equilibrium position. This is why it is sometimes called the Restoring

Force. The restoring force is proportional to the

displacement.

Simple Harmonic Motion

Simple Harmonic Motion – Vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium. This can be seen in most springs.

Hooke’s Law

In the case of a mass-spring system, the relationship between force and displacement, discovered by Robert Hooke in 1678, is known as Hooke’s Law.

Felastic = -kx

Spring Force = -(Spring Constant) (Displacement)

Explanation of Hooke’s Law

The negative sign is needed because the direction of the spring force is always opposite the direction of the displacement.

Spring Constant – “k” – Is a measure of the springs stiffness. Larger the “k”, the more force is needed to stretch

the spring. SI units – N/m

The law only holds true if it doesn’t pass the springs elastic limit.

Example Problem

A 76N crate is attached to a spring that has a spring constant of 450N/m. How much displacement is caused by the weight of this crate?

Example Problem Answer

-0.17m

Energy Within a Spring

A stretched or compressed spring has elastic potential energy.

Once the mass on the spring starts to move, the potential energy is transformed to kinetic energy.

The total mechanical energy remains constant through out the motion of the mass.

Elastic Potential Energy

Graphing PE, KE & ME as a Function of Time

Time (sec)

KE

(J) /

PE

(J) /

ME

(J)

The Simple Pendulum

Bob All the mass of the bob is concentrated at a point (center of

mass) Fixed string

Top attached to a fixed position and the bottom end attached to bob

There is no air resistance and the mass of the string is negligible.

θ

Restoring Force of a Pendulum The restoring force of a pendulum is a

component of the bob’s weight Perpendicular to the string. The perpendicular component of the weight force

is what pulls the bob towards the equilibrium position.

Hence the restoring force.

Force Diagram of a Pendulum

θ

Simple Harmonic Motion of a Pendulum A pendulum is a simple harmonic oscillator

as long as the angle of displacement is small. Less than 15 degrees

At any displacement, a simple pendulum has gravitational potential energy. Mechanical Energy is conserved. Potential energy is from gravity, not elastic like a

mass-spring system. Graphs of KE and PE are exactly the same for

both the pendulum and mass-spring system.

Simple Harmonic Motion Diagram