Energy Conservation – Spring

Simple Harmonic Motion

(SHM)

Is a form of Oscillation occur from the motion of an object.

It can occur in a various type of systems, in this

presentation we’ll use the most common ‘Simple

Harmonic Motion’: A mass on a spring.

Energy Conservation

Ideal Condition : Absent of friction

Total Energy (E) = Kinetic Energy (K) + Potential Energy

(U)

Kinetic Energy = (1/2)mv^2

Potential Energy of the spring = (1/2)kx^2

Therefore, E = (1/2)mv^2 + (1/2)kX^2

1st Case

• Considering these following

cases, where x = 0 is the initial

point.

• The object mass (m) is held

stationary at the position x = X

• By doing so the work has

been done on the object and

stored as a potential energy

(U)

• Because V = 0 (Object is

stationary)

• Kinetic energy (K) = 0

E = K + U = (1/2)mv^2 + (1/2)kX^2

K = 0

Therefore,

E = 0 + (1/2)kX^2

2nd Case

• After the object mass (m) get

released the object move due

to the force stored in the

potential energy (U)

• Since the object is moving

the energy is converted

from potential energy (U)

to kinetic energy (K)

• Therefore, Potential

Energy (U) = 0

E = K + U = (1/2)mv^2 + (1/2)kX^2

U = 0

Therefore,

E = (1/2)m(-v^2) + 0

E = (1/2)mv^2 + 0

3rd Case

• The object mass (m) is

stationary at the position x = -X

• By doing so the work has

been done on the object and

stored as a potential energy

(U)

• Because V = 0 (Object is

stationary)

• Kinetic energy (K) = 0

E = K + U = (1/2)mv^2 + (1/2)kX^2

K = 0

Therefore,

E = 0 + (1/2)k(-X^2)

E = 0 + (1/2)kX^2

Understanding Harmonic

Motion (Spring)

x(t) = A Cos (ωt + Φ)

x(t) = A Cos (ωt + Φ)

Amplitude

Angular Frequency

Initial Phase

Constant

Time

Amplitude (A)

• Amplitude – The height of the waves

• Indicate the y-axis (Displacement, Distance)

• Depend on the graph eg. Position-time graph

Angular Frequency (ω)

• Angular Frequency - A measurement of a rotation rate of a circle.

• Measure by:

ω = (2π / T) = 2πf

Initial Phase Constant (Φ)

• Initial Phase Constant – Indicate the initial phase of a graph, where and

how much graph shift with respect to the x-axis

Q1 : A mass of 10 kg oscillating on a spring passes

through its equilibrium point with a velocity of 15 m/s.

What is the energy of the system at this point?

Questions

Q2 : A box mass of m kg is moving to the left at a

velocity of v m/s due to the force from the spring. The

spring is not fully stretched. Express this in a formula.

Q3 : Sketch a graph of f where Amplitude = 5, Angular

Frequency = 2 and Initial Phase constant = 5

Q1 : K = (1/2)mv^2 = (1/2)(10)(15^2) = 1,125 Joules

Solutions

Q2 : E = (1/2)mv^2 + (1/2)kx^2 (v not equal to 0)

Q3 :