Simple Harmonic Motion

1. Energy Conservation – Spring

2. 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.

3. 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

4. 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

5. 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

6. 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

7. Understanding Harmonic
Motion (Spring)
x(t) = A Cos (ωt + Φ)

8. x(t) = A Cos (ωt + Φ)
Amplitude
Angular Frequency
Initial Phase
Constant
Time

9. Amplitude (A)
• Amplitude – The height of the waves
• Indicate the y-axis (Displacement, Distance)
• Depend on the graph eg. Position-time graph

10. Angular Frequency (ω)
• Angular Frequency - A measurement of a rotation rate of a circle.
• Measure by:
ω = (2π / T) = 2πf

11. 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

12. 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

13. 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 :