Download -Energy of SHM -Comparing SHM to Circular Motion

-Energy of SHM
-Comparing SHM
to
Circular Motion
Click here for skyscraper
http://www.youtube.com/watch?v=lFV6NBbjxME
Mrs. Coyle
Objectives
• Energy considerations for a spring- mass oscillator
• Analyzing energy graphs
• Calculating speed of oscillator using conservation of
energy
• Proving the period equation
• Relating SHM to circular motion
• Describing SHM motion using trig functions
• Analyzing motion diagrams
Potential Energy, U, of a SHM Oscillator
Fs
Fa
Fs
Remember:
Work Done by an applied force Fa =kx
to Increase the Potential Energy of a
Spring is
W= ½ k x2
This work increased the potential
energy of the spring by the same
amount.
So at point x, U= ½ k x2
Draw a graph of potential energy,
U= ½ k x2 vs displacement, x for a SHM
oscillator:
U
x
Potential Energy of a Spring
-Draw the
graph of
kinetic energy
vs
displacement
for a SHM
oscillator.
SHM
Graph of
Potential and
Kinetic Energy
vs
Displacement
SHM
Graph of
Potential and Kinetic
Energy
vs
Time
Finding the expression of maximum speed
When the spring is set to SHM,
the total mechanical energy of a SHM at any time is:
E= ½ mv2 + ½ kx2
Write the expression of conservation of energy:
a) At amplitude, A:
b) At maximum speed:
c) Write an expression for maximum speed in terms of
amplitude:
Deriving the expression for period
T=2π
𝑚
𝑘
from maximum speed.
• Hint: Use the previous expression for vmax and
remember the original definition of speed in terms
of period for circular motion.
Relating Circular Motion
and SHM
Assume the wheel is
rotating at a constant
frequency, f, and period T.
Click for animation
https://www.youtube.com/watch?v=9r0HexjGRE4
Expressing x using a trig function
a)Express x in terms of A and ϴ,
using a trig function, when the
object is at point P:
x=
b)Express ϴ in terms of
angular frequency, ω.
(Hint: ω=radians/time)
c)Express x in terms of A, angular
frequency ω, and time t, using a
trig function, when the object is at
point P:
Answer:
x = A cos (ω t)
ω relates to f by ω= 2πf
• Write x in terms of f:
x=
Draw a graph of displacement vs time for a
SHM oscillator whose displacement is given by
x = A cos (2πf t)
x
t
Particle Oscillating in SHM
• http://bcs.wiley.com/hebcs/Books?action=mininav&bcsId=3606&itemId=0
471758019&assetId=111700&resourceId=10211
Problem:
For an object undergoing SHM whose displacement is given
by:
x = 3 cos (628t) in meters,
determine the amplitude, the frequency and the period of
oscillation.
Period of a Spring-Mass SHM Oscillator
T=2π
𝑚
𝑘
a) If you take this oscillator to the moon will the
period change and why?
b) If you double the amplitude will the period
change and why?
c) If you want to double the period what can you
change and why?
Application
Click here for video of testing wooden building on shake table
https://www.youtube.com/watch?v=hSwjkG3nv1c
Summary
• Energy considerations for a spring- mass oscillator
• Analyzed energy graphs
• Calculated speed of oscillator using conservation of
energy
• Proved the period equation
• Relating SHM to circular motion
• Described SHM motion using trig functions
• Analyzed motion diagrams