1.
Damped
Oscillations
Author: Hantao Mai
Physics Learning Objective

2.
Key Questions Being Answered
 1) What are damped Oscillations?
 2) What variables are considered in calculating damped
oscillations?
 3) What is an underdamped oscillator?
 4) What is a critically damped oscillator?
 5) What is an over-damped oscillator
 6) What is the energy and energy loss in a Damped Harmonic
Oscillator?
 7) What questions might be asked on damped oscillation?

3.
Question 1: What are damped Oscillations?
 Definition: Oscillations In the presence of frictional forces
are called damped oscillations.
 The formula F=-bv is used to determine the drag force an
object experiences when moving though a medium.
 b = The damping constant or drag constant
 V = velocity of the object moving through the medium
 The negative sign in the formula indicates that the
direction of the drag is opposite of the velocity of the
object.

4.
Question 1: What are damped Oscillations?
(cont.)
 It has been shown that two facts are true about damped
oscillations:
 1) The amplitude of a damped oscillator decreases
exponentially with time.
 2) The oscillation frequency of a damped oscillator is lower
than the oscillation frequency of an undamped oscillator.
 The equation used to find the displacement of a damped
oscillator is:
 X(t) = Ae^(-bt/2m)cos(W(d)t + Φ)

5.
Question 2: What variables are considered in
calculating damped oscillations?
 As seen in the last slide, the equation to determine the displacement
of a damped oscillator is:
 X(t) = Ae^(-bt/2m)cos(W(d)t + Φ)
 The variables considered include:
 W(d) = (k/m) – (b^2/4m^2)
 K is the spring constant
 M is the mass
 B is the damping constant
 W(d) is known as the oscillation frequency in the presence of drag
force.
 W(0) (which is sqrt (k/m)) is called the natural frequency and is when
b = 0.

6.
Question 3: What is an underdamped
oscillator?
 An Underdamped oscillator is when the natural frequency
(W(0)) is larger than the fraction b/2m.
 W(0) > (b/2m)
 Underdamped oscillations are when the natural frequency
(W(0)) is close to the oscillation frequency (W(D)).
 The main effect of drag force in underdamped oscillations
is to decrease the amplitude exponentially with time.

7.
Question 4: What is a critically damped
oscillator?
 A critically damped oscillator is when the natural frequency is equal
to b/2m
 W(0) = (b/2m)
 In this case, the oscillation frequency (W(D)) is ) so there is no back-
and-forth motion in the oscillator.
 The question X(t) = (a(1) + a(2)t)e^(w(0)t) is used to find the position
of a critically damped oscillator where:
 A(1) and A(2) are determined by the initial conditions X(0) and X(1).
 Critically damped oscillators reach its equilibrium position faster than
any other value of the damping constant.
 I.e. These oscillators quickly return to their natural position.

8.
Question 5: What is an over-damped
oscillator?
 An over-damped oscillator is when the value of (b/2m)
surpasses that of the natural frequency.
 W(0) < (b/2m)
 As the value of b increases, it takes an increasing amount
of time for the oscillator to reach back to its natural
position.

9.
Question 6: What is the energy and energy
loss in a Damped Harmonic Oscillator?
 The Energy in a damped harmonic oscillator is given by
the equation:
 E(t) = (1/2)(kA^2e^(-bt/m))
 E(0) = (1/2)kA^2
 The fraction energy loss in one oscillation is given by the
equation:
 1 – e^(-bT(D)/m)
 B is the damping constant
 T(D) is the period 2pi/W(D)

10.
Question 7: What questions might be asked
on damped oscillation?
 Using the example given in the textbook, we are asked to
find the damping constant (b) given the information:
 M = 2kg
 K = 10N/m
 A= 25cm at t=0
 The amplitude falls to 75% of its initial value after 4
oscillations.

11.
Question 7: What questions might be asked
on damped oscillation?
(cont.)
 Step 1: Calculate the total energy
 E(0) = (1/2)kA^2 = 0.312J
 Step 2: Calculate the period
 T = 2pi/(W(0)) = 2pi*Sqrt(m/k) = 2.81s
 Step 3: Use the information that the amplitude falls off by
75% in for oscillation.
 A(4T) = 3/4A
 e^(-2bt/m)=3/4

12.
Question 7: What questions might be asked
on damped oscillation?
(cont.)
 Step 4: Solve for b.
 e^(-2bt/m)=3/4
 -2bt/m = ln(3/4)
 b = -m/2T*(ln3/4)
 b = -(2/2*2.81)*ln(3/4)
 b = 0.102kg/2

13.
Conclusion
 The question that were asked in this power point only
cover the main concepts of the ideas of damped
oscillation.
 If you have any questions that could be added to this
power-point, feel free to message me and I will add them
onto this slideshow.
Thank You for Watching