Physics for the survivors
6. lecture
...after the test
Goals of today:
• Understand the oscillations
and its relationship with the sinus function
Group work!!
• Groups of 2-3 people
• Answer questions (Plickers)
The pendulum – your favourite
𝜃
If we displace the
mass out of equilibrium, it s
wings
Let us denote with 𝜃 the
angle the current position
of the mass and
equilibrium
Exercise 1: Show graphically the dependence of
angle 𝜃 on time. The first point is given.
Exercise 2: Plot the time-dependency of the rate
of change of the angle 𝜽.
Exercise 3: Plot the dependence of the angle 𝜃 on time
if the initial displacement angle is 2 times greater than
in exercise 1.
Exercise 4: Plot the dependence of the angle 𝜃
vs time for a pendulum moving 2x faster than in
exercise 1.
Exercise 1: Show graphically the dependence of
angle 𝜃 on time. The first point is given.
Exercise 2: Plot the time-dependency of the rate
of change of the angle 𝜽.
Exercise 3: Plot the dependence of the angle 𝜃 on time
if the initial displacement angle is 2 times greater than
in exercise 1.
Exercise 4: Plot the dependence of the angle 𝜃
vs time for a pendulum moving 2x faster than in
exercise 1.
Let's re-familiarize ourselves with
3 concepts:
• Oscillation period
• the time interval during which the body performs a complete
oscillation and returns to its original position
• symbol: T
• SI unit: s (second)
• Oscillation frequency
• the number of oscillations per unit of time
• symbol: ν (greek “nu”) vai f
• SI units: Hz (hertz) or 1/s (1 over a second)
• Oscillation amplitude
• Maximum change in a quantity relative to equilibrium
• symbol: A (sometimes) depends on the type of oscillator
• Units: whatever are the units of the quantity undergoing
oscillation
T

1

1
T
What is the quantity shown?
A.Frequency
B.Period
C.Amplitude
D.None of the
above
What is the quantity shown?
A.Frequency
B.Period
C.Amplitude
D.None of the
above
What is the quantity shown?
A.Frequency
B.Period
C.Amplitude
D.None of the
above
The formula for describing the oscillation
𝜃 = 𝐴 ∙ sin(𝜔𝑡)
Sin(x)
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
315
330
345
360
0
0.258819
0.5
0.707107
0.866025
0.965926
1
0.965926
0.866025
0.707107
0.5
0.258819
0
-0.25882
-0.5
-0.70711
-0.86603
-0.96593
-1
-0.96593
-0.86603
-0.70711
-0.5
-0.25882
0
2*sin(x)
sin(2x)
How does the fluctuation graph change with A?
How does the fluctuation graph change when changing 𝜔?
Fill in the table to the end and plot it!
By increasing A, the amplitude of the oscillation _______________________
By increasing 𝜔, the amplitude of the oscillation ______________________
Increasing A, the oscillation period _________________________
Increasing 𝜔, the fluctuation period __________________________
The pendulum, a derivation. What are the
forces?
• Gravity (mg)
• Tension
The cosine part of mg is
counteracted by the tension
Arc length is equal to
radius times angle (in
radians)
Length of the pendulum
is constant so velocity is
just the instantaneous
change in the angle
Acceleration is the
change in velocity over
time (or the second order
change in position)
Rearrange terms
Small angle
approximation
A function with this property is the
sine (or cosine or both).
2
x  A sin( t   )  A sin(t   ) 
T
T  2
l
g
Suppose we have a pendulum of
length l
We displace it by an angle 𝜃 and measure that its
period is T. Next we turn displace it by an angle 2𝜃.
How will the oscillation period change?
A.The period will be 2T
B. The period will be 0.5 T
C. The period will be 4 T
D. The period will not change
Very important! (for small
deviations)
Period and frequency are properties of a
mathematical pendulum. There can be only one
oscillation frequency for a pendulum of a given
length.
Amplitude is something that we can change almost
arbitrarily (as long as the small deviation law holds).
• The pendulum has an amplitude A (maximum
displacement angle is A) and period T. What is the
displacement after time period T?
A.
0
B.
A
C.
2A
D.
4A
• The pendulum has an amplitude A and a
period T. After how long has the pendulum
traveled 6A?
A.
3/4 T
B.
1+1/4 T
C.
1+1/2 T
D.
2T
• Are there non-conservative forces
at play?
• What energies are present here?
• If you know the amplitude, can
you tell me the energy?
θ
l
In which position does the pendulum
have the maximum potential energy?
A. 1 and 3
B. 2 and 4
C. 1 and 4
D.Another answer
1
2
3
4
In which position does the pendulum
have the maximum kinetic energy?
A. 1 and 3
B. 2 and 4
C. 1 and 4
D. Another answer
1
2
3
4
In which position does the pendulum
have the maximum total energy?
A. 1 and 3
B. 2 and 4
C. 1 and 4
D. Another answer
1
2
3
4
• The pendulum has an amplitude A (maximum
displacement angle is A) and period T. What is the
distance travelled after time period T?
A.
B.
0
𝟐𝑻𝑨
𝟒𝑻𝟐
C.
𝑨
𝟐𝝅𝒈
𝟐𝝅
D.
/𝑻𝑨
𝒈