Graphs of Harmonic Motion
1. The following graph represents a 0.40 kg mass oscillating horizontally on a spring.
a) Determine the amplitude, period, and frequency from graph.
The amplitude of this graph is 4 cm. I determined this by looking and the maximum and minimum values,
which were 4 and -4. This meant the total distance traveled by the mass was 8 cm. The amplitude is half that
number, 4 cm.
The period of this graph is 2 s. I just went from the first max (t = .5) to the second max (t = 2.5).
Since f = 1/T, f = ½ Hz.
b) Write the equations for the sinusoidal curves using sine or cosine.
From Friday, and yesterday’s videos, I know y = Amplitude*trig(ωt).
Since, the graph starts at a minimum amplitude, it is a sine function. I learned this in precalc/alg II,
whatever. But, when t = 0, sin (0) = 0. This matches with the initial point on the graph. This is how I know
it is a sine function.
I found the amplitude was 4 cm in part (a).
To find ω, use the equation ω = 2πf. So, ω = 2π(1/2) = 3.14. I have cut and pasted the way this equation
shows up on the formula sheet.
The equation is: y = 4sin(3.14t). Or if you leave π in your answer, y = 4sin(πt).
c) Determine the spring constant for the spring.
0.4
2 = 2π√ . We found out the other day this is very hard for us to solve. The correct answer is 3.94 N/m.
𝑘
d) Calculate the maximum acceleration of the mass.
kx = ma
3.94(0.04) = 0.4a
a = 0.394 m/s2
4 cm = 0.04 m
Graphs of Harmonic Motion
2. The following graph represents a pendulum swinging to and fro.
a) Determine the amplitude, period, and frequency from graph.
To find the amplitude, I found the maximum values are 4 and the minimum values are –2. This is a
difference of 6 cm. Half of that is the amplitude, 3 cm. Also, if you see the red line I drew right through the
middle of the function? This is like the “new x-axis”. From that line, the amplitude is 3 cm each way.
Period is 3 s. Frequency is 1/3 Hz.
b) Write the equations for the sinusoidal curves using sine or cosine.
This curve begins at a maximum (or minimum) value. This represents a cosine curve.
To find ω, use the equation ω = 2πf. So, ω = 2π(1/3) = 2.09. I have cut and pasted the way this equation
shows up on the formula sheet.
2
The equation is: y = 3cos(2.09t). Or, if you leave π in your answer 𝑦 = 3cos⁡( 𝜋𝑡).
3
c) Determine the length of the pendulum.
𝐿
3 = 2π√
10
L = 2.282 m
Graphs of Harmonic Motion
3. A 0.80 kg mass vibrates according to the equation, 𝑥 = 0.3cos⁡(8.0𝑡), where x is in meters and t is in
seconds. Determine the
a) Amplitude
The amplitude is 0.3 m
b) Period
ω=8
T = 2π/8 = π/4 = 0.785 s
c) Frequency
f = 8/2π = 4/π = 1.2732
d) Spring constant
0.8
0.785= 2π√ . The correct answer is 51.2 N/m.
𝑘
e) Total distance the mass travels during 1 period.
For one cycle, the mass will travel 0.3m four times. The answer is 1.2 m.
Graphs of Harmonic Motion
𝜋
4. A pendulum swings according to the equation, 𝑥 = 2.4sin⁡( 𝑡), where x is in meters and t is in
4
seconds. Determine the
a) Amplitude
2.4 m
b) Period
ω = π/4
T = 2π/(π/4) = 8 s
c) Frequency
f = 1/8 Hz
d) Length of the pendulum.
𝐿
8 = 2π√
10
L = 16.2278 m
e) Total distance the bob travels during 1 period.
4(2.4) = 9.6 m