A pendulum swings uniformly back and forth, taking 2 seconds to move from the position directly above point A to the position directly above point B.
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A . B
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(SIMPLE DRAWING)
The distance from A to B si 20 centimeters. Let d(t) be the horizontal distance from the pendulum to the center line at time t seconds (with distances to the right of the line measured by positive numbers and distances to the left by negative ones.) Assume that the pendulum is on the center line at time t= 0 and moving to the right. Assume that the motion of the pendulum is simple and harmonic. Find the rule of d(t).
velocity = 20cm / 2sec. = 10 cm /sec.
A B
d(cm) -10 -8 -6 -4 -2 0 2 4 6 8 10.
t(sec) 1 .8 .6 .4 .2 0.2.4.6.8 1.
The period of the oscillations or
vibrations is the time required for
the pendulum to go from A to B and
back to A ( 40 cm ).
T = 40 cm / 10cm/sec. - 4 sec. =
Period
f = 1 / T = 1 / 4 = .25 cycles per
sec = 0.25 hertz.
A B
d(cm)-10 -8 -6 -4 -2 0 2 4 6 8 10
t(se) 1 .8 .6 .4 .2 0 .2 .4 .6.8 1
Christa, the data on the time line
should be shifted one space to the right.
d(t) = -10cospi/2(x-2)+10
a pendulum swings back and forth taking 2 seconds to move from the position directly above from point a to the position directly above point b, as shown in the figure below. the distance form a to b is 20 cm. let d be the horizontal distance from her pendulum to the (dashed) centre line at time t seconds (with distances to the right of the line measured by positive numbers and distances to the left by negative ones). assume that the pendulum is on the centre line at time t = 0 and moving to the right.
a) assume that the motion of the pendulum can be described by a sinusoidal function. Sketch the graph of d versus t for 0<\ t <\ 8.
b) write a sine equation that describes your graph
To find the rule of d(t), we need to determine how the horizontal distance from the pendulum to the center line changes with time.
Given that the pendulum takes 2 seconds to move from point A to point B, we can determine the average speed of the pendulum during this time. The total distance traveled by the pendulum is 20 centimeters (the distance from A to B), and it takes 2 seconds to cover this distance. Therefore, the average speed is:
Average speed = Total distance / Time taken
= 20 cm / 2 s
= 10 cm/s
Since the motion of the pendulum is simple harmonic, its velocity is constantly changing as it swings back and forth. However, we can determine the initial velocity at time t = 0, which is the velocity when the pendulum is at the center line and moving to the right. Since the pendulum covers 10 cm in 1 second, its initial velocity is +10 cm/s.
Now, let's define the rule of d(t) as the horizontal distance from the pendulum to the center line at time t seconds. Based on the information given, the pendulum starts at the center line moving to the right, so its initial position is 0 cm.
We can model the motion of the pendulum using a sine function, which represents simple harmonic motion:
d(t) = A * sin(ωt + φ) + C
In this equation:
- A represents the amplitude of the pendulum's motion (the maximum horizontal distance from the center line)
- ω (omega) represents the angular frequency of the pendulum's motion
- φ (phi) represents the phase angle or initial phase of the pendulum's motion
- C represents the vertical shift or offset of the pendulum's motion (in this case, representing the initial position at t = 0, which is 0 cm)
Since the pendulum moves symmetrically about the center line, we observe that at t = 1 second, it reaches the maximum distance to the right (A), and at t = 2 seconds, it reaches the maximum distance to the left (-A).
Given that the total distance from the pendulum's maximum displacement to the center line is 20 cm, we can determine the amplitude (A) by dividing the total distance by 2:
A = Total distance / 2
= 20 cm / 2
= 10 cm
To determine the angular frequency (ω), we can use the relationship between the period (T) and the angular frequency:
T = 2π / ω
The period is the time taken for a complete oscillation, which is 2 seconds in this case. Solving for ω:
2 = 2π / ω
ω = 2π / 2
ω = π
Now we can write the rule of d(t) with the values for A, ω, and C:
d(t) = 10 cm * sin(πt + φ) + 0 cm
Since the pendulum starts at the center line moving to the right, at t = 0, its initial phase angle (φ) is 0.
Therefore, the rule of d(t) for the given pendulum is:
d(t) = 10 cm * sin(πt)