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
I can't see the figure, but if A and B are both 10 cm from the center line, then the maximum value of d is 10. This is the amplitude of the sine function desired, so since the period is 2 seconds,
d = 10sin(πt)
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
what don't you like about what I wrote?
How would I draw this
geez - you know how to draw a sine curve, right?
So, draw your generic sine curve, but instead of -1 to 1 on d, label it -10 to 10
and on the t axis, instead of a period going from 0 to 2π, label the axis so it goes from 0 to 2.
Sine curves all look the same. Just label the axes as you need for the amplitude and period.
To sketch the graph of d versus t for 0 < t < 8, we need to understand the properties of a pendulum's motion and the sine function.
A pendulum swings back and forth in an arc, following a periodic motion. The motion can be described by a sinusoidal function, specifically a sine function.
Let's break down the given information:
- The pendulum takes 2 seconds to move from the position directly above point a to the position directly above point b.
- The distance from a to b is 20 cm.
- The pendulum starts at the center line (at time t = 0) and is moving to the right.
Based on this information, we can make the following observations:
- The period of the pendulum's motion is 2 seconds (the time it takes to move from a to b and back to a).
- The amplitude of the motion is half the distance from a to b, which is 10 cm.
Now, let's sketch the graph of d versus t:
1. Start by drawing a horizontal line representing the center line.
2. Mark the point on the center line as the starting position at t = 0.
3. To the right of the starting position, mark a point that is 10 cm away from the center line. This represents the maximum displacement to the right.
4. To the left of the starting position, mark a point that is 10 cm away from the center line. This represents the maximum displacement to the left.
5. Connect the points with smooth curves, resembling a sine wave.
6. Repeat the pattern as the pendulum swings back and forth for 8 seconds.
The resulting graph should show a symmetric sine wave, oscillating between the maximum displacements to the right and left of the center line.
Now, to write a sine equation that describes the graph:
The general equation for a sine function is:
d = A * sin(B * (t + C)) + D
where:
- A represents the amplitude (half the distance between the maximum and minimum values of d).
- B determines the period of the function (2π/B gives the period).
- C is a phase shift that affects the starting position of the function.
- D corresponds to a vertical shift along the y-axis.
Using the given information:
- Amplitude (A) = 10 cm
- Period (2π/B) = 2 seconds
- The pendulum starts at the center line, so there is no phase shift (C = 0).
- There is no vertical shift (D = 0).
Thus, the sine equation that describes the graph is:
d = 10 * sin((π/1) * t)