Question

The shadow of a pendulum cast on a flat board moves on a straight line. By placing the x-axis on the straight line with the origin at the middle of the total path, the x-coordinate of the shadow is given by the following function: x(t) = 34cos(πt), where t is in seconds and x is in centimeters.
How much distance, in centimeters, does the shadow travel in 20 sec?

Answer 1

Answer:

1360 cm

Explanation:

Oscillating Motion

The shadow of  a pendulum's x-coordinate is given by

x(t) = 34cos(πt)

Where 34 cm is the amplitude of the oscillation and $w=\pi$ is the angular frequency. Each time the shadow completes a whole cycle, it travels four times the amplitude: Twice from the peak to the valley and twice from the valley back to the peak. This means that each cycle the shadow travels 4*34=136 cm

Let's find the period of the oscillations. Given w, we can know the period by the formula

$\displaystyle T=\frac{2\pi}{w}=\frac{2\pi}{\pi}=2\ sec$

If we want to calculate the distance traveled by the shadow in 20 seconds, we can see it makes 20/2=10 full cycles, each one traveling 136 cm, thus the total distance traveled is 10*136=1360 cm