# Problem: The shadow of a pendulum cast on a 7t 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) = 52cos(πt), where t is in seconds and x is in centimetersPart (a) Find the speed, in centimeters per second, of the shadow at t = ¼ sPart (b) Find the speed, in centimeters per second, of the shadow at t = ½ sPart (c) Find the magnitude of the acceleration, in meters per second squared, of the shadow’s motion at t = 1/3 sPart (d) How much distance, in centimeters, does the shadow travel in 20 sec?

###### FREE Expert Solution

Velocity is expressed as:

$\overline{){\mathbf{v}}{\mathbf{=}}\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}}$

Acceleration:

$\overline{){\mathbf{a}}{\mathbf{=}}\frac{\mathbf{d}\mathbf{v}}{\mathbf{d}\mathbf{t}}}$

Part (a)

From our velocity equation:

$\begin{array}{rcl}\mathbf{v}& \mathbf{=}& \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}\mathbf{\left[}\mathbf{52}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{\pi }\mathbf{t}\mathbf{\right)}\mathbf{\right]}\\ & \mathbf{=}& \mathbf{-}\mathbf{52}\mathbf{\pi }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{\pi }\mathbf{t}\mathbf{\right)}\end{array}$

89% (12 ratings) ###### Problem Details

The shadow of a pendulum cast on a 7t 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) = 52cos(πt), where t is in seconds and x is in centimeters

Part (a) Find the speed, in centimeters per second, of the shadow at t = ¼ s

Part (b) Find the speed, in centimeters per second, of the shadow at t = ½ s

Part (c) Find the magnitude of the acceleration, in meters per second squared, of the shadow’s motion at t = 1/3 s

Part (d) How much distance, in centimeters, does the shadow travel in 20 sec?

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Simple Harmonic Motion of Pendulums concept. You can view video lessons to learn Simple Harmonic Motion of Pendulums. Or if you need more Simple Harmonic Motion of Pendulums practice, you can also practice Simple Harmonic Motion of Pendulums practice problems.