Simple Harmonic Motion of Pendulums Video Lessons

Concept

# 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:

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

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###### 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?