I need help on my math work

Question

The period of a simple pendulum is represented by the formula T=2\pi \sqrt(((3)/(-32))) where T is the period of the pendulum in seconds and L is the length of the pendulum in feet. Suppose a 3 foot pendulum is displaced a distance of 0.4 feet from its middle position

Write an equation of the form [y=Acos(wt)] that models the position of the pendulum with respect to time.

Answer 1

The equation for the period of a pendulum is T = 2π√(L/g) but to use this equation, just use the positive value of g. If you insist on making it negative, then consider the L as negative as well. Either way, you end up with T = 2π√(3/32) = π√6/4 which is approximately 1.924 seconds. Please note that the omega(ω) in the equation you wrote is 2π divided by the period so, in this case, ω = 4√6/3 or approx 3.266.

You ask for an equation in the form y = Acos(ωt) so I'm assuming you want an equation the models the height of the mass as a function of time. Often times, it is easier to model a pendulum using the angle as a function of time because the motion has both x and y components. But I'll assume you want to model height (y). To do that, we will need to make a slight change to the period. The period is defined as one complete cycle dropping down from the high point, to the middle, going up to the high point on the opposite side then returning again. Notice that when considering the height, you actually go through 2 complete periods during that time? So, to model y using this, we need to have the normal period represent 2 complete "height cycles". That means the omega we use will need to be divided by 2 becoming ω = 2√6/3.

Assuming (0, 0) is at the bottom of the pendulum with the pendulum not moving, to move it left be 0.4 feet means you would be moving it up as well. The amount you move it up can be solved by drawing some right triangles. With 3 as the hypotenuse and 0.4 as the x-dimension, then the y-dimension (length from the pivot point down) is √(3² - 0.4²) = √8.84. That makes the height above the origin at 3 - √8.84 (0.0268 feet)

So at t = 0, y = 3 - √8.84. The at t = 1/4 period (or π√6/16), y = 0. The at t = 1/2 period (or π√6/8), y = 3 - √8.84 again, and then the pendulum begins to swing back so that at t = 3/4 period (or or 3π√6/16) and then finally at t = 1 period, the height is again at y = 3 - √8.84.

Since we are using cosine, we want the max to be at time zero and the minimum to be at 1/4 of the period and to be at y = 0. To achieve that, we need to move the cosine function up by half that max height and the cut the amplitude in half.

So the equation becomes y(t) = [(3 - √8.84)/2]cos(2√6/3t) +(3 - √8.84)/2. If you plug that into your TI-84 or into demos, you will get a graph showing the height as a function of time.

This is a bit complicated. It makes me think the question really wanted you to write a function in terms of the angle θ as a function of time.