Question

Starting at its rightmost position, it takes 2 seconds for the pendulum of a grandfather clock to swing a horizontal distance of 18 inches from right to left and 2 seconds for the pendulum to swing back from left to right. Which of the following equations models d, the horizontal distance in inches of the pendulum from the center as a function of time, t, in seconds? Assume that right of center is a positive distance and left of center is a negative distance.
a) d=9cos(pi/4 t)
b) d=9cos(pi/2 t)
c) d=18cos(pi/4 t)
d) d=18cos(pi/2 t)

Answer 1

Answer:

The correct answer is b,  x = 9 cos (pi / 2 t)

Explanation:

The equation that describes a simple pendulum is

             θ  = θ₀  cos (wt + φ)

The angle is measured is radians

            θ = x / L

We replace

           d / L = x₀ / L cos (wt + φ)

            x₀ = 9 in

         

We replace

             d = 9 cos (wt + φ)

Angular velocity is related to frequency and period.

           w = 2π f = 2π / T

The period is the time of a complete oscillation T = 4 s

           w =2π / 4

           w = π / 2

Let's replace

             x = 9 cos (π/2 t + φ)

As the system is released from the root x = x₀ for t = 0 s

              x₀ = x₀ cos φ

             Cos φ = 1

             φ = 0°

The final equation is

             x = 9 cos (pi / 2 t)

The correct answer is b

Answer 2

Answer:

b) d=9cos(pi/2 t)

Explanation:

This is a cosine function in the such as: y = a cos bt...

a = (maximum distance - minimum distance)/2:

a = (max - min)/2

maximum distance = 18 inches minimum distance = 0

a = (18 - 0)/2 = 18/2 = 9

That is a = 9

To solve for b, similar to the period:

The period in radians:

P = 2pi/b, which is the amount of time it takes to revolve one full cycle...

multiply that time by time from minimum to maximum, 2 to give the period as 2 × 2 = 4.

Thus to find b..

4 = 2pi/b

4b = 2pi/b × b

4b = 2pi

or

b = pi/2

Also

a = 9

b = pi/2

So our cosine function is:

d = 9cos((pi/2)t)

Hence the equation that models d