Sinusoidal Equation of Pendulum
The pendulum on a grandfather clock moves in a SINUSOIDAL fashion. The length of the pendulum is 31 inches at its lowest point. The pendulum is 1.5 feet from the floor. The pendulum begins to move at t1 so that the interior angle between t2 and t3 is 90 degrees. It takes .875 seconds for the pendulum to swing from t2 to t3.
- Write the sinusoidal equation that models the movement of the pendulum as a COSINE FUNCTION
- How high will the pendulum be in one hour ?
- What is the minimum and maximum height the pendulum will reach?
- To model the movement of the pendulum as a cosine function, we need to identify the amplitude, period, and vertical shift of the function.
The amplitude of the function is half the distance between the highest and lowest points of the pendulum’s swing, which is (1.5 ft) – 31 inches = 1.0833 ft.
The period of the function is the time it takes for one complete swing of the pendulum, which can be calculated using the length of the pendulum and the acceleration due to gravity. The formula for the period is T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Plugging in the values, we get:
T = 2π√(31/12 / 32.2) = 1.974 seconds
Since the pendulum completes one full swing (from t2 to t3 and back) in 0.875 seconds, this is only half of one period. Therefore, the frequency of the function is f = 1/T = 0.506 Hz.
The vertical shift of the function is the average height of the pendulum, which is halfway between the lowest and highest points. Since the pendulum is symmetrical, this is also the same as the height of the pendulum when it is at rest, which is (1.5 ft + 31 inches)/2 = 1.2917 ft.
Putting all of this together, the equation that models the movement of the pendulum as a cosine function is:
h(t) = 1.0833 cos(2π(0.506)t) + 1.2917
