Equations of Simple Harmonic Motion

Question

A wooden ball is tied on string 30 cm long and is oscillating like a pendulum. It is initially pulled back at 900 angle with the vertical, and is released with a push so that the ball reaches its maximum height back and forth.

a. Draw the figure that illustrates the problem.

b. If it reaches its maximum height after 3 seconds, what is the equation of its motion y at time t in seconds?( Hint: h(t) is in the form h(t)= |asinb(t-c)| +d)

c. Find the height of the wooden ball 10 seconds after it was released.

In-depth solution for each question will be highly appreciated!

Answer 1

The ball is at a 90 degree angle with the vertical, and is given an initial push, which means the real answer to this question is that the behavior of the ball will evolve chaotically and its position cannot be precisely determined as an explicit function of time.

Part A:

Draw a ball swinging around starting from a 90 degree angle on the right, and ending at a 90 degree angle on the left.

Part B:

the ball starts at its maximum height at t=0, so the most convenient function to model this behavior is a cosine, which is simply a shifted version of a sine.

note that there is no height above the ground stated in the problem, so we can disregard the + d term.

If h(t) = A * |cos(w * t)|, then we need A = 30cm such that the ball’s initial height is 30cm, and we need w * 3seconds = pi such that the ball has revolved 180 degrees after 3 seconds.

This simplifies to w = pi / 3seconds

-> h(t) = 30 * |cos(pi/3 * t)|

C: plug h(10) into a calculator