Question

Due to tides mean sea level off of Newport Beach reaches a height of 1.3 meters during high

tide and 0.3 meters during low tide. Successive high tides occur every 12 hours (43,200

seconds). A buoy with mass m = 40 kg is floating in the ocean off of Newport Beach.

1) Relevant concepts/equations. (5 points.)

2) Assume we begin to measure the buoy’s displacement at High tide which occurs exactly

at 12:00 am (0 seconds). Also assume we can model the buoy’s displacement as a simple

undamped oscillation. What is the Amplitude and phase angle for the buoy’s

displacement? (10 points)

3) During one half cycle of six hours (21600 seconds), the buoy’s displacement passes

through an angle of 180 degrees. From this information, what is the angular frequency

ω of the buoy? (5 points)

4) Using your previous answer, what is the force constant ‘k’ acting on the buoy? (5 points)

5) What is the maximum velocity of the buoy? What is the maximum acceleration of the

buoy? (10 points)

6) What is the energy of the buoy due to tidal displacement? (5 points)

7) How much work is done during one low tide to high tide cycle? How much Power per

hour is required to accomplish this? (Assume g = 9.81 m/s^2 compare your answer to a

65W light bulb which uses 65 watts per hour). (10 points)

Answer 1

1) The vertical height of the buoy can be modeled as a sinusoidal function of time. We can define an equilibrium position and then the maximum vertical displacement with respect to this position. This peaks will repeat themselves when a full period goes by. The general equation for this type of motion is:

Where A is the amplitude of the motion and T the period.

2) If we use the SHO model and start measuring the motion of our buoy at high tide:

Where y is in meters and t in hours. The amplitude is A=0.5m and the period is T=12hs

3) Angular frequency:

4) The constant k for the buoy:

5) Maximum velocity and maximum acceleration:

6) The energy of the buoy is given by:

It can also be caclulated using

7) The work done during one low tide to high tide cycle is given by:

The power per hour required is: