Tides and Simple Harmonic Motion
Date: 08/11/98 at 22:08:27 From: William Liao Subject: Simple Harmonic Motion Dr. Maths, The question is as follows: It is accepted that the rise and fall of the tide at a particular inlet is simple harmonic, with the time difference between successive high tides being 10 hours. The entrance to the inlet has a depth of 20m at high tide and 8m at low tide. If the low tide occurs at 10:00 a.m. on a certain day, find the earliest that a vessel requiring a minimum water depth of 15 m can pass through the entrance. I have identified that since T = 2pi/omega, and T = 10 hours = 36000 seconds in this case, then omega must equal to pi/18000. Also I have found a, the amplitude, to be 6m. This is obtained by 20m - 8m = 12m, but since this is its total displacement, the amplitude must be 6m as it oscillates 6m up and down, which makes up the 12m. I also found the desired displacement in this case to be -1, since low tide + medium tide is 8 + 6 = 14m, but 15m means it has to oscillate backwards for 1m, which is -1. I then used the equation for displacement, x = a cos(omega*t), substituting all the value except for t (which is what I have to find), I got -1 = 6 cos ((pi/18000)t). However, the answer I have obtained is just simply not correct, and I don't know why, or which step I have done is incorrect. The answer given on the back of my textbook is 12:46 p.m., and I have tried to work backwards to see what I did wrong, but that didn't seem to help either. Can you please point out what I did wrong? Thank you. Sincerely, Will
Date: 08/12/98 at 09:05:43 From: Doctor Rick Subject: Re: Simple Harmonic Motion Hi, Will. Thanks for going through your thinking so clearly. You haven't done anything wrong as far as you have stated it. I think that all you need is to be a little more careful in understanding the meaning of quantities. In particular, it would help to write an equation. We need to decide how to define the time. You chose to express time in seconds, but I don't see anything about when time starts. Let's say that time t is 0 at 10:00 A.M., the time of low tide. Then the equation for water depth, h, at the entrance to the inlet (measured in meters), is: h = 14 - 6*cos(2pi * t / T) where T is the period, 10 hours, or 36000 seconds. Do you see how I got this? As you said, the mean tide is 14m, and the amplitude is 6m. I chose a negative cosine so that the sinusoid will have its minimum at t = 0 -- the low tide. And you understand the term inside the cosine, omega * t. Now if I set h = 15m and solve for t, I get: t = 36000/(2pi) * arccos(-1/6) = 5729.6 * 1.73824 = 9959.4 sec = 2.7665 hours = 2 hours + (0.7665 * 60) min = 2 hours + 45.99 min Finally, remember that t is the time after 10:00 A.M., so the time you're after is 12:46 P.M. In other words, I think the only thing you forgot is to add your number of seconds to 10:00 A.M. By writing the equation carefully, and spelling out at the beginning what t means, you can be sure you're doing it right. I hope this helps. - Doctor Rick, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 08/12/98 at 09:32:14 From: William Liao Subject: Re: Simple Harmonic Motion Dr. Rick, Thank you for your explanation. Your calculation seems perfectly correct to me. However, there is still one thing I don't quite understand. You stated that t = 36000/(2pi) * arccos(-1/6). 36000/(2pi) would be omega of course. According to my textbook, T = 2pi/omega, and T equals to 36000 in this case, so wouldn't 36000 = 2pi/omega, which means omega = 2pi/36000 = pi/18000? Why is it 36000/(2pi)? Sincerely Will
Date: 08/12/98 at 10:32:05 From: Doctor Rick Subject: Re: Simple Harmonic Motion Hi again. You are correct that omega = 2pi/T. This is why the equation contains cos((2pi/T) * t). The coefficient of t inside the cosine is omega. Let's examine in more detail how I solved the equation for t: h = 14 - 6*cos((2pi/T) * t) h - 14 = -6*cos((2pi/T) * t) 14 - h = 6*cos((2pi/T) * t) (14 - h)/6 = cos((2pi/T) * t) arccos((14 - h)/6) = (2pi/T) * t (T/2pi) * arccos((14 - h)/6) = t You see that I had to _divide_ the inverse cosine by omega (2pi/T) in the process of solving for t. I hope this makes it clearer. - Doctor Rick, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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