A buoy oscillates in simple harmonic motion as waves go by. The buoy moves a total of 3.5 feet from its low point to its high point and it returns to its high point every 10 seconds.
What is the equation that describes this motion if it reaches its high point is at t=0?
Well there are lots of equations but let's take a cosine function because that is a simple one which is maximum at t = 0.
Call it y = a cos ( w t )
that has maximum a and minimum -a and is a at t = 0
now if it is to be 3.5/2 = 1.75 at t = 0 then a = 1.75 so so far:
y = 1.75 cos (w t)
now w t is zero at zero but I want it to be 2 pi or a full cycle at t = 10
so
w (10) = 2 pi
so w = 2 pi/10 = pi/5
so
y = 1.75 cos (pi t /5)
Wow thank you
Thanks!!! I couldn’t figure this out myself, and your explanation really helped!!!
Why did the buoy join a band?
Because it wanted to hit all the high notes!
To describe the motion of the buoy, we can use the equation for simple harmonic motion:
x(t) = A * cos(ωt + φ)
Where:
- x(t) represents the displacement of the buoy at time t.
- A is the amplitude of the motion, which is half the total distance traveled by the buoy from its low point to its high point, so A = 3.5/2 = 1.75 feet.
- ω (omega) is the angular frequency, which can be calculated using the formula ω = 2π / T, where T is the time taken for one complete oscillation. In this case, T = 10 seconds, so ω = 2π / 10 = π/5 radians per second.
- φ (phi) represents the phase constant, which is determined by the initial conditions. Since the buoy reaches its high point at t=0, we know that φ = 0.
Therefore, the equation describing the motion of the buoy is:
x(t) = 1.75 * cos((π/5)t)
To find the equation that describes the motion of the buoy, we can start by understanding the properties of simple harmonic motion (SHM). In SHM, an object oscillates back and forth with a specific period (T) and amplitude (A).
In this case, the buoy moves a total of 3.5 feet from its low point to its high point. This is the amplitude (A) of the motion. We're also given that it returns to its high point every 10 seconds, which is the period (T) of the motion.
The equation that describes the motion of an object in SHM can be written as:
x(t) = A * cos(2πt / T + φ)
Here, x(t) represents the position of the object at time t, A is the amplitude, T is the period, and φ is the phase constant.
Since the buoy reaches its high point at t = 0, we can eliminate the phase constant φ from the equation. So, the equation for the buoy's motion is:
x(t) = A * cos(2πt / T)
Substituting the given values, the equation becomes:
x(t) = 3.5 * cos(2πt / 10)