A particle moving in simple harmonic motion passes through the equilibrium point (x=0)
8 times per second. At t=0 its velocity at x=0.015m is negative. It travels 0.6m in a complete cycle.
The particle's position as a function of time is described by the following function:
x(t)=15sin(8pit+phi)cm
I don't know how to find phi.And I don't know which phi constant we have to chose beause I think we are going to get 2 different answers
put t=0 into the equation. Then, you know x(0)
1.5=15sin(phi)
phi=aresin(.001)
now be careful, you also know that phi can be several values, but because velocity is negative at t=0, then phi is between PI/2 and 3PI/2
Lets say I get a phi of 0.5 to find the other value do I have to add 2pi or pi
http://www.wolframalpha.com/input/?i=15+sin(8%CF%80(t%2B.1250398))
Well, finding phi can be a bit tricky, but don't worry, I'm here to help you find some laughter along the way!
Now, let's solve this puzzle step by step. We know that the particle passes through the equilibrium point (x = 0) 8 times per second. So, we can say that the period of the motion, T, is 1/8 seconds.
Now, we also know that the particle's velocity at x = 0.015m is negative. This means that at t = 0, the particle is at its maximum displacement in the negative direction.
By plugging in t = 0 and x = 0.015 to the position equation x(t)=15sin(8πt+φ)cm, we get:
0.015 = 15sin(φ).
Now, let's solve for φ. Divide both sides of the equation by 15:
sin(φ) = 0.001.
To solve for φ, we can take the inverse sine of both sides:
φ = sin^(-1)(0.001).
Now, here's where multiple answers might come in because sin^(-1)(0.001) has multiple possible solutions. However, we can ignore most of them and only focus on the one that satisfies our initial conditions.
Since we know that the particle starts at its maximum displacement in the negative direction, we can say that φ needs to be in the fourth quadrant, where sine values are negative. In this case, φ≈ -0.573 degrees or -0.01 radians.
So, in this context, we can choose φ = -0.01 radians.
I hope this helps! And remember, even in the world of mathematics, sometimes there are multiple possible answers, just like in the world of comedy!
To find the value of the constant φ in the given equation x(t) = 15 sin(8πt + φ), we can use the given information about the particle's motion.
First, let's recall a few key concepts about simple harmonic motion (SHM). In SHM, the equation x(t) = A sin(ωt + φ) represents the position of a particle at time t, where A is the amplitude of the motion, ω is the angular frequency, t is the time, and φ is the phase constant.
We are given that the particle passes through the equilibrium point (x = 0) 8 times per second. Since the particle completes one full cycle (from the maximum positive displacement to the maximum negative displacement and back) 8 times per second, the time period (T) of the motion is 1/8 seconds.
The time period of SHM is given by T = 2π/ω. We can rearrange this equation to solve for ω: ω = 2π/T. Substituting the given value of T = 1/8 seconds, we find ω = 2π/(1/8) = 16π rad/s.
Since we now have the value of ω, we can rewrite the equation for x(t) as x(t) = 15 sin(16πt + φ). To determine the value of φ, we need additional information about the motion.
The given information states that at t = 0, the velocity at x = 0.015m is negative. We know that velocity is the time derivative of displacement, so we can find the velocity function v(t) by differentiating x(t) with respect to t.
v(t) = dx/dt = 15(cos(16πt + φ))(16π) = 240π cos(16πt + φ).
At t = 0 and x = 0.015m, we can substitute these values into the equation to find the corresponding velocity:
0.015 = 240π cos(φ).
Now, we can solve this equation for the value of φ. Dividing both sides of the equation by 240π gives:
0.015 / (240π) = cos(φ).
The right-hand side of the equation gives us the cosine of φ. In order to find the value of φ, we need to take the inverse cosine (or arccosine) of this value.
Using a calculator or a math software, we can find the inverse cosine of 0.015 / (240π) to get the value of φ. Keep in mind that the inverse cosine function typically returns a value between 0 and π (or 0 and 180 degrees). Therefore, there may be multiple solutions for φ that satisfy the equation.
By following these steps, you can find the value of the constant φ in the given equation for the particle's position as a function of time.
2PI is the same angle. You want sin(PHI) positive, and cos(PHI) negative. Sketch a sin and cosine graph, take a look at where the added angle needs to land...I think PI/2 is what you need to get it to cos negative. check my thinking.
x(t) = 15 sin(8πt+Ø)
at t=0, x=0.015, so
15 sin(0+Ø) = 0.015
sinØ = 0.001
for small Ø, sinØ = Ø so I'd say
x(t) = 15 sin(8πt+0.001)
= 15 sin(8π(t+.0000398))
so it appears that Ø=.0000398
But, x'(0) < 0, so Ø must be offset by 1/2 period (1/8), making
x(t) = 15 sin(8π(t+.1250398))
x(t) = 15 sin(8πt+3.14259)