The midpoint of a guitar string executes simple harmonic motion with motion following the form x(t) = A sin(ωt + φ).
Amplitude (A)=1.60mm, Angular Velocity (w)=2760 and phase constant (o) = pi/2.
How do I find the initial displacement, velocity and acceleration of the midpoint of the string?
x(t) = A sin(ωt + φ)
v(t) = A ω cos(ωt + φ)
a(t) = - A ω2 sin (ωt + φ) = -ω^2 x(t)
so
x(0) = A sin(φ)
v(0) = A ω cos( φ)
a(0) = - A ω2 sin ( φ) = -ω^2 x(0)
at t = 0
x(0) = 160 sin(φ)
v(0) = 160*2760 cos( φ)
a(0) = -2760^2 160 sin(φ)
if φ = pi/2
x(0) = A = 160
v(0) = A ω * 0 = 0
a(0) = -A ω^2 = -160*2760^2
By the way
2760/2 pi is about 440 Hz
to hear a 440 Hz test tone use:
https://www.youtube.com/watch?v=Awx8-mq8g6g
To find the initial displacement, velocity, and acceleration of the midpoint of the string, you can differentiate the equation for displacement with respect to time.
Given equation for displacement of the midpoint of the string:
x(t) = A * sin(ωt + φ)
1. Initial Displacement:
The initial displacement can be found by substituting t = 0 into the equation for displacement.
x(0) = A * sin(ω * 0 + φ)
x(0) = A * sin(φ)
2. Velocity:
Differentiate the equation for displacement with respect to time (t) to find the velocity.
v(t) = d/dt [ A * sin(ωt + φ) ]
v(t) = A * ω * cos(ωt + φ)
Substituting t = 0 into the equation for velocity will give the initial velocity.
v(0) = A * ω * cos(ω * 0 + φ)
v(0) = A * ω * cos(φ)
3. Acceleration:
Differentiate the equation for velocity with respect to time (t) to find the acceleration.
a(t) = d/dt [ A * ω * cos(ωt + φ) ]
a(t) = -A * ω^2 * sin(ωt + φ)
Substituting t = 0 into the equation for acceleration will give the initial acceleration.
a(0) = -A * ω^2 * sin(ω * 0 + φ)
a(0) = -A * ω^2 * sin(φ)
Now you can substitute the given values to find the initial displacement, velocity, and acceleration.
Note: Make sure to convert the value of φ from radians to degrees if necessary.
To find the initial displacement, velocity, and acceleration of the midpoint of the string, you need to differentiate the equation for x(t) with respect to time.
1. Initial Displacement (x₀):
The initial displacement (x₀) represents the position of the midpoint of the guitar string at time t = 0. To find this, substitute t = 0 into the equation x(t):
x₀ = A sin(ω(0) + φ)
Since sin(0) = 0, we can simplify it to x₀ = A sin(φ).
2. Initial Velocity (v₀):
The initial velocity (v₀) represents the velocity of the midpoint of the guitar string at time t = 0. To find this, differentiate the equation for x(t) with respect to time:
v(t) = dx(t)/dt = Aω cos(ωt + φ)
Substitute t = 0 into the equation:
v₀ = Aω cos(ω(0) + φ)
Since cos(0) = 1, we can simplify it to v₀ = Aω cos(φ).
3. Initial Acceleration (a₀):
The initial acceleration (a₀) represents the acceleration of the midpoint of the guitar string at time t = 0. To find this, differentiate the equation for velocity (v(t)) with respect to time:
a(t) = dv(t)/dt = -Aω² sin(ωt + φ)
Substitute t = 0 into the equation:
a₀ = -Aω² sin(ω(0) + φ)
Since sin(0) = 0, we can simplify it to a₀ = 0.
Therefore, the initial displacement (x₀) is A sin(φ), the initial velocity (v₀) is Aω cos(φ), and the initial acceleration (a₀) is 0.