The displacement of a particle in centimeters is given by y=8 sin (2πft).
If f=20 Hz, find the
displacement and the velocity of the particle at a time t=0.01 s and 0.07 s.
(ans: y=7.61 cm, v=3.11 m/s, y=4.70 cm, v= -8.13 m/s)
After I plug in f to get w and switch to this equation: Asin(wt - kx), plugging in the numbers doesn't give me the right answer.
Thank you
nevermind, got it, had to switch calculator from degrees to radian, hehe
To find the displacement and velocity of the particle at a given time t in this equation (y = 8 sin (2πft)), we can follow these steps:
Step 1: Find the angular frequency (ω):
Given, f = 20 Hz
The angular frequency (ω) is calculated using the formula:
ω = 2πf
Plugging in the values: ω = 2π(20) = 40π rad/s
Step 2: Plug in the time values and solve for y:
For t = 0.01 s:
y = 8 sin (ωt)
y = 8 sin (40π * 0.01)
y = 8 sin (0.4π)
y ≈ 7.61 cm
For t = 0.07 s:
y = 8 sin (ωt)
y = 8 sin (40π * 0.07)
y = 8 sin (2.8π)
y ≈ 4.70 cm
Step 3: Calculate the velocity using the derivative of displacement:
Given that the derivative of y with respect to t is velocity (v), we calculate v using the formula:
v = dy/dt
For t = 0.01 s:
v = d/dt (8 sin (ωt))
v = 8ω cos (ωt)
v = 8(40π) cos (40π * 0.01)
v ≈ 8(40π) cos (0.4π)
v ≈ 3.11 m/s
For t = 0.07 s:
v = d/dt (8 sin (ωt))
v = 8ω cos (ωt)
v = 8(40π) cos (40π * 0.07)
v ≈ 8(40π) cos (2.8π)
v ≈ -8.13 m/s
So, at t = 0.01 s, the displacement is approximately 7.61 cm, and the velocity is approximately 3.11 m/s.
At t = 0.07 s, the displacement is approximately 4.70 cm, and the velocity is approximately -8.13 m/s.
If you're not getting the same answers, please double-check your calculations and make sure you are using the correct values for the angular frequency (ω) and time (t).