A steel strip clamped at one end vibrate with frequency of 20Hz and an amplitude of 5mm at the free end, were a small of 2g is positioned.
What is the velocity and the acceleration ?
huh?
12
To calculate the velocity and acceleration, we need to know the equation of motion for the vibrating steel strip. Since it is vibrating with a frequency of 20Hz, we can assume that it is undergoing simple harmonic motion.
The equation for the displacement of an object undergoing simple harmonic motion is given by:
x = A * sin(2πft)
Where:
x = displacement
A = amplitude
f = frequency
t = time
In this case, the amplitude (A) is 5mm, the frequency (f) is 20Hz, and the time (t) is not specified. Let's assume that t = 0s.
Substituting the given values into the equation, we get:
x = 5mm * sin(2π * 20Hz * 0s)
Simplifying further:
x = 5mm * sin(0)
Since the sine of 0 is 0, the displacement at t = 0s is 0.
Now, let's find the velocity and acceleration.
Velocity is the rate of change of displacement with respect to time. Mathematically, it is given by the derivative of displacement with respect to time:
v = dx/dt
Taking the derivative of the displacement equation, we get:
v = (d/dt)(5mm * sin(2π * 20Hz * t))
Simplifying further:
v = 5mm * (2π * 20Hz) * cos(2π * 20Hz * t)
At t = 0s, the velocity is:
v = 5mm * (2π * 20Hz) * cos(0)
Since the cosine of 0 is 1, the velocity at t = 0s is:
v = 5mm * (2π * 20Hz) * 1
v = 5mm * (40π) = 200π mm/s
The acceleration is the rate of change of velocity with respect to time. Mathematically, it is given by the derivative of velocity with respect to time:
a = dv/dt
Taking the derivative of the velocity equation, we get:
a = (d/dt)(5mm * (2π * 20Hz) * cos(2π * 20Hz * t))
Simplifying further:
a = -5mm * (2π * 20Hz) * (2π * 20Hz) * sin(2π * 20Hz * t)
At t = 0s, the acceleration is:
a = -5mm * (2π * 20Hz) * (2π * 20Hz) * sin(0)
Since the sine of 0 is 0, the acceleration at t = 0s is:
a = 0
Therefore, the velocity at t = 0s is 200π mm/s and the acceleration at t = 0s is 0.
To determine the velocity and acceleration of the vibrating steel strip, we need to understand the basic principles of simple harmonic motion.
First, let's calculate the angular frequency (ω) of the vibrating steel strip. The angular frequency is given by the formula: ω = 2πf, where f represents the frequency of vibration.
Given that the frequency is 20 Hz, we can calculate the angular frequency as follows:
ω = 2π * 20 Hz = 40π radians/second.
Next, we need to determine the displacement (x) of the small mass attached to the steel strip. The displacement can be calculated using the formula: x = A * sin(ωt), where A represents the amplitude of vibration and t is the time.
Given that the amplitude is 5 mm, we need to convert it to meters: A = 5 mm = 0.005 m.
Now, let's calculate the velocity (v) by taking the derivative of the displacement function with respect to time. The velocity can be expressed as: v = dx/dt, where dx represents the displacement change with respect to time.
Differentiating the displacement equation, we have:
v = d(A * sin(ωt))/dt = Aω * cos(ωt).
Substituting the values, we get:
v = (0.005 m) * (40π radians/second) * cos(40πt).
Finally, to calculate the acceleration (a), we differentiate the velocity equation with respect to time. The acceleration can be expressed as: a = dv/dt.
Differentiating the velocity equation, we have:
a = d(Aω * cos(ωt))/dt = -Aω² * sin(ωt).
Substituting the values, we get:
a = -(0.005 m) * (40π radians/second)² * sin(40πt).
So, the velocity of the vibrating steel strip is given by v = (0.005 m) * (40π radians/second) * cos(40πt), and the acceleration is given by a = -(0.005 m) * (40π radians/second)² * sin(40πt).