a steel strip clamped vibrated at one end vibration with frequancy of 50hz and amplitude of 8mm at the free end find (a)velocity of the end passing through the zero position (b)acceleration at maximum displacement
I) 2.51ms-¹
ii) 0.08ms-²
Physics
Oh, vibrating steel strips, are we getting all metal and groovy now? Let's dive into it!
(a) The velocity of the end passing through the zero position can be found by using the formula v = 2πfA, where v is the velocity, f is the frequency, and A is the amplitude.
Substituting the given values, we get:
v = 2π * 50Hz * 8mm
v ≈ 800π mm/s
So, my metal-head friend, the velocity of the free end passing through the zero position is approximately 800π mm/s.
(b) Now, let's calculate the acceleration at maximum displacement. The acceleration can be found using the formula a = (2πf)²A, where a is the acceleration, f is the frequency, and A is the amplitude.
Substituting the given values, we get:
a = (2π * 50Hz)² * 8mm
a ≈ 250,000π² mm/s²
Hence, the acceleration at maximum displacement is about 250,000π² mm/s².
Keep rockin' those vibrations, my friend!
To find the velocity of the free end passing through the zero position and the acceleration at maximum displacement of a vibrating steel strip, you can use the basic formulas of simple harmonic motion.
(a) Velocity of the end passing through the zero position:
The velocity of an object undergoing simple harmonic motion can be calculated using the formula:
v = ω * A * cos(ωt)
Where:
v is the velocity of the object
ω is the angular frequency (2π times the frequency)
A is the amplitude of the motion
t is the time
In this case, the angular frequency (ω) can be calculated as:
ω = 2π * f
Where:
f is the frequency of the vibration
Given that the frequency of the vibration is 50 Hz and the amplitude (A) is 8 mm (0.008 m), we can substitute these values into the equations:
ω = 2π * 50 = 100π rad/s
A = 0.008 m
Now, to find the velocity at the zero position, we need to find the time when the displacement is zero. At the extreme points of simple harmonic motion, the displacement is maximum and the velocity is zero. The time taken to go from a maximum or minimum displacement to the zero position is a quarter of the time period.
T = 1/f = 1/50 = 0.02 s
t = T/4 = 0.02/4 = 0.005 s
Now, substituting the values into the formula:
v = ω * A * cos(ωt)
v = 100π * 0.008 * cos(100π * 0.005)
v ≈ 0.251 m/s
Therefore, the velocity of the free end passing through the zero position is approximately 0.251 m/s.
(b) Acceleration at maximum displacement:
The acceleration of an object undergoing simple harmonic motion is given by the formula:
a = -ω^2 * A * sin(ωt)
Using the same values of angular frequency (ω) and amplitude (A) as calculated above, let's find the acceleration at maximum displacement. At the maximum displacement, the sine function is equal to 1.
t = 0 (At maximum displacement)
a = -ω^2 * A * sin(ωt)
a = - (100π)^2 * 0.008 * sin(100π * 0)
a ≈ 0
Therefore, the acceleration at maximum displacement is approximately 0.