A sewing machine needle moves with a rapid vibratory motion, rather like SHM, as it sews a seam. Suppose the needle moves 8.4 mm from its highest to its lowest position and it makes 27 stitches in 8.5 s. What is the maximum needle speed?
To find the maximum needle speed, we need to calculate the velocity at the extreme positions of the needle's motion.
First, we can find the time period (T) of each complete oscillation using the formula:
T = Total time / Number of oscillations
In this case, the total time is 8.5 seconds, and the number of oscillations is 27. Therefore, the time period (T) is:
T = 8.5 s / 27 oscillations
Next, we can find the frequency (f) using the formula:
f = 1 / T
By substituting the value of T we calculated earlier, we get:
f = 1 / (8.5s / 27)
Now, let's calculate the maximum speed. In simple harmonic motion (SHM), the velocity at the extreme points is maximum. For SHM, we can calculate the maximum speed (v_max) using the formula:
v_max = 2πfA
Where A is the amplitude of the motion, which is given as 8.4 mm in this case.
We already calculated the value of f using the formula mentioned above, so we can substitute it into the formula:
v_max = 2π * (1 / (8.5s / 27)) * 8.4 mm
Converting mm to m:
v_max = 2π * (1 / (8.5s / 27)) * 0.0084 m
Calculating this final expression will give us the maximum needle speed.