A friction welding machine is butt-welding two steel pipes together. Initially, the spinning pipe is rotating against the stationary pipe at a rate of frpm=746 rpm (rounds per minute). 25 seconds later the spinning pipe comes to rest and the molten steel between the two pipes begins to cool. Assuming that the deceleration of the moving pipe was uniform, how many revolutions did the moving pipe make before coming to rest?
746 rev/min * 1min /60 s = 12.43 rev/s = Vi
v = Vi - a t = 12.43 rev/s + a t
0 = 12.43 revs /s + a *(25 seconds)
a = -12.43 revs/s / 25 s = -0.497 revs/s^2
revs = Vi t + (1/2) a t^2 = 12.43*25 - 0.248*625 = 311 - 155 = 156 revs
or I could have used the average speed = 12.43/2 = 6.22 revs/s * 25s = 155 revs
since speed is linear in time with constant a
To find the number of revolutions the moving pipe made before coming to rest, we need to calculate the initial angular velocity, final angular velocity, and the time taken for the pipe to come to rest.
Given:
Initial angular velocity (ω_i) = 746 rpm
Final angular velocity (ω_f) = 0 (since the spinning pipe comes to rest)
Time taken for deceleration (t) = 25 seconds
We know that the deceleration is negative acceleration (opposite to the direction of motion). The relationship between the angular velocity, angular acceleration, and time is as follows:
ω_f = ω_i + α * t
Since the deceleration is uniform, we can find the angular acceleration (α) using the formula:
α = (ω_f - ω_i) / t
Substituting the given values:
α = (0 - 746) rpm / 25 s
Now, we can calculate the angular displacement (θ) using the formula:
θ = ω_i * t + 0.5 * α * t^2
Substituting the given values:
θ = 746 rpm * 25 s + 0.5 * [(0 - 746) rpm / 25 s] * (25 s)^2
Simplifying the equation:
θ = 746 * 25 + 0.5 * (-746) * 25
θ = 18650 - 18650
θ = 0
Therefore, the moving pipe made 0 revolutions before coming to rest.