The motion of a piston in an automobile engine is nearly simple harmonic. If the 1-kg piston travels back and forth over a distance of 12.0 cm, what is its maximum speed when the engine is running at 3000 rpm?
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To find the maximum speed of the piston, we need to determine its amplitude and period from the given information.
1. First, we find the amplitude:
The piston travels back and forth over a distance of 12.0 cm. The amplitude (A) is half of this total distance, so A = 12.0 cm / 2 = 6.0 cm.
2. Next, we find the period:
The period (T) can be calculated using the formula:
T = 1 / f
where f is the frequency (number of complete cycles per second).
Given that the engine is running at 3000 rpm (revolutions per minute), we need to convert this into cycles per second:
f = 3000 rpm * (1 min / 60s) = 50 cycles/second
Therefore, the period is:
T = 1 / 50 = 0.02 seconds
3. Now, we can find the maximum speed using the equation for simple harmonic motion:
v_max = 2Ï€A / T
Substituting the values we found:
v_max = 2Ï€ * 6.0 cm / 0.02 seconds
To get the maximum speed in m/s, we need to convert centimeters to meters:
v_max = 2Ï€ * 0.06 m / 0.02 seconds
Simplifying the expression:
v_max = 6π m/s ≈ 18.85 m/s
Therefore, the maximum speed of the 1-kg piston is approximately 18.85 m/s when the engine is running at 3000 rpm.