The motion of a piston in an automobile engine is nearly simple harmonic.If the 1-kg piston travels back and forth over a total distance of 10.0 cm,what is its maximum speed when the engine is running at 3 000 rpm?
A)31.4 m/s
B)15.7 m/s
C)7.85 m/s
D)3.93 m/s

Question

Quizplus · Accepted Answer

We can use the equation for simple harmonic motion:

x(t) = Acos(ωt + φ)

where x is the position of the piston, A is the amplitude (half of the total distance traveled), ω is the angular frequency, and φ is the phase constant.

We know that the total distance traveled is 10 cm, so the amplitude is A = 5 cm = 0.05 m.
We also know that the motion is periodic with a period of T = 1/f, where f is the frequency of the engine. Since the engine is running at 3 000 rpm, its frequency is:

f = 3 000 rpm ÷ 60 s/min = 50 Hz

So the angular frequency is:

ω = 2πf = 2π × 50 Hz = 100π rad/s

At the maximum speed, the piston is passing through its equilibrium position, so its velocity is given by:

v_max = ωA

v_max = (100π rad/s) × (0.05 m)

v_max ≈ 15.7 m/s

Therefore, the answer is B.

The Motion of a Piston in an Automobile Engine Is