To solve the given problem, we need to use the formulae related to simple harmonic motion (SHM).
a) To calculate the maximum acceleration of the piston, we can use the formula for maximum acceleration in SHM:
amax = ω^2 * A
where amax is the maximum acceleration, ω is the angular frequency, and A is the amplitude.
First, we need to calculate the angular frequency using the given information about the maximum safe operating speed of the engine. The maximum speed of the engine is 6000 revolutions per minute. We can convert this to radians per second by multiplying it by 2π/60:
ω = (6000 rev/min) * (2π rad/1 rev) * (1 min/60 s)
Now, we can calculate the maximum acceleration:
amax = (ω^2) * A
b) To find the maximum speed of the piston, we can use the formula for maximum speed in SHM:
vmax = ω * A
where vmax is the maximum speed.
c) The maximum force acting on the piston is given by the equation:
Fmax = m * amax
where Fmax is the maximum force and m is the mass of the piston.
Let's calculate these values:
a) Maximum acceleration:
Convert the engine speed to rad/s:
ω = (6000 rev/min) * (2π rad/1 rev) * (1 min/60 s)
Calculate the maximum acceleration:
amax = (ω^2) * A
b) Maximum speed:
vmax = ω * A
c) Maximum force:
Fmax = m * amax
Let's substitute the given values into the equations and calculate the results.