A piston in a car engine has a mass of 0.75 kg and moves with motion which is approximately simple harmonic. If the amplitude speed of the engine is 6000 revolutions per minute,calculate
a) maximum acceleration of the piston
b) maximum speed of the piston
c) the maximum force acting on the piston
To calculate the maximum acceleration, maximum speed, and maximum force acting on the piston, we need to use the equation for simple harmonic motion:
a) Maximum Acceleration:
The formula for maximum acceleration in simple harmonic motion is given by:
amax = ω^2 * A
where,
amax = maximum acceleration
ω = angular frequency
A = amplitude
To find the angular frequency, we need to convert the amplitude speed from revolutions per minute (rpm) to radians per second (rad/s). We know that 1 revolution is equal to 2π radians, and 1 minute is equal to 60 seconds.
Step 1: Conversion:
Angular frequency (ω) = 2π * (revolutions per minute) / (60 seconds)
= 2π * (6000 revolutions per minute) / (60 seconds)
= 200π rad/s
Step 2: Calculation:
Maximum acceleration (amax) = (200π rad/s)^2 * (amplitude)
= (200π)^2 * (A)
= (200π)^2 * (0.75 kg)
So, you can now calculate the maximum acceleration by substituting the values for π and the mass of the piston (0.75 kg).
b) Maximum Speed:
To calculate the maximum speed of the piston, we use the formula:
vmax = ω * A
Here, vmax is the maximum speed, ω is the angular frequency, and A is the amplitude.
Maximum speed (vmax) = (angular frequency) * (amplitude)
= (200π rad/s) * (0.75 kg)
Again, you can calculate the maximum speed by substituting the values for π and the mass of the piston (0.75 kg).
c) Maximum Force:
The maximum force acting on the piston can be calculated using the formula:
Fmax = m * amax
where,
Fmax = maximum force acting on the piston
m = mass of the piston
amax = maximum acceleration
Substitute the values for the mass of the piston (0.75 kg) and the maximum acceleration which you calculated in part a.