A piston in a car engine has a mass of 0.75 kg and moves with motion wich was approximately simple harmonic. If the amplitude of zis oscillation is 10 cm and z max. safe operating speed of z engine is 6000 rev. per min. Calculate:

a. Max acceleration of z piston
b. Max.speed of the piston
c. The max. Force acting on the piston

6000 rpm = 628.3 radians/sec

Call that angular frequency "w"

Maximum acceleration = (Amplitide)*w^2
= 0.10 m * w^2 = 39480 m/s^2

Maximum velocity = (Amplitude)*w = 62.83 m/s.

Max force = (Mass)*(max acceleration)

To solve this problem, we need to convert the given quantities into the appropriate units.

Given:
Mass of the piston (m) = 0.75 kg
Amplitude (A) = 10 cm = 0.1 m
Maximum safe operating speed of the engine (ω) = 6000 rev/min

To calculate:

a. Maximum acceleration of the piston:
The maximum acceleration (a_max) can be calculated using the equation:
a_max = ω^2 * A

First, we need to convert the maximum safe operating speed from revolutions per minute (rev/min) to radians per second (rad/s):

1 revolution = 2π radians
1 minute = 60 seconds

ω = (6000 rev/min) * (2π rad/rev) * (1 min/60 s)
ω = 200π rad/s

Now we can calculate the maximum acceleration:
a_max = (200π rad/s)^2 * 0.1 m
a_max ≈ 40000π^2 m/s^2

b. Maximum speed of the piston:
The maximum speed (v_max) can be calculated using the equation:
v_max = ω * A

v_max = (200π rad/s) * 0.1 m
v_max ≈ 20π m/s

c. The maximum force acting on the piston:
The maximum force (F_max) can be calculated using the equation:
F_max = m * ω^2 * A

F_max = 0.75 kg * (200π rad/s)^2 * 0.1 m
F_max ≈ 30000π^2 N

Therefore, the answers are:
a. The maximum acceleration of the piston is approximately 40000π^2 m/s^2.
b. The maximum speed of the piston is approximately 20π m/s.
c. The maximum force acting on the piston is approximately 30000π^2 N.

To solve the given problem, we need to understand the concept of simple harmonic motion. Simple harmonic motion occurs when an object moves back and forth along a straight line with a restoring force that is proportional to its displacement from an equilibrium position. The motion of the piston in the car engine can be considered approximately as simple harmonic motion.

a. Max acceleration of the piston:
The maximum acceleration of an object undergoing simple harmonic motion can be calculated using the formula:
amax = ω^2 * A
where amax is the maximum acceleration, ω is the angular frequency, and A is the amplitude of oscillation.

To find the angular frequency (ω), we can use the formula:
ω = 2π * f
where f is the frequency of oscillation.

Since the frequency is given in rev. per min., we need to convert it to Hz (rev. per sec.) using the formula:
f = (frequency in rev. per min.) / 60

Given:
Amplitude (A) = 10 cm = 0.1 m
Max safe operating speed = 6000 rev. per min.

First, let's calculate the frequency:
f = 6000 / 60 = 100 Hz

Next, calculate the angular frequency:
ω = 2π * 100 = 200π rad/s

Finally, calculate the maximum acceleration:
amax = (200π)^2 * 0.1 = 40000π^2 * 0.1 ≈ 125663.71 m/s^2

So, the maximum acceleration of the piston is approximately 125663.71 m/s^2.

b. Max speed of the piston:
The maximum speed of the piston can be found using the formula:
vmax = ω * A

Given:
Angular frequency (ω) = 200π rad/s
Amplitude (A) = 0.1 m

vmax = 200π * 0.1 = 20π ≈ 62.83 m/s

Therefore, the maximum speed of the piston is approximately 62.83 m/s.

c. The max. force acting on the piston:
The maximum force acting on the piston can be calculated using the formula:
Fmax = m * amax
where m is the mass of the piston and amax is the maximum acceleration calculated in part (a).

Given:
Mass (m) = 0.75 kg
amax = 125663.71 m/s^2

Fmax = 0.75 * 125663.71 ≈ 94247.78 N

Therefore, the maximum force acting on the piston is approximately 94247.78 N.