A piston in a car engine has a mass of 0.75kg and moves with motion which is approximately simple harmonic. If the amplitude of this oscillation is 10 cm and the maximum safe operating 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.
period=1/6000 sec
A=.1m
mass=.75kg
max speed=A*2PI*6000 m/s
max acceleration=A*(2PI*6000)^2 m/s^2
maxforce=.75*maxaccleration.
http://physics.bu.edu/~duffy/py105/SHM.html
Good answer but not fineshed show step one by one
Answer
To solve this problem, let's break it down step by step:
a) Maximum Acceleration of the Piston:
The maximum acceleration of a particle undergoing simple harmonic motion can be found using the formula:
a = -ω^2 * x
where a is the acceleration, ω is the angular frequency, and x is the displacement from the equilibrium position.
Since we're given the amplitude (A) of the oscillation, we can calculate the displacement (x) as half of the amplitude:
x = A/2 = 10 cm / 2 = 5 cm = 0.05 m
To find the angular frequency (ω), we need the period (T) of the oscillation. The period is the time taken for one complete revolution, and it can be calculated using the formula:
T = 1 / (n/60)
where T is the period, n is the revolutions per minute, and 60 is the number of seconds in a minute.
Plugging in the values, we have:
T = 1 / (6000/60) = 0.01 seconds
Since ω = 2π / T, we can calculate:
ω = 2π / 0.01 ≈ 628.32 radians/second
Now we can find the maximum acceleration:
a = -ω^2 * x = -(628.32)^2 * 0.05 ≈ -19697.61 m/s^2
Therefore, the maximum acceleration of the piston is approximately -19697.61 m/s^2.
b) Maximum Speed of the Piston:
The maximum speed of a particle undergoing simple harmonic motion can be found using the formula:
v = ω * A
Plugging in the values:
v = 628.32 * 0.1 ≈ 62.832 m/s
Therefore, the maximum speed of the piston is approximately 62.832 m/s.
c) Maximum Force Acting on the Piston:
The maximum force acting on the piston can be calculated using Newton's second law:
F = m * a
where F is the force, m is the mass, and a is the acceleration.
Plugging in the values:
F = 0.75 * (-19697.61) ≈ -14773.21 N
Note that since the acceleration is negative, the force will also be negative, indicating that it acts in the opposite direction of the displacement.
Therefore, the maximum force acting on the piston is approximately -14773.21 N.