A piston of a car engine executes a simple harmonic motion.
the acceleration a of the piston is related to its displacement x by the equation:
a = -6.4*10^5x
a) Calculate the frequency of motion
b) The piston has a mass of 700 gms & a max displacement of 8 cm.
Calculate the max force on the pistion.
To calculate the frequency of motion, we can use the equation:
f = (1 / T)
where f is the frequency and T is the period of motion.
In this case, the motion is simple harmonic motion, and we know that the period T is related to the angular frequency ω by the equation:
T = 2π / ω
The angular frequency is related to the acceleration a by the equation:
a = -ω^2 * x
Comparing this equation with the given equation a = -6.4 * 10^5x, we can see that ω^2 = 6.4 * 10^5. Taking the square root of both sides, we get:
ω = √(6.4 * 10^5)
Now we can substitute this value of ω into the equation for T to find the period T. Then, we can use the equation for frequency f to calculate the frequency.
a) Calculate the frequency of motion:
In our case, ω = √(6.4 * 10^5)
T = 2π / ω
T = 2π / √(6.4 * 10^5)
f = 1 / T
Substituting the value of T, we get:
f = 1 / (2π / √(6.4 * 10^5))
Simplifying further, we get:
f = √(6.4 * 10^5) / (2π)
Evaluating this expression, we find:
f ≈ 89.44 Hz
So, the frequency of motion is approximately 89.44 Hz.
b) Calculate the max force on the piston:
The maximum force on the piston can be calculated using Hooke's Law, which states that the force (F) is equal to the mass (m) multiplied by the acceleration (a):
F = m * a
Given that the mass of the piston is 700 gms (or 0.7 kg) and the maximum displacement (x) is 8 cm (or 0.08 m), we can use the given equation for acceleration (a = -6.4 * 10^5x) to calculate the maximum force (F).
Substituting the given values into the equation, we get:
a (at maximum displacement) = -6.4 * 10^5 * 0.08
F = (0.7 kg) * (-6.4 * 10^5 * 0.08)
Evaluating this expression, we find:
F ≈ -322,560 N
The negative sign indicates that the force is in the opposite direction of the displacement. So, the maximum force on the piston is approximately 322,560 N.