The piston in a particular car engine moves in approximately simple harmonic motion with an amplitude of 8 cm. The mass of the piston is 8 kg and the piston makes 100 oscillations per second. Calculate:
a) the maximum value if the acceleration of the piston.
b) the force needed to produce this acceleration.
x = .08 sin 2 pi (100)t
= .08 sin 628 t
v = dx/dt = .08(628) cos 628 t
a = d^2x/dt^2 = -.08(628)^2 sin 628 t
so
max a = .08(628^2)
max f = m a = 8 * max a
To solve this problem, we need to apply the principles of simple harmonic motion (SHM) and Newton's second law of motion. Let's break down the problem step by step:
a) Finding the maximum acceleration:
In SHM, the acceleration of an object is given by the equation: a = -ω^2x, where ω is the angular frequency and x is the displacement from the equilibrium position.
To find the angular frequency (ω), we can use the formula: ω = 2πf, where f is the frequency in hertz (Hz). In this case, the frequency is given as 100 oscillations per second, so f = 100 Hz.
Plugging the frequency into the ω formula, we get:
ω = 2π(100) = 200π rad/s
Now, we can find the maximum acceleration (a_max) using the given amplitude (A) and the angular frequency (ω) as follows:
a_max = ω^2A
Plugging in the values:
a_max = (200π)^2(0.08) ≈ 3216.99 m/s^2
Therefore, the maximum value of the acceleration is approximately 3216.99 m/s^2.
b) Finding the force needed to produce this acceleration:
According to Newton's second law of motion, the force (F) required to produce an acceleration (a) on an object of mass (m) is given by the equation: F = ma.
Plugging in the values:
F = (8 kg)(3216.99 m/s^2) ≈ 25735.92 N
Therefore, the force needed to produce this acceleration is approximately 25735.92 Newtons.