The motion of a piston in an auto engine is simple harmonic. The piston travels back and forth over a distance of 23 cm, and the piston has a mass of 1.7 kg.
2515 rpm 23 cm
max speed of the piston 30.28757 m/s
What is the maximum force acting on the
piston when the engine is running at the same rate? Answer in units of N
find the frequency in Hz
f = 2515 rev/min (1 min/60 s) = 41.92 Hz
y = (0.23/2)sin (2 pi f t)
y = 0.115 sin 263 t
v = (0.115)(263) cos 263 t so yes max of 30.29 m/s
a = -30.29(263)sin 263 t
so max a = 7966 m/s^2
F = m a = 1.7*7966 = 13543 Newtons
thank you!!!!!
i must have messed up on my calculations
To find the maximum force acting on the piston, you can use the equation for maximum force in simple harmonic motion:
F = mω²x
Where:
F is the maximum force
m is the mass of the piston (1.7 kg)
ω is the angular velocity (converted from rpm to rad/s)
x is the amplitude of the motion (23 cm converted to meters)
First, let's convert the angular velocity from rpm to rad/s:
ω = (2π × RPM) / 60
ω = (2π × 2515) / 60
ω ≈ 263.24 rad/s
Next, calculate the maximum force:
F = (1.7 kg) × (263.24 rad/s)² × (0.23 m)
F ≈ 642.64 N
Therefore, the maximum force acting on the piston when the engine is running at the same rate is approximately 642.64 N.
To find the maximum force acting on the piston, we can use the equation for Simple Harmonic Motion:
F = -kx
where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the displacement of the piston is given as 23 cm, which is equal to 0.23 meters. The mass of the piston is given as 1.7 kg.
To calculate the spring constant, we need to find the angular frequency, ω, which is related to the frequency, f, by the equation:
ω = 2πf
The frequency can be calculated from the given engine speed in rpm (revolutions per minute). We need to convert rpm to revolutions per second (rps), and then divide by 2 to get the frequency in cycles per second (Hz).
Given that the engine speed is 2515 rpm:
f = 2515 rpm / 60 seconds = 41.92 rps / 2 = 20.96 Hz
Now we can calculate the angular frequency:
ω = 2π(20.96 Hz) = 131.95 radians/second
Using the equation for angular frequency:
ω^2 = k / m
we can solve for the spring constant, k:
k = ω^2 * m = (131.95 radians/second)^2 * 1.7 kg
k ≈ 30598.99 N/m
Finally, we can calculate the maximum force acting on the piston using the equation for Simple Harmonic Motion:
F = -kx = -(30598.99 N/m)(0.23 m)
F ≈ -7047.37 N
Since force is a vector quantity, the negative sign indicates that the maximum force acts in the opposite direction of the displacement of the piston. Therefore, the maximum force acting on the piston when the engine is running at the given rate is approximately 7047.37 N.