The motion of a piston in an automobile engine is nearly simple harmonic. If the 1-kg piston travels back and forth over a total distance of 10.0 cm, what is its maximum speed when the engine is running at 3 000 rpm? The answer is 15.7 m/s but I don't understand how to get there.
Maximum speed is w A, where w is the angular velocity and A is the amplitude. The amplitude A is half the displacement of the engine, or 5.0 cm.
The angular velocity in your case is
w = (3000 rev/min)*(2 pi rad/rev)/(60 sec/min) = 314 rad/s
Vmax = 5.0*314 = 1570 cm/s = 15.7 m/s
displacement=Amax(SIN W T)
VELOCITY= aMAX*W*COS WT
MAX VELOCITY=AMAX*W=.05M*2pi*3000RAD/60
MAX VELOCITY=.05*2pi*50
Well, let's have some fun with this problem!
To find the maximum speed of the piston, we can use the formula:
vmax = Aω
Here, A represents the amplitude of the motion (which is half the total distance traveled by the piston), and ω is the angular velocity (which we can calculate from the engine's rpm).
So let's start by finding the amplitude:
A = 10.0 cm / 2 = 5.0 cm = 0.05 m
And now, let's figure out the angular velocity:
ω = 2π * (n / 60)
Here, n is the number of revolutions per minute, which is given as 3000 rpm. So let's plug that in:
ω = 2π * (3000 / 60) = 100π rad/s
And now, we can find the maximum speed:
vmax = Aω = 0.05 m * 100π rad/s
Now, let's do some math:
vmax ≈ 15.7 m/s
And there you have it! The maximum speed of the piston is approximately 15.7 m/s. Keep in mind that this is a simplified approximation, and there may be other factors at play in a real-world scenario. But for now, let's just enjoy the idea of a speedy, harmonically oscillating piston in an engine!
To find the maximum speed of the piston, we can use the equation for simple harmonic motion, which relates displacement, velocity, and the angular frequency:
v = ω√(A^2 - x^2)
Where:
v is the velocity
ω is the angular frequency (ω = 2πf)
A is the amplitude (maximum displacement)
x is the position of the piston
In this case, the distance traveled by the piston is the total distance, which is double the amplitude:
A = 10.0 cm / 2 = 5.0 cm = 0.05 m
To find the angular frequency, we need to convert the engine speed from revolutions per minute (rpm) to radians per second (rad/s):
ω = 2πf
f = 3000 rpm = 3000/60 Hz = 50 Hz
Substituting the values into the equation for velocity:
v = 2πf√(A^2 - x^2)
v = 2π * 50 * √(0.05^2 - x^2)
Since we are looking for the maximum speed, we can substitute the maximum displacement to find the maximum value of v:
v_max = 2π * 50 * √(0.05^2 - 0)
v_max = 2π * 50 * 0.05
v_max ≈ 15.7 m/s
Therefore, the maximum speed of the piston when the engine is running at 3,000 rpm is approximately 15.7 m/s.
To find the maximum speed of the piston, we can use the equation for simple harmonic motion (SHM):
v_max = ω * A
Where:
v_max is the maximum speed of the piston,
ω (omega) is the angular frequency (2π times the frequency of oscillation),
A is the amplitude (half the total distance traveled by the piston).
We are given that the piston travels back and forth over a total distance of 10.0 cm, so the amplitude is half of that, which is 5.0 cm or 0.05 meters.
To find the angular frequency, we need to convert the engine speed from rpm (revolutions per minute) to radians per second.
The conversion is as follows:
1 revolution = 2π radians
1 minute = 60 seconds
Therefore, the angular frequency ω is given by:
ω = (2π * RPM) / 60
Substituting 3,000 rpm into the equation:
ω = (2π * 3000) / 60 = 100π rad/s
Finally, we substitute the values of ω and A into the equation for v_max:
v_max = (100π) * 0.05 = 15.7 m/s
So, the maximum speed of the piston is 15.7 m/s.