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MrMaterial
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Homework Statement
For a particle in simple harmonic motion, show that Vmax = (pi/2)*Vavg where Vavg is the average speed during one cycle of the motion.
Homework Equations
x(t) = A*cos(ωt) (SHM mathematical model)
v(t) = -Vmax*sin(ωt)
Fave = 1/(b-a)∫f(x)dx
The Attempt at a Solution
As soon as i get started with this problem, I hit a brick wall.
I don't know if this is due to me being brain-dead because of all the studying I've been doing today, but whatever it is I can't seem to wrap my head around it!
the problem: How do I calculate the average velocity?
I know v(t) is the derivative of x(t)
I also know the average value function shown above.
1.) find derivative of x(t) to get v(t)
v(t) = -Vmax*sin(ωt)
2.) use average value function on v(t) to find "average velocity"
since Vavg is defined as the average velocity of one cycle, and one cycle = 2pi
b = 2pi
a = 0
Fave = 1/(2pi - 0)*∫v(t)dt = 0
this makes the equation impossible to prove! no SHM has a Vmax of 0!
yet... how could the average velocity NOT be zero?
What am I doing wrong here?