- #1
erobz
Gold Member
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- Homework Statement
- Rolling without Slipping Demo - Walter Lewin
- Relevant Equations
- ## \sum F = ma ##
##\sum \tau = I \alpha##
So I'm watching this demo posted in another problem by @PeroK by Prof. Walter Lewin
and I can't help but see that in the part where he is demonstrating the independence of acceleration cylinder length that the (presumably) shorter aluminum cylinder is edging out the win...Not intended, but visible. So, think it must be drag? However, I come to the equation:
$$ \frac{3}{2} m \frac{dv}{dt} = mg \sin \theta - \frac{1}{2} C_d \rho_{air} A_{proj.} v^2 $$
Then use the mass of the cylinder to eliminate ##A_{proj.}##:
$$ A_{proj.} = \frac{4}{\rho_{cyl} \pi D} m $$
$$ \frac{3}{2} \cancel{m} \frac{dv}{dt} = \cancel{m} g \sin \theta - \frac{1}{2} C_d \rho_{air} \frac{4}{\rho_{cyl} \pi D} \cancel{m} v^2 $$
$$ \frac{3}{2} \frac{dv}{dt} = g \sin \theta - \frac{1}{2} C_d \rho_{air} \frac{4}{\rho_{cyl} \pi D} v^2 $$
And voila...I'm not left with an EOM dependent on ##L## (the length of the cylinder). Hmmm, so what is the explanation for what I'm seeing...given just this one observation (was it a fluke)?
and I can't help but see that in the part where he is demonstrating the independence of acceleration cylinder length that the (presumably) shorter aluminum cylinder is edging out the win...Not intended, but visible. So, think it must be drag? However, I come to the equation:
$$ \frac{3}{2} m \frac{dv}{dt} = mg \sin \theta - \frac{1}{2} C_d \rho_{air} A_{proj.} v^2 $$
Then use the mass of the cylinder to eliminate ##A_{proj.}##:
$$ A_{proj.} = \frac{4}{\rho_{cyl} \pi D} m $$
$$ \frac{3}{2} \cancel{m} \frac{dv}{dt} = \cancel{m} g \sin \theta - \frac{1}{2} C_d \rho_{air} \frac{4}{\rho_{cyl} \pi D} \cancel{m} v^2 $$
$$ \frac{3}{2} \frac{dv}{dt} = g \sin \theta - \frac{1}{2} C_d \rho_{air} \frac{4}{\rho_{cyl} \pi D} v^2 $$
And voila...I'm not left with an EOM dependent on ##L## (the length of the cylinder). Hmmm, so what is the explanation for what I'm seeing...given just this one observation (was it a fluke)?
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