- #1
redoxes
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There is a rod treating as rigid body, the rod which mass is m and lenghth is R rotate about one end of itself with the angular acceleration [tex]\alpha [/tex]. Apply a tangential external force f on another end to make the rod rotate. Now divide the rod into elements of mass. For each mass element dm with the distance r from the fixed end, it acted by an tangential component of resultant force dF which satisfy:
[tex]dF=dm\cdot a_{t} = dm\cdot \alpha r =\lambda dr \cdot \alpha r[/tex]([tex]\lambda[/tex] is the linear density)
[tex]\alpha =\frac{M}{I}=\frac{fR}{\frac{1}{3}mR^{2}}[/tex] (M is the moment of force,I is the moment of interia about the fixed end)
So the tangential component of resultant force F of the rod will be:
[tex]F = \int_{0}^{R}\lambda dr \cdot \alpha r=\frac{m}{R}\frac{fR}{\frac{1}{3}mR^{2}}\int_{0}^{R}r\cdot dr=\frac{3}{2}f[/tex]
As we see, F is not equal to f, is this a paradox? what is wrong in this argument?
[tex]dF=dm\cdot a_{t} = dm\cdot \alpha r =\lambda dr \cdot \alpha r[/tex]([tex]\lambda[/tex] is the linear density)
[tex]\alpha =\frac{M}{I}=\frac{fR}{\frac{1}{3}mR^{2}}[/tex] (M is the moment of force,I is the moment of interia about the fixed end)
So the tangential component of resultant force F of the rod will be:
[tex]F = \int_{0}^{R}\lambda dr \cdot \alpha r=\frac{m}{R}\frac{fR}{\frac{1}{3}mR^{2}}\int_{0}^{R}r\cdot dr=\frac{3}{2}f[/tex]
As we see, F is not equal to f, is this a paradox? what is wrong in this argument?