A uniform rod of mass m and length l is struck at an end by a force F perpendicular to the rod for a short time interval t. Calculate:
- The speed of the centre of mass.
- The angular speed of the rod about the centre of mass.
- The kinetic energy of the rod.
- The angular momentum of the rod about the centre of mass after the force has stopped to act. Assume that t is so small that the rod does not appreciably change its direction while the force acts.
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A uniform rod of mass m length $\ell$ is struck at an end by a force F. $\perp$ to the rod for a short time t:
$\Rightarrow\text{v}=\frac{\text{Ft}}{\text{m}}$
$\Rightarrow\Big(\frac{\text{m}\ell}{12}\Big)\times\omega=\Big(\frac{1}{2}\Big)\times\text{mv}$
$\Rightarrow\Big(\frac{\text{m}\ell}{12}\Big)\times\omega=\Big(\frac{1}{2}\Big)\ell\omega^2$
$\Rightarrow\omega=\frac{\text{m}\ell}{6\text{Ft}}$
$=\Big(\frac{1}{2}\Big)\times\text{m}\times\Big(\frac{\text{F}^2\text{t}^2}{\text{m}^2}\Big)+\Big(\frac{1}{2}\Big)\times\Big(\frac{\text{m}\ell^2}{12}\Big)\Big[36\times\Big(\frac{\text{F}^2\text{t}^2}{\text{m}^2\ell^2}\Big)\Big]$
$=\frac{\text{F}^2\text{t}^2}{\text{2m}}+\frac{3}{2}\Big(\frac{\text{F}^2\text{t}^2}{\text{m}}\Big)=2\Big(\frac{\text{F}^2\text{t}^2}{\text{m}}\Big)$
- Speed of the centre of mass:
$\Rightarrow\text{v}=\frac{\text{Ft}}{\text{m}}$
- The angular speed of the rod about the centre of mass:
$\Rightarrow\Big(\frac{\text{m}\ell}{12}\Big)\times\omega=\Big(\frac{1}{2}\Big)\times\text{mv}$
$\Rightarrow\Big(\frac{\text{m}\ell}{12}\Big)\times\omega=\Big(\frac{1}{2}\Big)\ell\omega^2$
$\Rightarrow\omega=\frac{\text{m}\ell}{6\text{Ft}}$
- $\text{K.E.}=\Big(\frac{1}{2}\Big)\text{mv}^2+\Big(\frac{1}{2}\Big)\ell\omega^2$
$=\Big(\frac{1}{2}\Big)\times\text{m}\times\Big(\frac{\text{F}^2\text{t}^2}{\text{m}^2}\Big)+\Big(\frac{1}{2}\Big)\times\Big(\frac{\text{m}\ell^2}{12}\Big)\Big[36\times\Big(\frac{\text{F}^2\text{t}^2}{\text{m}^2\ell^2}\Big)\Big]$
$=\frac{\text{F}^2\text{t}^2}{\text{2m}}+\frac{3}{2}\Big(\frac{\text{F}^2\text{t}^2}{\text{m}}\Big)=2\Big(\frac{\text{F}^2\text{t}^2}{\text{m}}\Big)$
- Angular momentum about the centre of mass:
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