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A small sphere of mass $m$ is suspended by a light and inextensible string of length $l$ from a point $O$ fixed on a smooth inclined plane of inclination $\theta $ with the horizontal. The sphere is moving in a circle on the incline as shown. If the sphere has a velocity $u$ at the top most position $A$ . Then,
the tension in the string as the sphere passes the $90^o$ position $B$ equal to $m\left( {\frac{{{u^2}}}{l} - 2g\sin \theta } \right)$
the tension in the string at the bottom most position $C$ equal to $m\left( {\frac{{{u^2}}}{l} + 5g\sin \theta } \right)$
the tension in the string as the sphere passes the $90^o$ position $B$ equal to $m\left( {\frac{{{u^2}}}{l} - 3g\sin \theta } \right)$
the tension in the string at the bottom most position $C$ equal to $m\left( {\frac{{{u^2}}}{l} - 5g\sin \theta } \right)$
Solution
$\mathrm{V}_{\mathrm{B}}=\sqrt{\mathrm{u}^{2}+2 \mathrm{g}\ell \sin \theta}$
$\mathrm{T}_{\mathrm{B}}=\frac{\mathrm{mv}_{\mathrm{B}}^{2}}{\ell} \mathrm{m}\left[\frac{\mathrm{u}^{2}}{\ell}+2 \mathrm{g} \sin \theta\right]$
$\mathrm{V}_{\mathrm{C}}^{2}=\left(\mathrm{u}^{2}+4 \mathrm{g}\ell \sin \theta)\right.$
$\mathrm{T}_{\mathrm{C}}-\mathrm{mg} \sin \theta=\frac{\mathrm{mv}_{\mathrm{C}}^{2}}{\ell} \Rightarrow \mathrm{T}_{\mathrm{C}}=\mathrm{m}\left[\frac{\mathrm{u}^{2}}{\ell}+5 \mathrm{g} \sin \theta\right]$
