Question 178: Work, Energy and Power & Circular Motion

Question

A point mass ' $m$ ' attached at one end of a massless, inextensible string of length ' $l$ ' performs a vertical circular motion and the string rotates in vertical plane, as shown in the diagram. The increase in the centripetal acceleration of the point mass when it moves from point A to point C is
[ $\mathrm{g}=$ acceleration due to gravity.]

Accepted Answer

$$ \begin{aligned} &	ext { By Conservation of energy, }\ &\begin{aligned} & \mathrm{KE}_{\mathrm{A}}+\mathrm{PE}_{\mathrm{A}}=\mathrm{KE}_{\mathrm{C}}+\mathrm{PE}_{\mathrm{C}} \ 	herefore \quad & 2 \mathrm{mgl}=\frac{1}{2} \mathrm{mv}_{\mathrm{c}}^2+\mathrm{mgl} \quad \ldots\left(\because \mathrm{At} \mathrm{~A}, \mathrm{v}_{\mathrm{A}}=0ight) \end{aligned} \end{aligned} $$
$$ 	herefore \quad \mathrm{v}_{\mathrm{C}}^2=2 \mathrm{gl} $$
We know,
Centripetal acceleration $=\frac{\mathrm{v}^2}{\mathrm{r}}$
$\ldots(\mathrm{r}=$ radius of circular motion $)$
$	herefore \quad$ At point A ,
$$ a_A=\frac{v_A^2}{l} \Rightarrow a_A=0 \ldots\left(\because v_A=0ight) $$
$	herefore \quad$ At point C ,
$$ \mathrm{a}_{\mathrm{C}}=\frac{\mathrm{v}_{\mathrm{C}}^2}{\mathrm{l}} \Rightarrow \mathrm{a}_{\mathrm{C}}=2 \mathrm{~g} $$
$	herefore \quad$ Increase in the centripetal acceleration of the point mass from point A to point C is 2 g

Answer

3 g

Answer

2 g

Answer

g

Answer

$\frac{\mathrm{g}}{2}$

Work, Energy and Power & Circular Motion

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