Question

A particle is given an initial speed $u$ inside a smooth spherical shell of radius $R=1\text{m}$ and it is just able to complete the circle. Acceleration of the particle when its velocity is vertical is:

• A. $g\sqrt{10}$
• B. $g$
• C. $\frac{g}{\sqrt{2}}$
• D. $\frac{g}{\sqrt{6}}$

Answer 1

The correct option is A $g\sqrt{10}$

Since the particle is given just enough speed to complete the vertical circle,
$u^2=5gR$


Velocity is vertical when block is at height $R$ from bottom (Point $B$)
Applying energy conservation between $A$ and $B$
$U_A+K_A=U_B+K_B$
$\Rightarrow 0+\frac{1}{2}m{u}^2=\frac{1}{2}m{v}^2+mgR$
$\Rightarrow u^2=v^2+2gR$
$\Rightarrow v^2=5gR-2gR$
$\Rightarrow v^2=3gR$
Centripetal acceleration,
$a_c=\frac{v^2}{R}=3g$
Tangential acceleration,
$a_T=g$

Therefore, $a_{\text{total}}=\sqrt{a_{T^2}+a_{c^2}}$
$=\sqrt{g^2+9g^2}$
$=g\sqrt{10}$