1. Set up the motion

A small ball slides बिना friction on the inside of a smooth hemispherical vessel.

• It is released from rest at point $P$.
• At a general point $Q$, let the angular position from $P$ be $\alpha$.

We need the relation between $A=\frac{\text{centripetal force}}{\text{normal reaction}}.$

2. Speed of the ball at angle $\alpha$

Take radius of the vessel as $R$ and mass of ball as $m$.

If the ball has moved through angle $\alpha$ from the top point $P$, then its vertical drop is $h=R(1-\cos\alpha).$

By energy conservation, $mgR(1-\cos\alpha)=\frac12 mv^2$ so $v^2=2gR(1-\cos\alpha).$

Hence the required centripetal force is $F_c=\frac{mv^2}{R}=2mg(1-\cos\alpha).$

3. Normal reaction at point $Q$

Since the ball is moving on the inside of the sphere, toward the center the radial force balance is $N-mg\cos\alpha=\frac{mv^2}{R}.$

So, $N=\frac{mv^2}{R}+mg\cos\alpha.$

Substitute $\dfrac{mv^2}{R}=2mg(1-\cos\alpha)$: $N=2mg(1-\cos\alpha)+mg\cos\alpha$ $N=mg(2-\cos\alpha).$

4. Find the ratio $A$

Therefore, A=\frac{F_c}{N}=\frac{\dfrac{mv^2}{R}}{N}=rac{2mg(1-\cos\alpha)}{mg(2-\cos\alpha)}.

Thus, $\boxed{A=\frac{2(1-\cos\alpha)}{2-\cos\alpha}}.$

5. Nature of the graph

Now examine how $A$ changes with $\alpha$ as the ball moves from $Q$ to $R$.

Let $x=\cos\alpha.$ Then $A=\frac{2(1-x)}{2-x}.$

As $\alpha$ increases from $0$ to $\pi/2$ (top to equator of hemisphere), $\cos\alpha$ decreases from $1$ to $0$.

Important values:

• At $\alpha=0$: $A=\frac{2(1-1)}{2-1}=0.$

• At $\alpha=\pi/2$: $A=\frac{2(1-0)}{2-0}=1.$

Now check monotonicity: $A(x)=\frac{2(1-x)}{2-x}$ Differentiate w.r.t. $x$: \frac{dA}{dx}=\frac{-2(2-x)-2(1-x)(-1)}{(2-x)^2}=rac{-4+2x+2-2x}{(2-x)^2}=-\frac{2}{(2-x)^2}<0. Since $x=\cos\alpha$ decreases with increasing $\alpha$, $A$ increases with $\alpha$.

Also, near $\alpha=0$, slope starts from zero and then rises smoothly; the curve is increasing and approaches $1$ at $\pi/2$.

So the correct graph is the one that:

• starts from $(0,0)$,
• increases monotonically,
• reaches $(\pi/2,1)$.

This corresponds to Option C.

6. Comparison with stored answer

Stored correct answer = C.

My derived answer = C.

So they agree.