1. Condition for just reaching the top point $B$

For the bob to just complete the semicircle and reach the topmost point $B$, the string must remain just taut at $B$.

At the top point, minimum condition is: $T=0$ So centripetal force is provided only by weight: $\frac{mv_B^2}{L}=mg$ Hence, $v_B^2=gL$

Therefore kinetic energy at $B$ is $(K.E.)_B=\frac12 m v_B^2=\frac12 m(gL)=\frac12 mgL$

2. Use conservation of mechanical energy between $A$ and $B$

Take the lowest point $A$ as reference of potential energy zero.

Height gained from $A$ to $B$ is: $2L$ So increase in potential energy is: $\Delta U = mg(2L)=2mgL$

By energy conservation, $(K.E.)_A = (K.E.)_B + 2mgL$ Substitute $(K.E.)_B=\frac12 mgL$: $(K.E.)_A = \frac12 mgL + 2mgL = \frac52 mgL$

3. Find the ratio

$\frac{(K.E.)_A}{(K.E.)_B} = \frac{\frac52 mgL}{\frac12 mgL} = 5$

So, $(K.E.)_A : (K.E.)_B = 5:1$

4. Option check

• A: $5:1$ ✅
• B: $3:2$ ❌
• C: $1:5$ ❌
• D: $2:5$ ❌

Hence the correct option is A.