A simple pendulum oscillates freely between points $A$ and $B$. We now put a peg (nail) at the point $C$ as shown in above figure. As the pendulum moves from $A$ to the right, the string will bend at $C$ and the pendulum will go to its extreme point $D$. Ignoring friction, the point $D$
KVPY 2011, Advanced
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(a)
Total length of a pendulum remains same, so extreme point $D$ lies on the line $A B$, as shown below.
This can be proved by applying energy conservation between extreme positions $A$ and $D$ (its given friction is abscent),
$K_A+U_A=K_B+U_B=K_D+U_D$
$\Rightarrow 0+U_A=0+U_B=0+U_D$
$\Rightarrow U_A=U_B=U_D \Rightarrow h_A=h_B=h_D$
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