JEE Advanced · Physics · 5. Laws of Motion

A bob of mass $m$ , suspended by a string of length $l_{1}$ , is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass $m$ , suspended by a string of length $l_{2}$ , which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $\frac{l_{1}}{l_{2}}$ is

Verified Solution

Answer & Solution

Correct Answer

(C) 5

Step-by-step Solution

Detailed explanation

As shown in the figure shown above, the minimum speed required by the bob of mass $m$ attached to the string of length $l_{1}$ in order to complete the full circle is $\sqrt{5 g l_{1}}$ .

When the bob goes to the highest point, its speed becomes $\sqrt{g l_{1}}$ . Here, it strikes with the identical mass, which is tied to the string of length $l_{2}$ . The speed delivered to the object is $\sqrt{g l_{1}}$ . This should be equal to $\sqrt{5 g l_{2}}$ so that this bob could complete a full circle.
$\sqrt{g l_{1}} = \sqrt{5 g l_{2}}$
$\Rightarrow g l_{1} = 5 g l_{2}$
$\Rightarrow \frac{l_{1}}{l_{2}} = 5$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

A right-angled prism of refractive index

μ_{1}

is placed in a rectangular block of refractive index

μ_{2}

, which is surrounded by a medium of refractive index

μ_{3}

, as shown in the figure. A ray of light

e

enters the rectangular block at normal incidence. Depending upon the relationships between

μ_{1}, μ_{2}

and

μ_{3}

, it takes one of the four possible paths

e f, e g, e h

e i

Match the paths in List

I

with conditions of refractive indices in List

II

and select the correct answer using the codes given below the lists:

	List I		List II
A.	$e \to f$	P.	$μ_{1} > \sqrt{2} μ_{2}$
B.	$e \to g$	Q.	$μ_{2} > μ_{1} and μ_{2} > μ_{3}$
C.	$e \to h$	R.	$μ_{1} = μ_{2}$
D.	$e \to i$	S.	$μ_{2} < μ_{1} < \sqrt{2} μ_{2} and μ_{2} > μ_{3}$

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