A uniform rod of mass $m$ and length $l$ suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is _____ . ($g$ acceleration due to gravity)

Question

Accepted Answer

C. 1. Immediately after one string is cut, the rod is acted upon by only two external forces:

- Tension $T$ acting upward at the remaining end.
- Weight $mg$ acting downward through the centre of mass.

2. Let the centre of mass accelerate downward with acceleration $a$.

Applying Newton's second law for translational motion,

$$
mg-T=ma.
$$

3. The tension also produces a torque about the centre of mass.

Since the tension acts at a distance

$$
\frac{l}{2},
$$

the torque is

$$
	au=T\cdot\frac{l}{2}.
$$

4. The moment of inertia of a uniform rod about its centre is

$$
I=\frac{1}{12}ml^2.
$$

Using

$$
	au=I\alpha,
$$

we obtain

$$
T\cdot\frac{l}{2}
=
\frac{1}{12}ml^2\alpha.
$$

5. Immediately after the string is cut, the end attached to the remaining string behaves as the instantaneous pivot.

Hence, the linear acceleration of the centre of mass and the angular acceleration are related by

$$
a=\alpha\cdot\frac{l}{2},
$$

or

$$
\alpha=\frac{2a}{l}.
$$

6. Substitute this value of $\alpha$ into the torque equation:

$$
T\cdot\frac{l}{2}
=
\frac{1}{12}ml^2
\left(\frac{2a}{l}ight).
$$

Simplifying,

$$
T=\frac{ma}{3},
$$

or equivalently,

$$
ma=3T.
$$

7. From the translational equation,

$$
mg-T=ma.
$$

Replacing $ma$ by $3T$,

$$
mg-T=3T.
$$

Therefore,

$$
mg=4T,
$$

giving

$$
\boxed{T=\frac{mg}{4}}.
$$

8. Hence, the correct answer is **Option C**.

A uniform rod of mass mmm and length lll suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is _____ . (ggg acceleration due to gravity)

A uniform rod of mass $m$ and length $l$ suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is _____ . ( $g$ acceleration due to gravity)