A bar magnet is hung by a thin cotton thread in a uniform horizontal magnetic field and is in' equilibrium state. The energy required to rotate it by `60^(circ)` is `W`. Now the torque required to keep the magnet in this new: position is `sqrt(k) W`. Find `k`.

Answer

To solve the problem, we need to analyze the situation of a bar magnet in a uniform magnetic field and calculate the torque required to keep it in a new position after being rotated by \(60^\circ\).

### Step 1: Understand the initial conditions
The bar magnet is initially in equilibrium in a uniform magnetic field. The magnetic moment of the magnet is denoted as \(M\) and the magnetic field as \(B\). The torque (\(\tau\)) acting on the magnet in equilibrium is given by:
\[ \tau = M \times B \times \sin(\theta) \]
where \(\theta\) is the angle between the magnetic moment and the magnetic field.

### Step 2: Calculate the work done \(W\)
The work done \(W\) to rotate the magnet by \(60^\circ\) can be expressed as:
\[ W = M \cdot B \cdot (1 - \cos(60^\circ)) \]
Since \(\cos(60^\circ) = \frac{1}{2}\), we have:
\[ W = M \cdot B \cdot \left(1 - \frac{1}{2}\right) = M \cdot B \cdot \frac{1}{2} = \frac{1}{2} M \cdot B \]

### Step 3: Calculate the torque in the new position
After rotating the magnet by \(60^\circ\), the angle between the magnetic moment and the magnetic field becomes \(60^\circ\). The torque required to keep the magnet in this new position is given by:
\[ \tau' = M \cdot B \cdot \sin(60^\circ) \]
Using \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), we have:
\[ \tau' = M \cdot B \cdot \frac{\sqrt{3}}{2} \]

### Step 4: Relate the new torque to \(W\)
From the previous steps, we have:
\[ \tau' = \sqrt{k} W \]
Substituting for \(W\):
\[ \tau' = \sqrt{k} \cdot \frac{1}{2} M \cdot B \]

### Step 5: Equate the two expressions for torque
Now we can equate the two expressions for torque:
\[ M \cdot B \cdot \frac{\sqrt{3}}{2} = \sqrt{k} \cdot \frac{1}{2} M \cdot B \]
We can cancel \(M \cdot B\) from both sides (assuming \(M \cdot B \neq 0\)):
\[ \frac{\sqrt{3}}{2} = \sqrt{k} \cdot \frac{1}{2} \]

### Step 6: Solve for \(k\)
Multiplying both sides by 2:
\[ \sqrt{3} = \sqrt{k} \]
Squaring both sides gives:
\[ 3 = k \]

### Final Answer
Thus, the value of \(k\) is:
\[ \boxed{3} \]