1. Model the conical pendulum

For a conical pendulum of string length $l$, making angle $\theta$ with the vertical (here Z-axis), the bob moves in a horizontal circle of radius $r$.

The forces on the bob are:

• Tension $T$ along the string
• Weight $mg$ downward

Resolving tension:

$T\cos\theta = mg$ $T\sin\theta = \frac{mv^2}{r}$

2. Eliminate $T$

Divide the second equation by the first:

$\frac{T\sin\theta}{T\cos\theta} = \frac{mv^2/r}{mg}$

$\tan\theta = \frac{v^2}{rg}$

So,

$v^2 = rg\tan\theta$

3. Substitute the given values

Given:

• $r = 0.4\,\text{m}$
• $g = 10\,\text{m/s}^2$
• $\theta = 45^\circ \Rightarrow \tan 45^\circ = 1$

Hence,

$v^2 = 0.4 \times 10 \times 1 = 4$

$v = 2\,\text{m/s}$

4. Check with options

• A: $0.4\,\text{m/s}$
• B: $4\,\text{m/s}$
• C: $0.2\,\text{m/s}$
• D: $2\,\text{m/s}$

Therefore, the correct option is:

$\boxed{\text{D: } 2\,\text{m/s}}$

5. Comparison with stored answer

Stored correct answer: $\text{D}$

Our derived answer is also $\text{D}$, so they agree.