The uncertainty in the position of a flying dust particle of mass 0.1 mg is found to be `10^(-4)`m. The uncertainty in the position of another flying dust particle of mass 0.5 mg is also found to be same
If a third flying dust particle of mass 1 mg had the same uncertainty in position as in its velocity, the uncertainty in its momentum would be

Options

A. `5.2 xx 10^(-28) kg ms^(-1)`
B. `5.2 xx 10^(-34) kg ms^(-1)`
C. `7.2 xx 10^(-15) kg ms^(-1)`
D. `7.2 xx 10^(-21) kg ms^(-1)`

Explanation

To solve the problem, we will use the Heisenberg Uncertainty Principle, which relates the uncertainties in position and momentum. The principle is given by the formula:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum,
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\)).

### Step 1: Identify the given values
- Mass of the third dust particle, \(m = 1 \, \text{mg} = 1 \times 10^{-3} \, \text{g} = 1 \times 10^{-6} \, \text{kg}\)
- Uncertainty in position, \(\Delta x = 10^{-4} \, \text{m}\)

### Step 2: Relate uncertainty in position and velocity
According to the problem, the uncertainty in position is equal to the uncertainty in velocity (\(\Delta v\)):
\[ \Delta v = \Delta x = 10^{-4} \, \text{m} \]

### Step 3: Calculate the uncertainty in momentum
The uncertainty in momentum (\(\Delta p\)) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]

Substituting the values:
\[ \Delta p = (1 \times 10^{-6} \, \text{kg}) \cdot (10^{-4} \, \text{m/s}) \]

### Step 4: Perform the calculation
\[ \Delta p = 1 \times 10^{-6} \times 10^{-4} = 1 \times 10^{-10} \, \text{kg m/s} \]

### Step 5: Conclusion
The uncertainty in momentum of the third dust particle is:
\[ \Delta p = 1 \times 10^{-10} \, \text{kg m/s} \]