Calculate the uncertainty in the position of a dust particle with mass equal to 1 mg if the uncertainty in its velocity is `5.5 xx 10^(-20) m s^(-1)`

Explanation

To solve the problem of calculating the uncertainty in the position of a dust particle, we will use the Heisenberg Uncertainty Principle. The principle states that the uncertainty in position (Δx) multiplied by the uncertainty in momentum (Δp) is equal to or greater than a constant (h/4π), where h is Planck's constant.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Mass of the dust particle (m) = 1 mg = 1 x 10^(-3) g = 1 x 10^(-6) kg
- Uncertainty in velocity (Δv) = 5.5 x 10^(-20) m/s
- Planck's constant (h) = 6.626 x 10^(-34) J·s

2. **Write the Heisenberg Uncertainty Principle Formula**:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where Δp (uncertainty in momentum) is given by:
\[ \Delta p = m \cdot \Delta v \]

3. **Calculate the Uncertainty in Momentum (Δp)**:
\[ \Delta p = m \cdot \Delta v = (1 \times 10^{-6} \text{ kg}) \cdot (5.5 \times 10^{-20} \text{ m/s}) = 5.5 \times 10^{-26} \text{ kg m/s} \]

4. **Substitute Δp into the Uncertainty Principle Formula**:
\[ \Delta x \cdot (5.5 \times 10^{-26}) \geq \frac{6.626 \times 10^{-34}}{4\pi} \]

5. **Calculate the Right Side of the Inequality**:
- First, calculate \(4\pi\):
\[ 4\pi \approx 12.566 \]
- Now calculate \(\frac{h}{4\pi}\):
\[ \frac{6.626 \times 10^{-34}}{12.566} \approx 5.280 \times 10^{-35} \text{ J·s} \]

6. **Rearrange to Solve for Δx**:
\[ \Delta x \geq \frac{5.280 \times 10^{-35}}{5.5 \times 10^{-26}} \]

7. **Calculate Δx**:
\[ \Delta x \geq \frac{5.280 \times 10^{-35}}{5.5 \times 10^{-26}} \approx 9.6 \times 10^{-10} \text{ m} \]

### Final Answer:
The uncertainty in the position of the dust particle is approximately \(9.6 \times 10^{-10}\) meters.

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