The uncertainty in position and velocity of the particle are `0.2mm` and `10.54xx10^(-27)ms^(-1)` respectively then the mass of the particle is `: ( h=6.625xx10^(-34)Js)`

Options

A. `48.86g`
B. `30.86g`
C. `24.86gm`
D. `100g`

Explanation

To solve the problem of finding the mass of a particle given the uncertainties in position and velocity, we will use the Heisenberg Uncertainty Principle. Here’s a step-by-step solution:

### Step 1: Identify the Given Values
- Uncertainty in position (Δx) = 0.2 mm = 0.2 × 10^(-3) m = 0.0002 m
- Uncertainty in velocity (Δv) = 10.54 × 10^(-27) m/s
- Planck's constant (h) = 6.626 × 10^(-34) Js

### Step 2: Write the Heisenberg Uncertainty Principle Formula
The Heisenberg Uncertainty Principle states:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where Δp is the uncertainty in momentum. Since momentum (p) is given by \( p = mv \), we can express the uncertainty in momentum as:
\[ \Delta p = m \cdot \Delta v \]
Thus, the equation becomes:
\[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]

### Step 3: Rearrange the Formula to Solve for Mass (m)
Rearranging the formula to solve for mass (m):
\[ m \geq \frac{h}{4\pi \Delta x \cdot \Delta v} \]

### Step 4: Substitute the Values into the Formula
Now, substituting the known values into the rearranged formula:
\[ m \geq \frac{6.626 \times 10^{-34}}{4 \cdot \pi \cdot 0.0002 \cdot 10.54 \times 10^{-27}} \]

### Step 5: Calculate the Denominator
Calculating the denominator:
\[ 4 \cdot \pi \cdot 0.0002 \cdot 10.54 \times 10^{-27} \approx 4 \cdot 3.14 \cdot 0.0002 \cdot 10.54 \times 10^{-27} \]
\[ \approx 0.0008 \cdot 10.54 \times 10^{-27} \approx 8.432 \times 10^{-30} \]

### Step 6: Calculate the Mass (m)
Now substituting back into the mass equation:
\[ m \geq \frac{6.626 \times 10^{-34}}{8.432 \times 10^{-30}} \approx 7.85 \times 10^{-5} \text{ kg} \]

### Step 7: Convert to Grams (if necessary)
To convert kg to grams:
\[ m \approx 7.85 \times 10^{-5} \text{ kg} \times 1000 \approx 0.0785 \text{ g} \]

### Final Answer
The mass of the particle is approximately **7.85 × 10^(-5) kg** or **0.0785 g**.

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