The uncertainties in position and the velocity of a particle are `10^(10)` m and 1×`10^(−34).sec^(−1)` respectively. The mass of the particle in kg is

Options

A. h/4π×`10^(−24)`
B. `h/
(4π)`×`10 ^(−34)`
C. h×4×`10^33`
D. None of these

Answer

To solve the problem, we will use the Heisenberg Uncertainty Principle, which is given by the formula:

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

Where:
- \(\Delta x\) = uncertainty in position
- \(m\) = mass of the particle
- \(\Delta v\) = uncertainty in velocity
- \(h\) = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))

### Step-by-Step Solution:

**Step 1: Identify the given values.**
- \(\Delta x = 10^{10} \, \text{m}\)
- \(\Delta v = 1 \times 10^{-34} \, \text{s}^{-1}\)

**Step 2: Write down the uncertainty principle equation.**
\[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]

**Step 3: Rearrange the equation to solve for mass \(m\).**
\[ m \geq \frac{h}{4\pi \Delta x \Delta v} \]

**Step 4: Substitute the known values into the equation.**
- Substitute \(h = 6.626 \times 10^{-34} \, \text{Js}\), \(\Delta x = 10^{10} \, \text{m}\), and \(\Delta v = 1 \times 10^{-34} \, \text{s}^{-1}\).

\[ m \geq \frac{6.626 \times 10^{-34}}{4\pi \cdot (10^{10}) \cdot (1 \times 10^{-34})} \]

**Step 5: Calculate the denominator.**
\[ 4\pi \cdot (10^{10}) \cdot (1 \times 10^{-34}) = 4\pi \cdot 10^{-24} \]

**Step 6: Substitute the denominator back into the equation.**
\[ m \geq \frac{6.626 \times 10^{-34}}{4\pi \cdot 10^{-24}} \]

**Step 7: Calculate the mass \(m\).**
\[ m \geq \frac{6.626 \times 10^{-34}}{4\pi \cdot 10^{-24}} = \frac{6.626}{4\pi} \times 10^{-10} \]

**Step 8: Calculate the numerical value.**
Using \(\pi \approx 3.14\):
\[ m \geq \frac{6.626}{4 \times 3.14} \times 10^{-10} \approx \frac{6.626}{12.56} \times 10^{-10} \approx 0.528 \times 10^{-10} \, \text{kg} \]

Thus, the mass of the particle is approximately:
\[ m \approx 5.28 \times 10^{-11} \, \text{kg} \]

### Final Answer:
The mass of the particle is approximately \(5.28 \times 10^{-11} \, \text{kg}\).