The uncertainties in position and the velocity of a particle are `10^(-10)` m and 10×`10^(−22).sec^(−1)` respectively. The mass of the particle in kg is

Options

A. h×4×`10^23`
B. h/4π×`10^(−31)`
C. `h/
(4π)`×`10 ^(−24)`
D. None of these

Explanation

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states that:

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

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

### Step 1: Identify the given values
- Uncertainty in position, \(\Delta x = 10^{-10} \, \text{m}\)
- Uncertainty in velocity, \(\Delta v = 10 \times 10^{-22} \, \text{s}^{-1} = 10^{-21} \, \text{s}^{-1}\)

### Step 2: Rearrange the uncertainty principle formula to solve for mass \(m\)
From the uncertainty principle, we can rearrange the formula to find the mass \(m\):

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

### Step 3: Substitute the known values into the equation
Now we will substitute the values of \(h\), \(\Delta x\), and \(\Delta v\):

\[ m \geq \frac{6.626 \times 10^{-34} \, \text{Js}}{4\pi (10^{-10} \, \text{m})(10^{-21} \, \text{s}^{-1})} \]

### Step 4: Calculate the denominator
First, calculate the denominator:

\[ 4\pi (10^{-10})(10^{-21}) = 4\pi \times 10^{-31} \]

### Step 5: Calculate mass \(m\)
Now substitute this back into the equation for mass:

\[ m \geq \frac{6.626 \times 10^{-34}}{4\pi \times 10^{-31}} \]

### Step 6: Calculate the value
Using the approximate value of \(\pi \approx 3.14\):

\[ 4\pi \approx 12.56 \]

Thus, we have:

\[ m \geq \frac{6.626 \times 10^{-34}}{12.56 \times 10^{-31}} = \frac{6.626}{12.56} \times 10^{-34 + 31} = \frac{6.626}{12.56} \times 10^{-3} \]

Calculating \(\frac{6.626}{12.56} \approx 0.528\):

\[ m \geq 0.528 \times 10^{-3} \, \text{kg} = 5.28 \times 10^{-4} \, \text{kg} \]

### Final Answer
Thus, the mass of the particle is approximately:

\[ m \approx 5.28 \times 10^{-4} \, \text{kg} \]