The uncertainties in position and the velocity of a particle are `10^(-10)` m and 10×`10^(−24).sec^(−1)` respectively. The mass of the particle in kg is

Options

A. h×4×`10^23`
B. `h/
(4π)`×`10 ^(−33)`
C. h/4π×`10^(−33)`
D. None

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, \(\Delta v = 10 \times 10^{-24} \, \text{s}^{-1} = 10^{-23} \, \text{s}^{-1}\)

### 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:

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

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

\[ m \geq \frac{6.626 \times 10^{-34} \, \text{Js}}{4\pi \cdot (10^{-10} \, \text{m}) \cdot (10^{-23} \, \text{s}^{-1})} \]

### Step 4: Calculate the denominator
First, calculate \(4\pi \cdot (10^{-10}) \cdot (10^{-23})\):

\[ 4\pi \approx 12.566 \]
\[ 4\pi \cdot (10^{-10}) \cdot (10^{-23}) = 12.566 \cdot 10^{-33} \, \text{m s}^{-1} \]

### Step 5: Substitute back into the mass equation
Now substitute this back into the mass equation:

\[ m \geq \frac{6.626 \times 10^{-34}}{12.566 \times 10^{-33}} \]

### Step 6: Perform the division
Calculating the right side gives:

\[ m \geq \frac{6.626}{12.566} \times 10^{-34 + 33} = 0.528 \times 10^{-1} \, \text{kg} = 5.28 \times 10^{-2} \, \text{kg} \]

### Conclusion
Thus, the mass of the particle is approximately:

\[ m \geq 5.28 \times 10^{-2} \, \text{kg} \]