What will be the mass of a particle if uncertainty in its position is `10^(-8)` m and velocity is `5.26xx10^(-25)ms^(-1)`?

Options

A. 0.01 kg
B. 0.1 kg
C. 1 kg
D. 10 kg

Explanation

To find the mass of the particle given the uncertainty in its position and velocity, we can use Heisenberg's uncertainty principle. Here’s a step-by-step solution:

### Step 1: Write down the given values
- Uncertainty in position, \( \Delta x = 10^{-8} \, \text{m} \)
- Uncertainty in velocity, \( \Delta v = 5.26 \times 10^{-25} \, \text{m/s} \)

### Step 2: Use Heisenberg's uncertainty principle
According to Heisenberg's uncertainty principle, the relationship between the uncertainty in position and the uncertainty in momentum is given by the formula:
\[ \Delta x \cdot \Delta v = \frac{h}{4 \pi m} \]
where \( h \) is Planck's constant and \( m \) is the mass of the particle.

### Step 3: Rearrange the formula to solve for mass \( m \)
Rearranging the formula gives:
\[ m = \frac{h}{4 \pi \Delta x \Delta v} \]

### Step 4: Substitute the known values
Planck's constant \( h \) is approximately \( 6.626 \times 10^{-34} \, \text{Js} \). Now substituting the values into the equation:
\[ m = \frac{6.626 \times 10^{-34}}{4 \pi (10^{-8}) (5.26 \times 10^{-25})} \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ 4 \pi \approx 12.566 \]
Now substituting:
\[ m = \frac{6.626 \times 10^{-34}}{12.566 \times 10^{-8} \times 5.26 \times 10^{-25}} \]

Calculating \( 12.566 \times 10^{-8} \times 5.26 \times 10^{-25} \):
\[ 12.566 \times 5.26 \approx 66.1 \quad \text{and} \quad 10^{-8} \times 10^{-25} = 10^{-33} \]
So,
\[ 12.566 \times 10^{-8} \times 5.26 \times 10^{-25} \approx 66.1 \times 10^{-33} \]

### Step 6: Final calculation for mass
Now substituting back:
\[ m = \frac{6.626 \times 10^{-34}}{66.1 \times 10^{-33}} \approx 0.01 \, \text{kg} \]

### Conclusion
The mass of the particle is approximately \( 0.01 \, \text{kg} \).