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/4π×`10^(−13)`
​
B. h×4×`10^33`
C. `h/
(4π)`×`10 ^(−13)`
D. None

Explanation

To find the mass of the particle using the uncertainty principle, we can follow these steps:

### Step 1: Understand the Uncertainty Principle
The uncertainty principle states that the product of the uncertainties in position (Δx) and momentum (Δp) is greater than or equal to a constant, which is given by:

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

where \( h \) is Planck's constant.

### Step 2: Relate Momentum to Mass and Velocity
Momentum (p) is defined as the product of mass (m) and velocity (v):

\[ p = m \cdot v \]

Thus, the uncertainty in momentum (Δp) can be expressed as:

\[ \Delta p = m \cdot \Delta v \]

### Step 3: Substitute into the Uncertainty Principle
Substituting Δp into the uncertainty principle gives us:

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

### Step 4: Rearrange to Solve for Mass
Rearranging the equation to solve for mass (m) yields:

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

### Step 5: Substitute the Given Values
We are given:

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

Now, we can substitute these values into the equation. First, we need to calculate \( \Delta x \cdot \Delta v \):

\[ \Delta x \cdot \Delta v = 10^{10} \cdot (10 \times 10^{-24}) = 10^{10} \cdot 10^{-23} = 10^{-13} \]

### Step 6: Use the Value of Planck's Constant
Planck's constant \( h \) is approximately \( 6.626 \times 10^{-34} \, \text{Js} \). Now substituting this into our equation:

\[ m \geq \frac{6.626 \times 10^{-34}}{4\pi \cdot 10^{-13}} \]

### Step 7: Calculate the Mass
Calculating the denominator:

\[ 4\pi \approx 12.566 \]

Thus,

\[ m \geq \frac{6.626 \times 10^{-34}}{12.566 \times 10^{-13}} \approx \frac{6.626}{12.566} \times 10^{-34 + 13} \approx 0.528 \times 10^{-21} \, \text{kg} \]

### Final Result
The mass of the particle is approximately:

\[ m \approx 0.528 \times 10^{-21} \, \text{kg} \]