The uncertainty in the position and velocity of a particle are `10^(-10)" m and "5.27 xx 10^(-24)" m s"^(-1)` respectively. Calculate the mass of the particle `(h = 6.625 xx 10^(-34)" Js").`

Explanation

To solve the problem of calculating the mass of the particle using the given uncertainties in position and velocity, we will apply the Heisenberg Uncertainty Principle. Here’s the step-by-step solution:

### Step 1: Understand the Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is greater than or equal to a constant value, which is given by:
\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]
where:
- Δx = uncertainty in position
- Δv = uncertainty in velocity
- h = Planck's constant
- m = mass of the particle

### Step 2: Rearrange the Formula to Solve for Mass
We can rearrange the equation to solve for the mass (m):
\[ m \geq \frac{h}{4\pi \Delta x \cdot \Delta v} \]

### Step 3: Substitute the Given Values
Given:
- Δx = \(10^{-10}\) m
- Δv = \(5.27 \times 10^{-24}\) m/s
- h = \(6.625 \times 10^{-34}\) Js

Now substitute these values into the rearranged formula:
\[ m \geq \frac{6.625 \times 10^{-34}}{4 \cdot \pi \cdot (10^{-10}) \cdot (5.27 \times 10^{-24})} \]

### Step 4: Calculate the Denominator
First, calculate the denominator:
\[ 4 \cdot \pi \cdot (10^{-10}) \cdot (5.27 \times 10^{-24}) \approx 4 \cdot 3.14 \cdot 10^{-10} \cdot 5.27 \times 10^{-24} \]
Calculating this step-by-step:
1. Calculate \(4 \cdot 3.14 \approx 12.56\)
2. Multiply \(12.56 \cdot 10^{-10} \cdot 5.27 \times 10^{-24} \approx 12.56 \cdot 5.27 \times 10^{-34}\)
3. Calculate \(12.56 \cdot 5.27 \approx 66.24\)

So, the denominator is approximately:
\[ 66.24 \times 10^{-34} \]

### Step 5: Calculate the Mass
Now substitute this back into the mass equation:
\[ m \geq \frac{6.625 \times 10^{-34}}{66.24 \times 10^{-34}} \approx \frac{6.625}{66.24} \]
Calculating this gives:
\[ m \approx 0.1 \text{ kg} \]

### Final Answer
The mass of the particle is approximately \(0.1 \text{ kg}\).

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