Two spheres, each of mass 625 kg, are placed with their centres 50 cm apart. The gravitational force between them is

Options

A. 10.42 dyne
B. 15.42 dyne
C. 20.42 dyne
D. 5.42 dyne

Answer

To solve the problem of finding the gravitational force between two spheres of mass 625 kg each, placed 50 cm apart, we can use Newton's law of universal gravitation. Here’s the step-by-step solution:

### Step 1: Write down the formula for gravitational force.
The gravitational force \( F \) between two masses is given by the formula:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( G \) is the universal gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \),
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.

### Step 2: Identify the values given in the problem.
From the problem, we have:
- Mass of the first sphere \( m_1 = 625 \, \text{kg} \)
- Mass of the second sphere \( m_2 = 625 \, \text{kg} \)
- Distance between the centers \( r = 50 \, \text{cm} = 0.5 \, \text{m} \) (converted from centimeters to meters).

### Step 3: Substitute the values into the formula.
Now we can substitute the known values into the gravitational force formula:
\[ F = \frac{(6.67 \times 10^{-11}) \cdot (625) \cdot (625)}{(0.5)^2} \]

### Step 4: Calculate \( r^2 \).
First, calculate \( r^2 \):
\[ r^2 = (0.5)^2 = 0.25 \, \text{m}^2 \]

### Step 5: Substitute \( r^2 \) back into the formula.
Now substitute \( r^2 \) into the equation:
\[ F = \frac{(6.67 \times 10^{-11}) \cdot (625) \cdot (625)}{0.25} \]

### Step 6: Calculate the numerator.
Calculate the numerator:
\[ 625 \cdot 625 = 390625 \]
Now, multiply this by \( G \):
\[ 6.67 \times 10^{-11} \cdot 390625 = 2.6046875 \times 10^{-5} \]

### Step 7: Divide by \( r^2 \).
Now divide by \( 0.25 \):
\[ F = \frac{2.6046875 \times 10^{-5}}{0.25} = 1.041875 \times 10^{-4} \]

### Step 8: Final answer.
Thus, the gravitational force \( F \) is approximately:
\[ F \approx 1.04 \times 10^{-4} \, \text{N} \]

### Summary of the solution:
The gravitational force between the two spheres is approximately \( 1.04 \times 10^{-4} \, \text{N} \).

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