The acceleration due to gravity on the surface of a pulsar of mass `M = 1 . 98 xx 10^(30)` kg and radius R = 12 km rotating with time period T =0 . 0 41 seconds is ` x xx 10^(11) m s^(-2) ` where the value of x is _____
`(G = 6 . 67 xx 10^(-11) MKS)`

Explanation

To find the acceleration due to gravity on the surface of a pulsar, we can use the formula for gravitational acceleration:

\[ g = \frac{G \cdot M}{R^2} \]

where:
- \( g \) is the acceleration due to gravity,
- \( G \) is the universal gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \),
- \( M \) is the mass of the pulsar, \( M = 1.98 \times 10^{30} \, \text{kg} \),
- \( R \) is the radius of the pulsar in meters.

### Step 1: Convert the radius from kilometers to meters
The radius of the pulsar is given as \( R = 12 \, \text{km} \). To convert this to meters:

\[ R = 12 \, \text{km} \times 1000 \, \text{m/km} = 12000 \, \text{m} \]

### Step 2: Substitute the values into the formula
Now, we can substitute the values of \( G \), \( M \), and \( R \) into the formula for gravitational acceleration:

\[ g = \frac{6.67 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \cdot 1.98 \times 10^{30} \, \text{kg}}{(12000 \, \text{m})^2} \]

### Step 3: Calculate \( R^2 \)
First, calculate \( R^2 \):

\[ R^2 = (12000 \, \text{m})^2 = 144000000 \, \text{m}^2 = 1.44 \times 10^8 \, \text{m}^2 \]

### Step 4: Substitute \( R^2 \) back into the equation
Now substitute \( R^2 \) back into the equation for \( g \):

\[ g = \frac{6.67 \times 10^{-11} \times 1.98 \times 10^{30}}{1.44 \times 10^8} \]

### Step 5: Calculate the numerator
Calculate the numerator:

\[ 6.67 \times 10^{-11} \times 1.98 \times 10^{30} = 1.32066 \times 10^{20} \]

### Step 6: Calculate \( g \)
Now divide the numerator by \( R^2 \):

\[ g = \frac{1.32066 \times 10^{20}}{1.44 \times 10^8} = 9.166 \times 10^{11} \, \text{m/s}^2 \]

### Step 7: Express \( g \) in the required form
The problem states that \( g \) can be expressed as \( x \times 10^{11} \, \text{m/s}^2 \). Thus, we have:

\[ g \approx 9.17 \times 10^{11} \, \text{m/s}^2 \]

### Step 8: Identify the value of \( x \)
From the above calculation, we can see that \( x \approx 9.17 \). If we round this to the nearest integer, we get:

\[ x \approx 9 \]

### Final Answer
The value of \( x \) is approximately **9**.

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