A body has a weight 90 kg on the earth's surface, the mass of the moon is 1/9 that of the earth's mass and its radius is 1/2 that of the earth's radius. On the moon the weight of the body is

Options

A. 45 kg
B. 202.5 kg
C. 90 kg
D. 40 kg

Answer

To find the weight of a body on the Moon given its weight on Earth, we can follow these steps:

### Step 1: Understand the relationship between weight, mass, and gravitational acceleration.
Weight is defined as the force exerted on an object due to gravity, which can be expressed as:
\[ W = m \cdot g \]
where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity.

### Step 2: Determine the weight of the body on Earth.
Given that the weight of the body on Earth is 90 kg, we can denote this as:
\[ W_E = 90 \, \text{kg} \]

### Step 3: Find the gravitational acceleration on Earth.
The gravitational acceleration on Earth (\( g_E \)) is approximately \( 9.8 \, \text{m/s}^2 \). However, for this problem, we will use the ratio of gravitational accelerations rather than their actual values.

### Step 4: Find the mass of the Earth and the Moon.
Let:
- \( M_E \) = Mass of the Earth
- \( M_M \) = Mass of the Moon = \( \frac{1}{9} M_E \)
- \( R_E \) = Radius of the Earth
- \( R_M \) = Radius of the Moon = \( \frac{1}{2} R_E \)

### Step 5: Write the formula for gravitational acceleration on the Moon.
The gravitational acceleration on the Moon (\( g_M \)) can be expressed as:
\[ g_M = \frac{G \cdot M_M}{R_M^2} \]
Substituting \( M_M \) and \( R_M \):
\[ g_M = \frac{G \cdot \left(\frac{1}{9} M_E\right)}{\left(\frac{1}{2} R_E\right)^2} \]
\[ g_M = \frac{G \cdot \frac{1}{9} M_E}{\frac{1}{4} R_E^2} \]
\[ g_M = \frac{4G \cdot M_E}{9R_E^2} \]

### Step 6: Relate the gravitational accelerations.
Since \( g_E = \frac{G \cdot M_E}{R_E^2} \), we can express \( g_M \) in terms of \( g_E \):
\[ g_M = \frac{4}{9} g_E \]

### Step 7: Calculate the weight of the body on the Moon.
The weight of the body on the Moon (\( W_M \)) can be calculated using the relationship:
\[ W_M = m \cdot g_M \]
We know that:
\[ W_E = m \cdot g_E \]
Thus, we can express the mass \( m \) as:
\[ m = \frac{W_E}{g_E} \]
Substituting this into the equation for \( W_M \):
\[ W_M = \frac{W_E}{g_E} \cdot g_M \]
Substituting \( g_M = \frac{4}{9} g_E \):
\[ W_M = \frac{W_E}{g_E} \cdot \left(\frac{4}{9} g_E\right) \]
The \( g_E \) cancels out:
\[ W_M = \frac{4}{9} W_E \]

### Step 8: Substitute the known weight from Earth.
Now substituting \( W_E = 90 \, \text{kg} \):
\[ W_M = \frac{4}{9} \cdot 90 \]
\[ W_M = 40 \, \text{kg} \]

### Final Answer:
The weight of the body on the Moon is **40 kg**.

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