A body weights 100 kg on earth . Find its weight on mars . The mass and radius of mars are 1/10 and 1/2 of the mass and radius of earth .

Answer

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

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

### Step 2: Determine the weight of the body on Earth
Given that the body weighs 100 kg on Earth, we can express this as:
\[ W_{Earth} = 100 \, \text{kg} \]

### Step 3: Find the acceleration due to gravity on Mars
We know that the mass and radius of Mars are given as:
- Mass of Mars (\( M_{Mars} \)) = \( \frac{1}{10} \) of the mass of Earth (\( M_{Earth} \))
- Radius of Mars (\( R_{Mars} \)) = \( \frac{1}{2} \) of the radius of Earth (\( R_{Earth} \))

The formula for acceleration due to gravity at the surface of a planet is:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant.

Using this, we can express the acceleration due to gravity on Mars (\( g_{Mars} \)):
\[ g_{Mars} = \frac{G \cdot M_{Mars}}{R_{Mars}^2} \]
Substituting the values of \( M_{Mars} \) and \( R_{Mars} \):
\[ g_{Mars} = \frac{G \cdot \left(\frac{1}{10} M_{Earth}\right)}{\left(\frac{1}{2} R_{Earth}\right)^2} \]
\[ g_{Mars} = \frac{G \cdot \left(\frac{1}{10} M_{Earth}\right)}{\left(\frac{1}{4} R_{Earth}^2\right)} \]
\[ g_{Mars} = \frac{G \cdot M_{Earth}}{R_{Earth}^2} \cdot \frac{1}{10} \cdot 4 \]
\[ g_{Mars} = 4 \cdot \frac{G \cdot M_{Earth}}{R_{Earth}^2} \cdot \frac{1}{10} \]
\[ g_{Mars} = \frac{4}{10} g_{Earth} = \frac{2}{5} g_{Earth} \]

### Step 4: Calculate the weight of the body on Mars
Now we can find the weight of the body on Mars using the mass and the new acceleration due to gravity:
\[ W_{Mars} = m \cdot g_{Mars} \]
Since the weight on Earth is 100 kg, we can find the mass:
\[ m = \frac{W_{Earth}}{g_{Earth}} \]
Assuming \( g_{Earth} \) is approximately 10 m/s² for simplicity:
\[ m = \frac{100 \, \text{kg}}{10 \, \text{m/s}^2} = 10 \, \text{kg} \]

Now substituting back to find the weight on Mars:
\[ W_{Mars} = m \cdot g_{Mars} = 10 \, \text{kg} \cdot \left(\frac{2}{5} \cdot 10 \, \text{m/s}^2\right) \]
\[ W_{Mars} = 10 \, \text{kg} \cdot 4 \, \text{m/s}^2 = 40 \, \text{kg} \]

### Final Answer
The weight of the body on Mars is 40 kg.

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