Calculate the weight of a body on the surface of Mars whose mass is 1/9 of the mass of the earth and radius is half the radius of the earth, and the body weighs 60 kg-f on the surface of the earth.

Explanation

To calculate the weight of a body on the surface of Mars, we can follow these steps:

### Step 1: Understand the given information
- The mass of Mars is \( \frac{1}{9} \) of the mass of Earth.
- The radius of Mars is \( \frac{1}{2} \) of the radius of Earth.
- The weight of the body on Earth is \( 60 \, \text{kg-f} \).

### Step 2: Write the formula for weight
The weight of a body is given by the formula:
\[ W = m \cdot g \]
where \( W \) is the weight, \( m \) is the mass of the body, and \( g \) is the acceleration due to gravity.

### Step 3: Calculate the mass of the body
Given that the weight of the body on Earth is \( 60 \, \text{kg-f} \), we can express this in terms of mass and gravitational acceleration on Earth (\( g_e \)):
\[ W_e = m \cdot g_e \]
From this, we can find the mass \( m \):
\[ m = \frac{W_e}{g_e} \]
Assuming \( g_e \approx 9.8 \, \text{m/s}^2 \), we can calculate \( m \):
\[ m = \frac{60 \, \text{kg-f}}{9.8 \, \text{m/s}^2} \approx 6.12 \, \text{kg} \]

### Step 4: Calculate the acceleration due to gravity on Mars
The acceleration due to gravity on Mars (\( g_m \)) can be calculated using the formula:
\[ g_m = \frac{G \cdot M_m}{R_m^2} \]
Where:
- \( G \) is the universal gravitational constant,
- \( M_m \) is the mass of Mars,
- \( R_m \) is the radius of Mars.

Given:
- \( M_m = \frac{1}{9} M_e \)
- \( R_m = \frac{1}{2} R_e \)

Substituting these values into the equation for \( g_m \):
\[ g_m = \frac{G \cdot \left(\frac{1}{9} M_e\right)}{\left(\frac{1}{2} R_e\right)^2} = \frac{G \cdot \frac{1}{9} M_e}{\frac{1}{4} R_e^2} = \frac{4G \cdot M_e}{9R_e^2} \]
We know that \( g_e = \frac{G \cdot M_e}{R_e^2} \), thus:
\[ g_m = \frac{4}{9} g_e \]

### Step 5: Calculate the weight of the body on Mars
Now we can calculate the weight of the body on Mars:
\[ W_m = m \cdot g_m = m \cdot \left(\frac{4}{9} g_e\right) \]
Substituting \( m \) and \( g_e \):
\[ W_m = 6.12 \, \text{kg} \cdot \left(\frac{4}{9} \cdot 9.8 \, \text{m/s}^2\right) \]
Calculating \( W_m \):
\[ W_m \approx 6.12 \cdot \left(\frac{4 \cdot 9.8}{9}\right) \approx 6.12 \cdot 4.36 \approx 26.67 \, \text{kg-f} \]

### Final Answer:
The weight of the body on the surface of Mars is approximately \( 26.67 \, \text{kg-f} \).

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