A man weighs 54 kg on the surface of earth. How much will he weigh on the surface of a planet whose mass is one-fifth and radius is half of that of earth ?

Answer

To find out how much a man weighing 54 kg on the surface of the Earth would weigh on a different planet with a mass that is one-fifth and a radius that is half of that of Earth, we can follow these steps:

### Step 1: Understand the weight on Earth
The weight of an object on the surface of a planet is given by the formula:
\[ W = mg \]
where:
- \( W \) is the weight,
- \( m \) is the mass of the object (54 kg in this case),
- \( g \) is the acceleration due to gravity on that planet.

On Earth, \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

### Step 2: Calculate the acceleration due to gravity on the new planet
The formula for the acceleration due to gravity \( g \) on the surface of a planet is given by:
\[ g = \frac{GM}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

For the new planet:
- The mass \( M_p \) is \( \frac{M}{5} \) (one-fifth of Earth's mass),
- The radius \( R_p \) is \( \frac{R}{2} \) (half of Earth's radius).

Substituting these values into the formula for \( g \):
\[ g_p = \frac{G \left(\frac{M}{5}\right)}{\left(\frac{R}{2}\right)^2} \]

### Step 3: Simplify the expression for \( g_p \)
Substituting the radius into the equation:
\[ g_p = \frac{G \left(\frac{M}{5}\right)}{\frac{R^2}{4}} \]
This simplifies to:
\[ g_p = \frac{4GM}{5R^2} \]

### Step 4: Relate \( g_p \) to \( g \)
We know that the acceleration due to gravity on Earth is:
\[ g = \frac{GM}{R^2} \]
Thus, we can express \( g_p \) in terms of \( g \):
\[ g_p = \frac{4}{5} g \]

### Step 5: Calculate the weight on the new planet
Now, substituting \( g_p \) into the weight formula for the new planet:
\[ W' = mg_p = m \left(\frac{4}{5} g\right) \]
Substituting \( m = 54 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \):
\[ W' = 54 \cdot \left(\frac{4}{5} \cdot 9.8\right) \]

### Step 6: Perform the calculation
Calculating \( W' \):
\[ W' = 54 \cdot \left(\frac{4 \cdot 9.8}{5}\right) \]
\[ W' = 54 \cdot \left(\frac{39.2}{5}\right) \]
\[ W' = 54 \cdot 7.84 \]
\[ W' = 422.56 \, \text{N} \]

### Final Answer
The weight of the man on the surface of the new planet is approximately \( 422.56 \, \text{N} \).

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