If a planet existed whose mass was twice that of Earth and whose radius 3 times greater, how much will a 1kg mass weigh on the planet?

Options

A. 25 N
B. 2.17 N
C. 1.1 N
D. 5 N

Explanation

To find the weight of a 1 kg mass on a hypothetical planet with a mass twice that of Earth and a radius three times greater, we can follow these steps:

### Step 1: Understand the formula for weight
The weight \( W \) of an object is given by the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity at that location.

### Step 2: Identify the values for the new planet
- The mass of the new planet \( M_p = 2M_e \) (where \( M_e \) is the mass of Earth).
- The radius of the new planet \( R_p = 3R_e \) (where \( R_e \) is the radius of Earth).

### Step 3: Write the formula for acceleration due to gravity
The acceleration due to gravity \( g \) on the surface of a planet is given by:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

### Step 4: Calculate the new acceleration due to gravity \( g' \)
For the new planet, we can express the acceleration due to gravity \( g' \) as:
\[ g' = \frac{G \cdot M_p}{R_p^2} \]
Substituting the values we identified:
\[ g' = \frac{G \cdot (2M_e)}{(3R_e)^2} \]
This simplifies to:
\[ g' = \frac{2G \cdot M_e}{9R_e^2} \]

### Step 5: Relate \( g' \) to Earth's gravity \( g \)
We know that:
\[ g = \frac{G \cdot M_e}{R_e^2} \]
Thus, we can express \( g' \) in terms of \( g \):
\[ g' = \frac{2}{9} \cdot g \]

### Step 6: Substitute the value of \( g \)
The standard value of \( g \) on Earth is approximately \( 9.8 \, \text{m/s}^2 \):
\[ g' = \frac{2}{9} \cdot 9.8 \approx 2.18 \, \text{m/s}^2 \]

### Step 7: Calculate the weight on the new planet
Now, we can find the weight \( W' \) of the 1 kg mass on the new planet:
\[ W' = m \cdot g' = 1 \, \text{kg} \cdot 2.18 \, \text{m/s}^2 = 2.18 \, \text{N} \]

### Step 8: Conclusion
The weight of a 1 kg mass on the new planet is approximately \( 2.18 \, \text{N} \).