A planet whose size is the same and mass is `4` times as that of Earth, find the amount of energy needed to lift a `2 kg` mass vertically upwards through `2m` distance on the planet. The value of `g` on the surface of Earth is `10 ms^(-2)`.

Options

A. `80J`
B. `160 J`
C. `100J`
D. `24J`

Explanation

To solve the problem, we need to find the amount of energy required to lift a 2 kg mass vertically upwards through a distance of 2 meters on a planet that has the same radius as Earth but has a mass that is 4 times that of Earth.

### Step-by-Step Solution:

1. **Determine the acceleration due to gravity on the new planet (g')**:
- We know that the acceleration due to gravity on Earth (g) is given as \(10 \, \text{m/s}^2\).
- The formula for gravitational acceleration is:
\[ g' = \frac{GM'}{R^2} \]
- Since the mass of the new planet (M') is 4 times the mass of Earth (M), we can write:
\[ M' = 4M \]
- The radius (R) remains the same as that of Earth. Thus, we can substitute:
\[ g' = \frac{G(4M)}{R^2} = 4 \cdot \frac{GM}{R^2} = 4g \]
- Substituting the value of g:
\[ g' = 4 \cdot 10 \, \text{m/s}^2 = 40 \, \text{m/s}^2 \]

2. **Calculate the work done (energy required) to lift the mass**:
- The work done (W) in lifting an object is given by the formula:
\[ W = m \cdot g' \cdot h \]
- Where:
- \(m\) is the mass being lifted (2 kg),
- \(g'\) is the acceleration due to gravity on the new planet (40 m/s²),
- \(h\) is the height through which the mass is lifted (2 m).
- Plugging in the values:
\[ W = 2 \, \text{kg} \cdot 40 \, \text{m/s}^2 \cdot 2 \, \text{m} \]
- Calculating:
\[ W = 2 \cdot 40 \cdot 2 = 160 \, \text{Joules} \]

### Final Answer:
The amount of energy needed to lift the 2 kg mass vertically upwards through a distance of 2 meters on the planet is **160 Joules**.