On a planet (whose size is the same as that of earth and mass 4 times that of the earth ) the energy needed to lift a 2 kg mass vertically upwards through 2 m distance on the planet is (g = `10m// sec^(2)` on surface of earth )

Options

A. 16 J
B. 32 j
C. 160 J
D. 320 J

Explanation

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

### Step-by-Step Solution:

1. **Identify the known values:**
- Mass of the object (m) = 2 kg
- Height (h) = 2 m
- Gravitational acceleration on Earth (g) = 10 m/s²
- Mass of the planet (M) = 4 * Mass of Earth (M_E)

2. **Calculate the gravitational acceleration (g') on the new planet:**
The formula for gravitational acceleration at 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.

Since the radius of the planet is the same as that of Earth, we can express this as:
\[ g' = \frac{G \cdot (4 \cdot M_E)}{R_E^2} \]

We know that:
\[ g = \frac{G \cdot M_E}{R_E^2} \]
Therefore, substituting for \(M_E\) in terms of \(g\):
\[ g' = 4 \cdot g \]
Substituting \(g = 10 \, \text{m/s}^2\):
\[ g' = 4 \cdot 10 = 40 \, \text{m/s}^2 \]

3. **Calculate the energy required to lift the mass:**
The energy (E) required to lift an object against gravity is given by:
\[ E = m \cdot g' \cdot h \]
Substituting the known values:
\[ E = 2 \, \text{kg} \cdot 40 \, \text{m/s}^2 \cdot 2 \, \text{m} \]
\[ E = 2 \cdot 40 \cdot 2 = 160 \, \text{Joules} \]

4. **Final Result:**
The energy needed to lift the 2 kg mass vertically upwards through a distance of 2 m on the planet is **160 Joules**.