Answer :
To determine on which body the gravitational potential energy (GPE) is the least for a [tex]\(100 \text{ kg}\)[/tex] mass lifted to a height of [tex]\(3 \text{ m}\)[/tex], we can use the formula for gravitational potential energy:
[tex]\[ \text{GPE} = m \cdot g \cdot h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass in kilograms (kg),
- [tex]\( g \)[/tex] is the acceleration due to gravity in meters per second squared ([tex]\( m/s^2 \)[/tex]),
- [tex]\( h \)[/tex] is the height in meters (m).
We need to calculate the GPE for each of the given bodies.
1. Earth:
- Acceleration due to gravity, [tex]\( g = 9.8 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 9.8 \times 3 = 2940 \, J \)[/tex]
2. Mercury:
- Acceleration due to gravity, [tex]\( g = 3.59 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 3.59 \times 3 = 1077 \, J \)[/tex]
3. Neptune:
- Acceleration due to gravity, [tex]\( g = 14.07 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 14.07 \times 3 = 4221 \, J \)[/tex]
4. Pluto:
- Acceleration due to gravity, [tex]\( g = 0.42 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 0.42 \times 3 = 126 \, J \)[/tex]
Comparing the gravitational potential energies:
- Earth: [tex]\( 2940 \, J \)[/tex]
- Mercury: [tex]\( 1077 \, J \)[/tex]
- Neptune: [tex]\( 4221 \, J \)[/tex]
- Pluto: [tex]\( 126 \, J \)[/tex]
The body on which a [tex]\(100 \text{ kg}\)[/tex] man has the least gravitational potential energy when lifted to a height of [tex]\(3 \text{ m}\)[/tex] is Pluto.
Therefore, the answer is:
B. Pluto
[tex]\[ \text{GPE} = m \cdot g \cdot h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass in kilograms (kg),
- [tex]\( g \)[/tex] is the acceleration due to gravity in meters per second squared ([tex]\( m/s^2 \)[/tex]),
- [tex]\( h \)[/tex] is the height in meters (m).
We need to calculate the GPE for each of the given bodies.
1. Earth:
- Acceleration due to gravity, [tex]\( g = 9.8 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 9.8 \times 3 = 2940 \, J \)[/tex]
2. Mercury:
- Acceleration due to gravity, [tex]\( g = 3.59 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 3.59 \times 3 = 1077 \, J \)[/tex]
3. Neptune:
- Acceleration due to gravity, [tex]\( g = 14.07 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 14.07 \times 3 = 4221 \, J \)[/tex]
4. Pluto:
- Acceleration due to gravity, [tex]\( g = 0.42 \, m/s^2 \)[/tex]
- Mass, [tex]\( m = 100 \, kg \)[/tex]
- Height, [tex]\( h = 3 \, m \)[/tex]
- Gravitational Potential Energy, [tex]\( \text{GPE} = 100 \times 0.42 \times 3 = 126 \, J \)[/tex]
Comparing the gravitational potential energies:
- Earth: [tex]\( 2940 \, J \)[/tex]
- Mercury: [tex]\( 1077 \, J \)[/tex]
- Neptune: [tex]\( 4221 \, J \)[/tex]
- Pluto: [tex]\( 126 \, J \)[/tex]
The body on which a [tex]\(100 \text{ kg}\)[/tex] man has the least gravitational potential energy when lifted to a height of [tex]\(3 \text{ m}\)[/tex] is Pluto.
Therefore, the answer is:
B. Pluto
