Question

Which of the following changes to the earth-sun system would reduce the magnitude of the force between them to one-fourth the value found in Part A?

Reduce the mass of the earth to one-fourth its normal value.
Reduce the mass of the sun to one-fourth its normal value.
Reduce the mass of the earth to one-half its normal value and the mass of the sun to one-half its normal value.
Increase the separation between the earth and the sun to four times its normal value.

part A was:

Consider the earth following its nearly circular orbit (dashed curve) about the sun. The earth has mass mearth=5.98

Answer 1

Reduce the mass of the earth to one-fourth its normal value.

Reduce the mass of the sun to one-fourth its normal value.

Reduce the mass of the earth to one-half its normal value and the mass of the sun to one-half its normal value.

Step-by-step explanation:

Every particle in the universe attracts any other particle with a force that is directly proportional to the product of its masses and inversely proportional to the square of the distance between them. So, in this case we have:

`F=(Gm_Em_S)/(d^2)`

If
`m'_E=(m_E)/(4)`:

`F'=(Gm'_Em_S)/(d^2)\\F'=(G((m_E)/(4))m_S)/(d^2)\\F'=(1)/(4)(Gm_Em_S)/(d^2)\\F'=(1)/(4)F`

If
`m'_S=(m_S)/(4)`

`F'=(Gm_Em'_S)/(d^2)\\F'=(Gm_E((m_S)/(4)))/(d^2)\\F'=(1)/(4)(Gm_Em_S)/(d^2)\\F'=(1)/(4)F`

If
`m'_E=(m_E)/(2)` and
`m'_S=(m_S)/(2)`:

`F'=(Gm'_Em'_S)/(d^2)\\F'=(G((m_E)/(2))((m_S)/(2)))/(d^2)\\F'=(1)/(4)(Gm_Em_S)/(d^2)\\F'=(1)/(4)F`