Question

(b) An example of an inverse square field is the gravitational field F = −(mMG)r/|r|3. Use part (a) to find the work done by the gravitational field when the earth moves from aphelion (at a maximum distance of 1.52 108 km from the sun) to perihelion (at a minimum distance of 1.47 108 km). Use the values m = 5.97 1024 kg, M = 1.99 1030 kg, and G = 6.67 ✕ 10−11 N·m2/kg2. (Round your answer to two decimal places.)

Answer 1

To solve this, we need to recall that Work done W,

W = F x dₐ Eqn 1

where F is the gravitational field force and d is the distance travelled by the earth.

But F = −(mMG)r/|r|³ Eqn 2

then, W = −∫ (mMG)r/|r|³ x dₐ

Integrating this expression gives us

W = -4mMG ((1/d1) -(1/d2))

m = 5.971024 kg, M = 1.991030 kg G = 6.67 ✕ 10⁻¹¹ N·m²/kg²

d2 = 1.52x10¹¹m, d1 = 1.47x10¹¹m make the units in S.I units and similar

Substituting these value in to the eqn above gives

Now , W = 1.77 x 10³²J (2 Decimal Places)