Question

Prove that the Gravitational force is conservative force ​

Answer 1

Answer:

conservative force exists when the work done by that force on an object is independent of the object's path. Instead, the work done by a conservative force depends only on the end points of the motion. An example of a conservative force is gravity.

Answer 2

Need to Prove :-

• The gravitational force is conservative in nature.

$\red{\frak{Given}}\Bigg\{ \sf The \ Gravitational\ Force$

The forces whose work done is independent of the path taken by the object is called a conservative force . Gravitational force is an example of that . For proving it , let us assume a inclined plane , inclined at an angle θ , ( Refer to attachment ) .Let the height be h . Then the angle between s and the mg will be (90° - θ ) . Here we will be moving the object in first case from A to B , and in second case A to B via C .

• Case 1 : Moving object from A to B :-

• In this case , the angle between the force and the Displacement is 0° , since both are in downwards dirⁿ .

$\sf\dashrightarrow Work = F.s \\\\\sf\dashrightarrow Work_{(A \ to \ B)} = Fs \ cos\theta \\\\\sf\dashrightarrow Work_{(A \ to \ B)} = (mg) (h) cos 0^{\circ} \\\\\sf\dashrightarrow Work_{(A \ to \ B)} = mgh \times 1 \\\\\sf\dashrightarrow \boxed{\pink{\frak{ Work_{(A \ to \ B)} = mgh }}}$

$\rule{200}2$

• Case 2: Moving object A to C :-

• In this case , the angle between the force and Displacement is 90° - θ . And the measure of AC is s .

$\sf\dashrightarrow Work = F.s \\\\\sf\dashrightarrow Work_{(A C)} = Fs \ cos\theta \\\\\sf\dashrightarrow Work_{(AC)} = (mg)(s) cos(90^{\circ} -\theta) \\\\\sf\dashrightarrow Work_{(A C)} = mgs \times sin\theta \\\\\sf\dashrightarrow Work_{(A C)} = mgs \times \dfrac{h}{s} \\\\\sf\dashrightarrow \boxed{\pink{\frak{ Work_{(A C)} = mgh }}}$

$\rule{200}2$

• Case 3 : Moving object C to B :-

• Here the angle between the force and Displacement is 90°. We know that value of cos90° is 0 . So the work done will be 0 .

$\sf\dashrightarrow Work = F.s \\\\\sf\dashrightarrow Work_{( CB)} = Fs \ cos\theta \\\\\sf\dashrightarrow Work_{(CB)} = mgx \ cos 90^{\circ} \\\\\sf\dashrightarrow Work_{(CB)} = mgx \times 0 \\\\\sf\dashrightarrow \boxed{\pink{\frak{ Work_{(CB)} = 0 }}}$

Hence the work done to move object from A to B via C is mgh + 0 = mgh .

Hence we see that the work done via both paths is same , i.e . mgh . Hence the work done by gravitational force is independent of path taken , henceforth Gravitational force is Conservative force .