On the surface of earth acceleration due to gravity is \(\mathrm{g}\) and gravitational potential is \(\mathrm{V}\) match the followingTable - 1 Table \(-2\) (A) At height \(\mathrm{h}=\mathrm{R}\) value of \(\mathrm{g}\) (P) decrease by a factor \((1 / 4)\) (B) At depth \(\mathrm{h}=(\mathrm{R} / 2)\) (Q) decrease by a factor \((1 / 2)\) (C) At height \(\mathrm{h}=\mathrm{R}\) value of \(\mathrm{v}\) (R) increase by a factor \((11 / 8)\) (D) At depth \(\mathrm{h}=(\mathrm{R} / 2)\) value of \(\mathrm{v}\) (S) increase by a factor 2 (T) None

To determine the change in gravitational acceleration at height h = R, we can use the formula: \(g'=g\frac{R^2}{(R+h)^2}\) Where: \(g'\) = gravitational acceleration at height h \(g\) = gravitational acceleration on Earth's surface \(R\) = radius of Earth \(h\) = height above Earth's surface Given, \(h=R\), so we have: \(g'=g\frac{R^2}{(R+R)^2} = g\frac{R^2}{(2R)^2} = g\frac{1}{4}\) The gravitational acceleration at height \(h=R\) decreases by a factor of \(\frac{1}{4}\). So, the correct match for option A is P.

To determine the change in gravitational acceleration at depth h = R/2, we can use the formula: \(g''=g(1-\frac{h}{R})\) Where: \(g''\) = gravitational acceleration at depth h \(g\) = gravitational acceleration on Earth's surface \(R\) = radius of Earth \(h\) = depth below Earth's surface Given, \(h=\frac{R}{2}\), so we have: \(g''=g(1-\frac{R/2}{R}) = g(1-\frac{1}{2}) = g\frac{1}{2}\) The gravitational acceleration at depth \(h=R/2\) decreases by a factor of \(\frac{1}{2}\). So, the correct match for option B is Q.

To determine the change in gravitational potential at height h = R, we can use the formula: \(V'=-\frac{GM}{R+h}\) Where: \(V'\) = gravitational potential at height h \(G\) = gravitational constant \(M\) = mass of Earth \(R\) = radius of Earth \(h\) = height above Earth's surface Given, \(h=R\), so we have: \(V'=-\frac{GM}{R+R} = -\frac{GM}{2R}\) Initially, the gravitational potential on Earth's surface is: \(V = -\frac{GM}{R}\) Now let's find the factor by which the gravitational potential increases: \(\frac{V'}{V} = \frac{-GM/2R}{-GM/R} = \frac{1}{2}\) This means the gravitational potential at height \(h=R\) increases by a factor of \(\frac{1}{2}\). However, there is no matching option for this case, so the correct match for option C is T (None).

To determine the change in gravitational potential at depth h = R/2, we can use the formula: \(V''=-GM\frac{3}{2R}\) Where: \(V''\) = gravitational potential at depth h \(G\) = gravitational constant \(M\) = mass of Earth \(R\) = radius of Earth \(h\) = depth below Earth's surface Given, \(h=\frac{R}{2}\), so we have: \(V''=-GM\frac{3}{2R}\) Now let's find the factor by which the gravitational potential increases: \(\frac{V''}{V} = \frac{-GM(3/2R)}{-GM/R} = \frac{3}{2}\) This means the gravitational potential at depth \(h=R/2\) increases by a factor of \(\frac{3}{2}\). However, there is no matching option for this case, so the correct match for option D is T (None). So, the final matching is: A -> P B -> Q C -> T D -> T