પૃથ્વીની સપાટીથી કેટલી ઊંચાઈ \(h\) માટે ગુરુત્વ પ્રવેગ પૃથ્વીની સપાટીથી \(h\) ઊંડાઈ જેટલો થાય. પૃથ્વીની ત્રિજ્યા \(r\) અને પૃથ્વીના ભ્રમણની અસર અવગણો.

A \(\frac{\sqrt{5} R - R }{2}\)
B \(\frac{\sqrt{5}}{2} R - R\)
C \(\frac{ R }{2}\)
D \(\frac{\sqrt{3} R - R }{2}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{\sqrt{5} R - R }{2}\)

Step-by-step Solution

Detailed explanation

\(M =\) mass of earth \(M _{1}= mass\) of shaded portion \(R =\) Radius of earth \(M_{1}=\frac{M}{\frac{4}{3} \pi R^{3}} \cdot \frac{4}{3} \pi(R-h)^{3}\) \(=\frac{M(R-h)^{3}}{R}\) Weight of body is same at \(P\) and \(Q\) i.e. \(mg _{ P }= mg _{ Q }\) \(g _{ P }= g _{ Q }\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

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જો ગણ \(A\,=\,\{\,x\,\in \,R\,:\,x\) એ ધન પૃણાંક નથી \(\}\) પર વિધેય \(f\,:\,A\,\to \,R\) એ \(f\,(x)\, = \frac{{2x}}{{x - 1}}\) આપેલ હોય તો \(f\) એ . . .

JEE Mains 2019 Hard

Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) ચુંબકીય પ્રેરણ	\((i)\) \({ML}^{2} {T}^{-2} {A}^{-1}\)
\((b)\) ચુંબકીય ફ્લક્સ	\((ii)\) \({M}^{0} {L}^{-1} {A}\)
\((c)\) ચુંબકીય પરમીએબીલીટી	\((iii)\) \({MT}^{-2} {A}^{-1}\)
\((d)\) મેગ્નેટાઇઝેશન	\((iv)\) \({MLT}^{-2} {A}^{-2}\)

આપેલ વિકલ્પો માંથી સાચો વિકલ્પ પસંદ કરો.

JEE Mains 2021 Medium

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