એક \(\mathrm{a}\) ત્રિજ્યા ધરાવતી તકતીનું એકમ ક્ષેત્રફળ દીઠ દળ તેના કેન્દ્રથી \(r\) અંતરે \(\sigma(\mathrm{r})=\mathrm{A}+\mathrm{Br}\) મુજબ બદલાય છે. તો તકતીના સમતલને લાંબા અને કેન્દ્રમાથી પસાર થતી અક્ષને અનુલક્ષીને જડત્વની ચાકમાત્રા કેટલી થાય?

A \(2 \pi \mathrm{a}^{4}\left(\frac{\mathrm{A}}{4}+\frac{\mathrm{aB}}{5}\right)\)
B \( \pi \mathrm{a}^{4}\left(\frac{\mathrm{A}}{4}+\frac{\mathrm{aB}}{5}\right)\)
C \(2 \pi \mathrm{a}^{4}\left(\frac{\mathrm{aA}}{4}+\frac{\mathrm{B}}{5}\right)\)
D \(2 \pi \mathrm{a}^{4}\left(\frac{\mathrm{A}}{4}+\frac{\mathrm{B}}{5}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(2 \pi \mathrm{a}^{4}\left(\frac{\mathrm{A}}{4}+\frac{\mathrm{aB}}{5}\right)\)

Step-by-step Solution

Detailed explanation

\(\mathrm{dI}=\mathrm{dm}^{2}\) \(\mathrm{dI}=\sigma 2 \pi \mathrm{r}\) dr \(\mathrm{r}^{2}\) \(\mathrm{dI}=2 \pi(\mathrm{A}+\mathrm{Br}) \mathrm{r}^{3} \mathrm{dr}\) \(\int \mathrm{dI}=2 \pi \int_{0}^{\mathrm{a}}\left(\mathrm{Ar}^{3}+\mathrm{Br}^{4}\right) \mathrm{dr}\)…

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

Explore more questions on app

From JEE Mains

Explore more questions on app