NEET-XII-Physics
11: Gravitation
- #12A tunnel is dug along a chord of the earth at a perpendicular distance R/2 from the earth’s centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.Ans : Let d be distance of the particle from the centre of the Earth.
`` \,\mathrm{\,Now\,},{d}^{2}={x}^{2}+\left(\frac{{R}^{2}}{4}\right)=\frac{4{x}^{2}+{R}^{2}}{4}``
`` \Rightarrow d=\left(\frac{1}{2}\right)\sqrt{4{x}^{2}+{R}^{2}}``
Let M be the mass of the Earth and M' be the mass of the sphere of radius d.
Then we have:
`` M=\left(\frac{4}{3}\right)\pi {R}^{3}\,\mathrm{\,\rho \,}``
`` {M}^{1}=\left(\frac{4}{3}\right)\,\mathrm{\,\pi \,}{d}^{3}\,\mathrm{\,\rho \,}``
`` \therefore \frac{{M}^{\mathit{1}}}{M}=\frac{{d}^{3}}{{R}^{3}}``
Gravitational force on the particle of mass m is given by
`` F=\frac{G{M}^{1}m}{{d}^{2}}``
`` \Rightarrow F=\frac{\,\mathrm{\,G\,}{d}^{3}Mm}{{R}^{3}{d}^{2}}``
`` \Rightarrow F=\frac{GMm}{{R}^{3}}d``
∴ Normal force exerted by the wall, FN = F cos θ`` =\frac{\,\mathrm{\,GM\,}md}{{\,\mathrm{\,R\,}}^{3}}\times \frac{\,\mathrm{\,R\,}}{2d}=\frac{\,\mathrm{\,GM\,}md}{2{\,\mathrm{\,R\,}}^{2}}``
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