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सही विकल्प का चयन कीजिए :
$(a)$ बढ़ती तुंगता के साथ गुरुत्वीय त्वरण बढ़ता/घटता है।
$(b)$ बढ़ती गहराई के साथ (पृथ्वी को एकसमान घनत्व को गोला मानकर) गुरुत्वीय त्वरण बढ़ता/घटता है।
$(c)$ गुरुत्वीय त्वरण पृथ्वी के द्रव्यमान/पिण्ड के द्रव्यमान पर निर्भर नहीं करता।
$(d)$ पृथ्वी के केन्द्र से $r_{2}$ तथा $r_{1}$ दूरियों के दो बिन्दुओं के बीच स्थितिज ऊर्जा-अन्तर के लिए सूत्र $-G M m\left(1 / r_{2}-1 / r_{1}\right)$ सूत्र $m g\left(r_{2}-r_{1}\right)$ से अधिक/कम यथार्थ है।
Solution
$(a)$ Decreases, $(b)$ Decreases, $(c)$ Mass of the body, $(d)$ More
Acceleration due to gravity at depth $h$ is given by the relation:
$g _{h}=\left(1-\frac{2 h}{R_{ c }}\right) g$
Where,
$R_{e}=$ Radius of the Earth $g =$ Acceleration due to gravity
on the surface of the Earth
It is clear from the given relation that acceleration due to gravity decreases with an increase in height.
Acceleration due to gravity at depth $d$ is given by the relation:
$g _{d}=\left(1-\frac{d}{R_{e}}\right) g$
It is clear from the given relation that acceleration due to gravity decreases with an increase in depth.
Acceleration due to gravity of body of mass $m$ is given by the relation:
$g =\frac{ G M}{R^{2}}$
Where,
$G =$ Universal gravitational constant
$M=$ Mass of the Earth
$R=$ Radius of the Earth
Hence, it can be inferred that acceleration due to gravity is independent of the mass of the body.
Gravitational potential energy of two points $r_{2}$ and $r_{1}$ distance away from the centre of the Earth is respectively given by:
$V\left(r_{1}\right)=-\frac{ G m M}{r_{1}}$
$V\left(r_{2}\right)=-\frac{G m M}{r_{2}}$
Difference in potential energy, $V=V\left(r_{2}\right)-V\left(r_{1}\right)=-\operatorname{GmM}\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$
Hence, this formula is more accurate than the formula $m g\left(r_{2}-r_{1}\right)$
