8.2 Choose the correct alternative:

(a) Acceleration due to gravity increases/decreases with increasing altitude

(b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density)

(c) Acceleration due to gravity is independent of mass of the earth/mass of the body

(d) The formula –G Mm(1/r₂ – 1/r₁) is more/less accurate than the formula mg(r₂ – r₁) for the difference of potential energy between two points r₂ and r₁ distance away from the centre of the earth

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Vishal Baghel | Contributor-Level 10

5 months ago

(a) Decreases - Acceleration due to gravity at depth h is given by $g_{h}$ = (1 – $\frac{2 h}{R_{e}}$ )g, where $R_{e} = R a d i u s o f t h e e a r t h,$ g = acceleration due to gravity on the surface of the Earth. From this equation, it is clear that acceleration due to gravity decreases with increase in height

(b) Decreases – Acceleration due to gravity at depth d is given by $g_{d}$ = (1- $\frac{d}{R_{e}}$ )g. So the acceleration due to gravity decreases with increase in depth.

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(b) Decreases – Acceleration due to gravity at depth d is given by $g_{d}$ = (1- $\frac{d}{R_{e}}$ )g. So the acceleration due to gravity decreases with increase in depth.

(c) Mass of the body – Acceleration due to gravity of body mass m is given by the relation g = $\frac{G M}{R^{2}}$ , where G = Universal gravitation constant, M = mass of the Earth and R = radius of the Earth. Hence, it can be inferred that acceleration due to gravity is independent of the mass of the body.

(d) More – Gravitational potential energy of two points $r_{1}$ and $r_{2}$ distance away from the centre of the Earth is respectively given by:

V( $r_{1}$ ) = $-$ - $\frac{G m M}{r_{1}}$ and V( $r_{2}$ ) = $-$ - $\frac{G m M}{r_{2}}$

Difference in potential energy, V = V( $r_{2}$ ) $-$ V( $r_{1}$ ) = $- G m M (\frac{1}{r_{2}}$ $- \frac{1}{r_{1}}$ )

Hence this formula is more accurate than mg( $r_{2}$ - $r_{1}$ )

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(a) Decreases - Acceleration due to gravity at depth h is given by <math><msub><mrow><mrow><mi>g</mi></mrow></mrow><mrow><mrow><mi>h</mi></mrow></mrow></msub></math> = (1 – <math><mfrac><mrow><mrow><mn>2</mn><mi>h</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mi>e</mi></mrow></mrow></msub></mrow></mrow></mfrac></math> )g, where <math><msub><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mi>e</mi></mrow></mrow></msub><mo>=</mo><mi>R</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>u</mi><mi>s</mi><mi></mi><mi>o</mi><mi>f</mi><mi></mi><mi>t</mi><mi>h</mi><mi>e</mi><mi></mi><mi>e</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>h</mi><mo>,</mo></math> g = acceleration due to gravity on the surface of the Earth. From this equation, it is clear that acceleration due to gravity decreases with increase in height (b) Decreases – Acceleration due to gravity at depth d is given by <math><msub><mrow><mrow><mi>g</mi></mrow></mrow><mrow><mrow><mi>d</mi></mrow></mrow></msub></math> = (1- <math><mfrac><mrow><mrow><mi>d</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mi>e</mi></mrow></mrow></msub></mrow></mrow></mfrac></math> )g. So the acceleration due to gravity decreases with increase in depth. (c) Mass of the body – Acceleration due to gravity of body mass m is given by the relation g = <math><mfrac><mrow><mrow><mi>G</mi><mi>M</mi></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow></mfrac><mi></mi></math> , where G = Universal gravitation constant, M = mass of the Earth and R = radius of the Earth. Hence, it can be inferred that acceleration due to gravity is independent of the mass of the body. (d) More – Gravitational potential energy of two points <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math> and <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math> distance away from the centre of the Earth is respectively given by:V( <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math> ) = <math><mo>-</mo></math> - <math><mfrac><mrow><mrow><mi>G</mi><mi>m</mi><mi>M</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></mrow></mrow></mfrac></math> and V( <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math> ) = <math><mo>-</mo></math> - <math><mfrac><mrow><mrow><mi>G</mi><mi>m</mi><mi>M</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></mrow></mrow></mfrac></math>Difference in potential energy, V = V( <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math> ) <math><mo>-</mo></math> V( <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math> ) = <math><mo>-</mo><mi></mi><mi>G</mi><mi>m</mi><mi>M</mi><mo>(</mo><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></mrow></mrow></mfrac></math> <math><mo>-</mo><mi></mi><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></mrow></mrow></mfrac></math> )Hence this formula is more accurate than mg( <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math> - <math><msub><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math> )

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