13. If the gravitational force had varied as r 2 insteadof r-2. the po

1.	13. If the gravitational force had varied as r 2 insteadof r-2. the potential energy of a particle at adistance from centre of earth would beproportional to(1)r-1(2) r-5/2(3)r-3/2(4)r-2
Answer» Answer: None of the given OPTION is correct. The correct answer is $\rm r^3.$ Explanation: The Gravitational force $\rm F_g$ and the potential ENERGY $\rm U$ of a PARTICLE are related as $\rm F_g = -\dfrac{dU}{dr}\\\therefore U=-\int\limits^r_0F_g\ dr'.$ The gravitational force on the particle at r distance from the surface of the EARTH is given as $\rm F_g = \dfrac{GMm}{r^2}.$ G = Universal Gravtitational constant. M = MASS of the Earth. m = mass of the particle. $\rm r^2$ $\rm r^{-2}$ If the gravitational force had varied as instead of , then, $\rm F_g = GMmr^2.$ In that case, $\rm U=-\int\limits^r_0(GMmr'^2)\ dr'\\=-GMm\int\limits^r_0r'^2\ dr'\\=-GMm\left ( \dfrac{r'^3}{3}\right )\limits^r_0\\=-GMm\left ( \dfrac{r^3}{3}-0\right )\\=-GMm\left ( \dfrac{r^3}{3}\right ).\\\Rightarrow U\propto r^3.$

13. If the gravitational force had varied as r 2 insteadof r-2. the potential energy of a particle at adistance from centre of earth would beproportional to(1)r-1(2) r-5/2(3)r-3/2(4)r-2

Answer»

Answer:

None of the given OPTION is correct. The correct answer is $\rm r^3.$

Explanation:

The Gravitational force $\rm F_g$ and the potential ENERGY $\rm U$ of a PARTICLE are related as

$\rm F_g = -\dfrac{dU}{dr}\\\therefore U=-\int\limits^r_0F_g\ dr'.$

The gravitational force on the particle at r distance from the surface of the EARTH is given as

$\rm F_g = \dfrac{GMm}{r^2}.$

G = Universal Gravtitational constant.
M = MASS of the Earth.
m = mass of the particle.

$\rm r^2$ $\rm r^{-2}$ If the gravitational force had varied as instead of , then, $\rm F_g = GMmr^2.$

In that case,

$\rm U=-\int\limits^r_0(GMmr'^2)\ dr'\\=-GMm\int\limits^r_0r'^2\ dr'\\=-GMm\left ( \dfrac{r'^3}{3}\right )\limits^r_0\\=-GMm\left ( \dfrac{r^3}{3}-0\right )\\=-GMm\left ( \dfrac{r^3}{3}\right ).\\\Rightarrow U\propto r^3.$

13. If the gravitational force had varied as r 2 insteadof r-2. the potential energy of a particle at adistance from centre of earth would beproportional to(1)r-1(2) r-5/2(3)r-3/2(4)r-2

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment

13. If the gravitational force had varied as r 2 insteadof r-2. the potential energy of a particle at adistance from centre of earth would beproportional to(1)r-1(2) r-5/2(3)r-3/2(4)r-2​

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment

13. If the gravitational force had varied as r 2 insteadof r-2. the potential energy of a particle at adistance from centre of earth would beproportional to(1)r-1(2) r-5/2(3)r-3/2(4)r-2