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Question

(A) The time period of a communication satellite is $T_c = 24 \, hrs$.According to Kepler's third law of planetary motion,the square of the time perio

Vedclass · Accepted Answer

$3 \, hrs$

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(A) The time period of a communication satellite is $T_c = 24 \, hrs$.According to Kepler's third law of planetary motion,the square of the time period is directly proportional to the cube of the orbital radius,i.e.,$T^2 \propto r^3$.Let $r_c$ be the radius of the communication satellite's orbit and $r_x$ be the radius of satellite $X$'s orbit. Given $r_x = \frac{1}{4} r_c$,or $\frac{r_c}{r_x} = 4$.Using the ratio form of Kepler's law: $\frac{T_c}{T_x} = \left( \frac{r_c}{r_x} \right)^{3/2}$.Substituting the values: $\frac{24}{T_x} = (4)^{3/2}$.Since $(4)^{3/2} = (2^2)^{3/2} = 2^3 = 8$,we have $\frac{24}{T_x} = 8$.Therefore,$T_x = \frac{24}{8} = 3 \, hrs$.

Column-$I$	Column-$II$
$(1)$ Kepler's first law	$(a)$ Law of periods
$(2)$ Kepler's second law	$(b)$ Law of orbits
$(3)$ Kepler's third law	$(c)$ Law of areas

An earth satellite $X$ is revolving around the earth in an orbit whose radius is one-fourth of the radius of the orbit of a communication satellite. The time period of revolution of $X$ is ..........

Similar Questions

Match Column-$I$ with Column-$II$.

Column-$I$ Column-$II$

$(1)$ Kepler's first law $(a)$ Law of periods

$(2)$ Kepler's second law $(b)$ Law of orbits

$(3)$ Kepler's third law $(c)$ Law of areas

$A$ planet is revolving around the sun in an elliptical path as shown in the figure. Which of the following is the correct option?

$A$ planet takes $200$ days to complete one revolution around the Sun. If the distance of the planet from the Sun is reduced to one-fourth of the original distance,how many days will it take to complete one revolution?

$Assertion$ : For the planets orbiting around the sun,angular speed,linear speed and $K.E.$ change with time,but angular momentum remains constant.
$Reason$ : No torque is acting on the rotating planet. So its angular momentum is constant.

In a solar system, the time-period of revolution of a planet tracing a circular orbit of radius $R$ is proportional to :

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