Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II : The time period of revolution of the satellite is $T = 2\pi \sqrt{\frac{R_e}{g}}$ (for satellite very close to the earth surface), where $R_e$ radius of earth and g acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below :
Statement I Analysis: Satellite Time Period and Earth Density
For a satellite orbiting very close to the Earth's surface, the orbital radius $r$ is approximately equal to the Earth's radius $R_e$. The time period $T$ is given by the formula:
$T = 2\pi \sqrt{\frac{r^3}{GM_e}}$
where $M_e$ is the mass of the Earth and $G$ is the gravitational constant.
We know the acceleration due to gravity at the surface is $g = \frac{GM_e}{R_e^2}$, which implies $GM_e = gR_e^2$. Substituting this into the time period formula for $r \approx R_e$:
$T \approx 2\pi \sqrt{\frac{R_e^3}{gR_e^2}} = 2\pi \sqrt{\frac{R_e}{g}}$
Now, let's express $g$ in terms of Earth's density ($\rho_e$). The mass of the Earth is $M_e = \rho_e \times (\frac{4}{3}\pi R_e^3)$.
So, $g = \frac{G(\rho_e \frac{4}{3}\pi R_e^3)}{R_e^2} = \frac{4}{3}\pi G \rho_e R_e$.
Substituting this expression for $g$ back into the time period formula:
$T \approx 2\pi \sqrt{\frac{R_e}{\frac{4}{3}\pi G \rho_e R_e}} = 2\pi \sqrt{\frac{3}{4\pi G \rho_e}}$
This derivation shows that the time period $T$ is inversely proportional to the square root of the Earth's density ($\rho_e$). Therefore, Statement I is true.
Statement II Analysis: Satellite Time Period Formula
Statement II provides the formula $T = 2\pi \sqrt{\frac{R_e}{g}}$ for a satellite very close to the Earth's surface.
As derived above, this formula is an approximation obtained by setting the orbital radius $r$ equal to the Earth's radius $R_e$ in the general time period equation $T = 2\pi \sqrt{\frac{r^3}{GM_e}}$.
However, for any satellite in orbit, the orbital radius $r$ must be strictly greater than the Earth's radius $R_e$ ($r > R_e$). Even for orbits "very close" to the surface, $r$ is slightly larger than $R_e$. Using $R_e$ directly instead of $r$ makes the formula an approximation, not a strictly accurate representation for an orbit.
Because the formula uses $R_e$ instead of the actual orbital radius $r$, it is considered technically inaccurate. Therefore, Statement II is false.
Conclusion
Based on the analysis, Statement I is true, and Statement II is false.
The correct option is: Statement I is true but Statement II is false.
Similar Questions
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
A cylindrical tube AB of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to AB and passing through the edge at end A, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then
(Consider the temperature is same at all points in the tube)
- A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is _________ $\text{kg.m}^2$.
- A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is _________.
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
- Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is ______ $\text{kg.m}^2$. ($g = 9.8 \text{ m/s}^2$)
Important Questions from Mechanics
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
A cylindrical tube AB of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to AB and passing through the edge at end A, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then
(Consider the temperature is same at all points in the tube)- A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is _________ $\text{kg.m}^2$.
- A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is _________.
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
