Formulae [1]
Formula: Gravity with Altitude
The formulas for acceleration due to gravity (g) are provided below:
On the Earth's Surface:
\[g = \frac{G M}{R^2}\]
At height $h$ above the Earth's surface:
\[g_h = g \frac{R^2}{(R+h)^2} \quad \text{or} \quad g_h = g \left(I + \frac{h}{R}\right)^{-2}\]
Simplified Formula for Small Altitudes ($h \ll R$):
\[g_h = g \left(1 - \frac{2h}{R}\right)\]
Definition of Terms:
- g: Acceleration due to gravity on the Earth's surface.
- gh: Acceleration due to gravity at height h above the Earth's surface.
- G: Universal Gravitational Constant.
- M: Mass of the Earth.
- R: Radius of the Earth.
- h: Altitude or height above the Earth's surface.
Theorems and Laws [1]
Write the Kepler’s laws.
- Kepler’s first law: The orbit of a planet is an ellipse with the sun at one of the foci.
- Kepler’s second law: The line joining the planet and the sun sweeps equal areas in equal intervals of time.
- Kepler’s third law: The square of the orbital period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the planet from the Sun.
Key Points
Key Points: Comparison of Two SHMs (Same Period, Same Path)
- When a particle is subjected simultaneously to two SHMs having the same period and along the same path, the resultant motion is also SHM along the same path.
- The resultant displacement of the particle at any instant is equal to the vector sum of its individual displacements due to both the SHMs at that instant.
Important Questions [19]
- Calculate the Period of Revolution of Jupiter Around the Sun. the Ratio of the Radius of Jupiter’S Orbit to that of the Earth’S Orbit is 5.
- The Dimensions of Universal Gravitational Constant Are
- State Kepler'S Laws of Planetary Motion.
- State Kepler'S Law of Orbit and Law of Equal Areas.
- Define Binding Energy and Obtain an Expression for Binding Energy of a Satellite Revolving in a Circular Orbit Round the Earth.
- Determine the Binding Energy of Satellite of Mass 1000 Kg Revolving in a Circular Orbit Around the Earth When It is Close to the Surface of Earth.
- Find the Total Energy and Binding Energy of an Artificial Satellite of Mass 800 Kg Orbiting at a Height of 1800 Km Above the Surface of the Earth.
- The Escape Velocity of a Body from the Surface of the Earth is 11.2 km/s. If a Satellite Were to Orbit Close to the Surface, What Would Be Its Critical Velocity?
- If the Earth Completely Loses Its Gravity, Then for Any Body
- Explain Why an Astronaut in an Orbiting Satellite Has a Feeling of Weightlessness.
- Derive an Expression for Critical Velocity of a Satellite Revolving Around the Earth in a Circular Orbit.
- Derive an Expression for Acceleration Due to Gravity at Depth ‘D’ Below the Earth’S Surface.
- A hole is drilled half way to the centre of the Earth. A body is dropped into the hole. How much will it weigh at the bottom of the hole if the weight of the body on the Earth’s surface is 350 N?
- Prove that gh=g(1-"2h"/R) where gh is the acceleration due to gravity at altitude h and h << R (R is the radius of the earth).
- A sonometer wire vibrates with frequency n1 in air under suitable load of specific gravity of . When the load is immersed in water, the frequency of vibration of wire n2 will be
- ‘G’ is the Acceleration Due to Gravity on the Surface of the Earth and ‘R’ is the Radius of the Earth Show that Acceleration Due to Gravity at Height ‘H’ Above the Surface of the Earth
- What is the Decrease in Weight of a Body of Mass 500 Kg When It is Taken into a Mine of Depth 1000 Km?
- Find the Velocity at the Highest Point.
- What is the Decrease in Weight of a Body of Mass 600kg When It is Taken in a Mine of Depth 5000m?
Concepts [9]
- Newton’s Law of Gravitation
- Periodic Time
- Kepler’s Laws
- Binding Energy and Escape Velocity of a Satellite
- Weightlessness
- Variation of ‘G’ Due to Lattitude and Motion
- Variation in the Acceleration>Variation in Gravity with Altitude
- Communication satellite and its uses
- Composition of Two S.H.M.’S Having Same Period and Along Same Line
