## Unit – II D |

## Problems on Acceleration Due to Gravity:

### Formulae:

- Acceleration due to gravity on the surface of the earth / Planet is
- Relation between acceleration due to gravity and radius of the earth is
- Acceleration due to gravity at latitude Φ is
- Acceleration due to gravity at altitude h is
- Acceleration due to gravity at depth d is
- If a problem is related to the weight of a body multiply both sides by the mass of the body m and get relevant formulae.

### Problems:

- Assuming the earth to be a homogeneous sphere, find the density of the material of the earth from the following data. g = 9.8 m/s², G = 6.676 x 10
^{-11}N m²/kg² , R = 6400 km, - Three 5 kg masses are kept at the vertices of an equilateral triangle each of side of 0.25 m. Find the resultant gravitational force on any one mass. G=6.67 x 10
^{-11}N m²/kg². - Taking G = 6.67 x 10
^{-11}N m²/kg², the radius of the earth as 6400 km and mean density of earth as 5500 kg/m³, calculate g at the surface of the earth. - At what height will be acceleration due to the gravity of the earth fall off to one-half that at the surface? At what height will the acceleration due to gravity be 8 m/s²? Take radius of earth = 6400 km.
- How far from the centre of the earth does the acceleration due to gravity reduce by 5 per cent of its value at the surface of the earth? Take radius of the earth as 6.4 x 10
^{6}m. - At a certain height above the surface of the earth, the gravitational acceleration is 90 % of its value on the surface of the earth. Determine the height if the radius of the earth is 6400 km.
- At what height above the earth’s surface will the acceleration due to gravity be 4% of the value at the surface of the earth? g= 9.8 m/s², R= 6400 km.
- How far away from the centre of the earth does the acceleration due to gravity reduce by 1 % of its value on earth’s surface? The radius of the earth is 6400 km.
- At what height above the earth’s surface will the acceleration due to gravity be 25% of the value at the surface of the earth? g= 9.8 m/s², R= 6400 km.
- The mass of the body on the surface of the earth is 100 kg. What will be its mass and weight at an altitude of 1000 km?
- A body weights 1.8 kg on the surface of the earth. How much will it weigh on the surface of a planet whose mass is 1/9 that of earth and whose radius us half that of the earth?
- A body weights 4.5 kg on the surface of the earth. How much will it weigh on the surface of a planet whose mass is 1/9 that of earth and whose radius us half that of the earth?
- The radius of a planet is half that of earth. The acceleration due to gravity on the planet’s surface is half that on earth’s surface. Find the mass of the planet in terms of mass M of earth.
- Find the acceleration due to gravity on the surface of the moon. Given that the mass of the moon is 1/80 times that of the earth and the diameter of the moon is 1/4 times that of the earth. g = 9.8 m/s².
- The moon revolves around the earth in 27 days in an orbit of radius 4 x 10
^{5}km. Find the linear velocity of the moon in its orbit and the normal acceleration. - The mass of mars is1/9 times that of the earth and radius is 1/2 times that of the earth. What is the weight of the body on the surface of the mars whose mass on the earth’s surface is 1000 kg.
- A star having a mass 2.5 times that of the sun and collapsed to a size of 12 km rotates with a speed of 1.5 rev/s (Extremely compact stars of this kind are called neutron stars. Astronomical objects pulsars belong to this category). Will object placed on its equator remain stuck to its surface due to gravity. Mass of sun is 2 x 10
^{30}kg - The mass of the Hubble telescope is 11600 kg. What is its weight when it is in an orbit 598 km above the surface of the earth. Mass of earth is 5.98 10
^{24}kg, Radius of earth = 6400 km, G = 6.67 x 10^{-11}N m²/kg²

### Answers/Solutions:

- 5483 kg/m³
- 4.621 x 10-8 N towards the centroid
- 9.83 m/s²
- 2650 km; 684 km
- 6566 km
- 346 km
- 25600 km
- 6432 km
- 6400 km
- 100 kg, 733 N
- 0.8kgWt
- 2 kg
- M/8
- 1.96m/s²
- 1.077 km/s; 2.90 10
^{-3}m/s² - 4356 N
- It will remain stuck
- 9.447 104 N