Gravitation (11Acd11).

Sheet 03 Variation of Acceleration Due to Gravity.

1. At what height above the surface of the earth the value of \(g\) is half of its value on surface of earth? Given: Radius of the earth \(R_{e}=\left(6.4\times10^{6}\right)m\).
   
   
   Ans: \(h=2649.6km\).
   
   
2. A body of mass \(m\) is raised to a height from the surface of the earth where the acceleration due to gravity is \(g\). Prove that: loss of weight due to variation in \(g\) is approximately \(\frac{2mgh}{R}\), where \(R\) is radius of the planet Earth.
   
   
3. A body is taken to height of \(16km\) from the surface of the earth. Find: the percentage decrease in the weight of the object. Radius of Earth = 6400km.
   
   
   Ans: \(0.5%\).
   
   
4. A body weighing \(196N\) on the surface of the earth. Calculate: the weight of the body at \(h=R\) where \(R=6400km\) is the radius of the Earth.
   
   
   Ans: \(W=49N\).
   
   
5. A body hanging from a spring stretches it by \(3cm\) at the earth’s surface. How much it will the same body stretches the spring at a place \(800km\) above the the surface of the earth? Given: Radius of the earth \(R_{e}=\left(6.4\times10^{4}\right)km\).
   
   
   Ans: \(X’=2.37cm\).
   
   
6. A body of mass \(5kg\) is weighed in a balance at the top of a tower \(20m\) high. Then the mass is suspended from the pan of the balance by a fine string of \(20m\) long and reweighted. What will be the change in its weight, assume that the radius of the earth \(=6400km\).
   
   
   Ans: \(\Delta W=\left(3.06\times10^{-4}\right)kgf\).
   
   
7. What will be the percentage change in weight of an object, when it is taken to \(32km\) below the surface of the earth. Take: Radius of the Earth \(R_{e}=6400km\) .
   
   
   Ans: \(\frac{\Delta W}{W}\times100=0.5%\).
   
   
8. Assuming the Earth to be a sphere of uniform density, how much a body weighs half way down the center of the earth if it weighed \(400N\) on the surface of earth?
   
   
   Ans: \(W_{d}=200N\).
   
   
9. Compare the weight of a body when it is (i) \(1km\) above the surface of the Earth. (ii) \(1km\) below the the surface of the earth. Given: Radius of the earth is \(6300km\).
   
   
10. At what depth below the surface of the Earth does the acceleration due to gravity becomes \(1%\) of its value at the surface of the earth? Earth is considered as sphere of radius \(\left(6.3\times10^{6}\right)m\), rotating about its polar axis with a period of \(1day\left(8.64\times10^{4}s\right)\). How much would the acceleration due to gravity differ from the poles to the equator?
    
    
    Ans: \(g_{p}-g_{e}=\left(3.37\times10^{-2}\right)\frac{m}{s^{2}}\).
    
    
11. How much faster than its present rate should the earth rotate about its axis so that the weight of the body on equator may be zero? What will be the duration of a day? Given: Radius of the Earth \(R_{e}=6400km\).
    
    
    Ans: \(\omega_{1}=17\times\omega\) and \(T=1.412hours\).
    
    
12. Calculate: the imaginary angular velocity of the earth for which the effective acceleration due to gravity at the equator becomes zero. In this case find the length (in hours) of a day? Given: Radius of the Earth \(=6400km\) and \(g=10\frac{m}{s^{2}}\).
    
    
    Ans: \(\omega=\left(1.25\times10^{-3}\right)\frac{rad}{sec}\) and \(T=1.4hours\).
    
    
13. An objects weighs \(10N\) at north pole of earth. In a geostationary satellite distance \(7R\) from center of the earth (of radius R), what will be its (i) True Weight and (ii) Apparent Weight?
    
    Ans: (i) \(W_{true}=0.2N\) and (ii) \(W_{apparent}=0N\).
    
    
14. Determine the speed with which the earth would have to rotate on its axis so that a person on the equator would weigh \(\frac{3}{5}^{th}\) as much as present. Take equatorial radius \(=6400km\).
    
    Ans: \(\omega=\left(7.8\times10^{-4}\right)\frac{rad}{s}\).