A body weighs \(900\text{ N}\) on the surface of earth. How much will it weigh at a height double the radius of earth?
Using the formula for acceleration due to gravity at height \(h\): \(g' = g\left(\frac{R}{R+h}\right)^2\). For \(h = 2R\), \(g' = \frac{g}{9}\) . Hence, weight at this height is \(W' = \frac{W}{9} = \frac{900\text{ N}}{9} = 100\text{ N}\).
An object is weighed at the equator using a physical balance and a spring balance. When the same object is taken to the pole, then corresponding readings on the physical balance and spring balance (also taken there) will respectively:
A physical balance measures mass, which is constant everywhere. A spring balance measures weight, \(W = mg\). Since gravity \(g\) is greater at the poles, the spring balance reading increases.
Two masses each equal to \(M\) are moving on a circular path of radius \(R\) about a common centre. The gravitational force of attraction between the masses has magnitude
For two identical masses to move on a circular path of radius \(R\) about a common centre, they must always be diametrically opposite. The distance between them is \(2R\). Thus, \(F = \frac{GM^2}{(2R)^2} = \frac{GM^2}{4R^2}\).
By what percentage will the acceleration due to gravity at a height of 1600 km from the surface of the Earth differ from that on the surface of the Earth? (Take radius of Earth to be 6400 km)
Acceleration due to gravity at height \(h\) is \(g' = g\left(\frac{R}{R+h}\right)^2 = g\left(\frac{6400}{8000}\right)^2 = 0.64g\). The percentage difference is \(\frac{g - 0.64g}{g}\times 100% = 36%\).
Assertion (A): At the centre of the earth, a body has centre of mass, but no centre of gravity.
Reason (R): Acceleration due to gravity is zero at the centre of the earth.
Assertion (A) is true. A body always has a center of mass.
Centre of gravity is the point where the net gravitational torque is zero. At the center of the Earth, the acceleration due to gravity \( g \) is zero. Thus, there is no gravitational force, and consequently, no center of gravity in the usual operational sense.
Reason (R) is true. The acceleration due to gravity \( g \) is indeed zero at the centre of the earth. Since the absence of gravity leads to no definable centre of gravity, (R) is the correct explanation for (A).
Assertion (A): If earth stops rotating about its axis, then the value of acceleration due to gravity increases everywhere, except at the poles.
Reason (R): The value of acceleration due to gravity is maximum at the poles.
Assertion (A): The effective gravity is given by \(g' = g - Romega^2cos^2lambda\). If \(omega = 0\), then \(g' = g\). This causes \(g\) to increase everywhere except at poles (where \(coslambda = 0\)). So (A) is true.nReason (R): Due to rotation and equatorial bulge, \(g\) is maximum at poles and minimum at the equator. So (R) is true.n(R) correctly explains (A) as the effect of rotation explains the variation.
The fractional change in acceleration due to gravity on surface of earth (g) when the body is taken to a depth h from the surface of earth, is (where R = radius of Earth)
The acceleration due to gravity at depth \(h\) is \(g_d = g\left(1 - \frac{h}{R}\right)\). The change in gravity is \(\Delta g = g - g_d = g\frac{h}{R}\). Thus, the fractional change \(\frac{\Delta g}{g}\) is \(\frac{h}{R}\).
By what percentage will the acceleration due to gravity at a height of \( 1600\text{ km} \) from the surface of the Earth differ from that on the surface of the Earth? (Take radius of Earth to be \( 6400\text{ km} \))
Acceleration due to gravity at height \( h \) is \( g' = g\left(\frac{R}{R+h}\right)^2 = g\left(\frac{6400}{8000}\right)^2 = 0.64g \). The percentage difference is \( \frac{g - g'}{g} \times 100 = (1 - 0.64) \times 100 = 36% \).
Assertion (A): If a body is taken from earth to moon, its gravitational mass becomes one-sixth on moon.
Reason (R): Gravitational mass depends upon acceleration due to gravity.
Mass is an intrinsic property of a body and does not change with location or acceleration due to gravity. Only weight \(W = mg\) changes. Therefore, both Assertion (A) and Reason (R) are false.