Question 161:
easyThe 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 \(\frac{h}{R}\).
Question 162:
easyStatement A: If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attracts the particle.
Statement B: If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero.
Statement C: If a particle is inside a uniform solid sphere, the gravitational force on the particle is zero.nIn light of above statements choose the correct option.
By shell theorems, the gravitational force inside a uniform spherical shell is zero (Statement B is correct) and outside it behaves like a point mass (Statement A is correct). Inside a solid sphere, the force is non-zero (Statement C is incorrect).
Question 163:
easyBy 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 \( \frac{g - g'}{g} \times 100 = (1 - 0.64) \times 100 = 36% \).
Question 164:
easyAssertion (A): Escape velocity from surface of a planet is \(V_e\). If a tunnel is made inside the surface, the escape velocity from a point inside the tunnel must be greater than \(V_e\).
Reason (R): Gravitational force is a conservative central force.
Assertion (A) is true; the gravitational potential energy is more negative (deeper well) inside a uniform planet, requiring greater escape velocity.
Reason (R) is true and foundational: gravity being a conservative central force allows the definition and calculation of potential energy and escape velocity.
Question 165:
easyAssertion (A): Total energy is conserved in moving a satellite to higher orbit.
Reason (R): Sum of change in potential energy and kinetic energy is same in magnitude and opposite in nature.
Assertion (A) is false; moving a satellite to a higher orbit requires external work, increasing its total energy. Reason (R) is false in this context; \(\Delta K = -\Delta U\) applies only when mechanical energy is conserved, which is not the case when external work is done.
Question 166:
easyAssertion (A): An artificial satellite is moving in a circular orbit of the earth. If the gravitational pull suddenly disappears, then it moves with the same speed tangential to the original orbit.
Reason (R): The orbital speed of a satellite decreases with the increase in radius of the orbit.
Assertion (A) is true due to inertia (Newton's First Law). Reason (R) is true as orbital speed \(v = \sqrt{GM/r}\) decreases with increasing \(r\).
However, R does not explain A; A is a consequence of inertia, not the relation between orbital speed and radius.
Question 167:
easyAssertion (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.
Question 168:
easyAssertion (A): A person in an artificial satellite revolving around the earth feels weightlessness.
Reason (R): There is no gravitational force on the satellite.
Assertion (A) is true because weightlessness in orbit is due to continuous freefall. Reason (R) is false; gravitational force *is* present and provides the centripetal force for orbit.
Question 169:
easyAssertion (A): A spherically symmetric shell produces no gravitational field anywhere.
Reason (R): The field due to various mass elements cancel out, everywhere for a spherically symmetric shell.
A spherically symmetric shell produces zero gravitational field *inside* itself, but a non-zero field *outside*. Thus, "anywhere" and "everywhere" make both Assertion (A) and Reason (R) are false.
Question 170:
easyAssertion (A): The plane of the orbit of an artificial satellite must contain the centre of the earth.
Reason (R): For the orbital motion of satellite, the necessary centripetal force is provided by gravitational pull of earth on satellite.
Both Assertion (A) and Reason (R) are true. The gravitational force being a central force (R) ensures angular momentum conservation, which mandates that the orbital plane must contain the center of the earth (A).