The escape velocity from the Earth’s surface is \(v\). The escape velocity from the surface of another planet having a radius, four times that of Earth and same mass density is
Escape velocity is given by \(v_e = R\sqrt{\frac{8\pi G\rho}{3}}\). Since the mass density \(rho\) is the same, \(v_e\) is directly proportional to radius \(R\). Therefore, \(v_e' = 4v\).
A particle of mass \(m\) is projected with a velocity \(v = k V_e\) (\(k < 1\)) from the surface of the earth. (\(V_e = \text{escape velocity}\)) The maximum height above the surface reached by the particle is
By conservation of mechanical energy: \(-\frac{GMm}{R} + \frac{1}{2}mv^2 = -\frac{GMm}{R+h}\). Since \(v = k \sqrt{\frac{2GM}{R}}\), we substitute to get \(-\frac{1}{R}(1 - k^2) = -\frac{1}{R+h}\), which yields \(h = \frac{R k^2}{1-k^2}\).
Two masses each equal to \(M\) are moving on a circular path of radius \(R\) about another fixed mass \(M\) (at the centre of the circular path). The gravitational potential energy of the system is:
The total GPE of the three-mass system is \(U = -\frac{GMM}{R} - \frac{GMM}{R} - \frac{GMM}{2R} = -\frac{5GM^2}{2R}\) since the outer masses are at a distance of \(2R\) from each other and \(R\) from the center.
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.
If potential energy is assumed to be zero at infinity, then
For an orbiting satellite, Potential Energy \(U = -\frac{GMm}{r}\), Kinetic Energy \(K = \frac{GMm}{2r}\), and Total Energy \(E = -\frac{GMm}{2r}\). This shows that \(U = 2E\).
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, \(F = \frac{GM^2}{(2R)^2} = \frac{GM^2}{4R^2}\).
The ratio of escape velocity at earth (\(v_e\)) to the escape velocity at a planet (\(v_p\)) whose radius and mean density are twice as that of earth is
Escape velocity is given by \(v_e = R\sqrt{\frac{8}{3}\pi G\rho}\). Thus, \(v_e \propto R\sqrt{\rho}\). The ratio is \(frac{v_e}{v_p} = \frac{R_e}{R_p}\sqrt{\frac{\rho_e}{\rho_p}} = \frac{1}{2}\sqrt{\frac{1}{2}} = \frac{1}{2\sqrt{2}}\).
Statement 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.
In light of above statements choose the correct option.
Statements A and B are correct (shell theorem). Inside a uniform solid sphere, the gravitational force is non-zero (except at the center) and varies linearly with distance from the center.
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): The mechanical energy of earth-moon system remains same when a heavenly body passes nearby the earth-moon system.
Reason (R): Force exerted by heavenly body on the earth-moon system is non-conservative.
Assertion (A) is false:
An external heavenly body exerts a gravitational force on the Earth-Moon system, causing its mechanical energy to change.
Reason (R) is false: Gravitational force is a conservative force by nature. Therefore, both Assertion (A) and Reason (R) are false.