Two satellites of masses m and 2m are orbiting the earth at the same distance from the center of the earth. If the gravitational force on the satellite of mass m is F, what is the gravitational force on the satellite of mass 2m?
AF
B2F
CF/2
D4F
The gravitational force between two objects is given by F = G * (M1 * M2) / r^2. Here, the mass of the earth (M1) and the distance (r) are the same for both satellites. The force is directly proportional to the mass of the satellite (M2). Therefore, for the satellite of mass 2m, the gravitational force will be 2F.
PhysicsGravitationHard#22
A planet has a mass M and radius R. If a tunnel is dug along the diameter of the planet, and a small mass m is dropped into it, what will be the time period of its oscillation?
A2π√(R³/GM)
B2π√(GM/R³)
Cπ√(R³/GM)
D2π√(R/GM)
Inside a uniform sphere, the gravitational field at a distance r from the center is (GM/R³)r. The force acting on the mass m is a restoring force proportional to displacement, leading to simple harmonic motion. The time period T = 2π√(R³/GM), which is the same as the time period of a satellite in a low orbit.
PhysicsGravitationHard#23
If the radius of the earth were to shrink by 1% while its mass remains the same, by what percentage would the acceleration due to gravity on the surface of the earth increase?
A1%
B2%
C0.5%
D3%
Acceleration due to gravity g = GM/R². If R decreases by 1%, new R' = 0.99R. So, g' = GM/(0.99R)² = GM/(0.9801R²) ≈ 1.02g. Thus, g increases by approximately 2%. Mathematically, using differentiation, dg/g = -2(dR/R). Since dR/R = -0.01, dg/g = 0.02 or 2%.
PhysicsGravitationHard#24
Two stars each of mass M are approaching each other under mutual gravitational attraction. If they are very far apart initially, what will be their relative speed when they are at a distance r apart?
A√(GM/r)
B√(2GM/r)
C√(3GM/r)
D√(4GM/r)
Using conservation of energy, initially, potential energy is zero (since r → ∞) and kinetic energy is zero. At distance r, potential energy = -GM²/r. By conservation of energy, total energy remains zero. Kinetic energy of the system (considering reduced mass μ = M/2 for two equal masses) leads to relative speed v = √(2GM/r).
PhysicsGravitationHard#25
What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R into a circular orbit at an altitude equal to R?
AGMm/(2R)
BGMm/(4R)
C3GMm/(4R)
DGMm/R
Orbit radius = R + R = 2R. Total energy in orbit = -GMm/(2*2R) = -GMm/(4R). Initial energy on surface = -GMm/R. Energy required = Final energy - Initial energy = [-GMm/(4R)] - [-GMm/R] = GMm/R - GMm/(4R) = 3GMm/(4R).
PhysicsGravitationHard#26
The gravitational potential at a point on the axis of a thin uniform ring of mass M and radius R, at a distance x from the center, is given by which of the following expressions?
A-GM/√(R² + x²)
B-GM/(R + x)
C-GM/x
D-GM/R
For a thin ring, the gravitational potential at a point on its axis at distance x from the center is V = -GM/√(R² + x²). This is derived by integrating the potential due to each differential mass element of the ring, considering the distance from the point to each element is constant and equal to √(R² + x²).
PhysicsGravitationHard#27
A satellite is in a circular orbit around the earth. If the radius of its orbit is increased by a factor of 8, by what factor does its orbital speed change?
A1/2
B1/4
C1/√8
D1/8
Orbital speed v = √(GM/r). If r increases by a factor of 8, new v' = √(GM/(8r)) = (1/√8) * √(GM/r) = v/√8. So, the orbital speed decreases by a factor of 1/√8.
PhysicsGravitationHard#28
The escape velocity from a planet is v. If a tunnel is dug to the center of the planet and a body is dropped into it, what will be its speed at the center?
Av
Bv/√2
Cv/2
D√2 v
Escape velocity v = √(2GM/R). At the center, potential energy is -3GMm/(2R). Initial energy at surface = -GMm/R. By conservation of energy, kinetic energy at center = [-GMm/R] - [-3GMm/(2R)] = GMm/(2R). Speed = √(GM/R) = v/√2.
PhysicsGravitationHard#29
A binary star system consists of two stars of masses M and 2M orbiting about their common center of mass. If the distance between them is d, what is the orbital speed of the star of mass M?
A√(GM/(3d))
B√(2GM/(3d))
C√(GM/d)
D√(2GM/d)
Center of mass is closer to 2M. Distance of M from center of mass is 2d/3, and for 2M it is d/3. For mass M, centripetal force is provided by gravitational force. So, Mv²/(2d/3) = G(M)(2M)/d². Solving, v = √(2GM/(3d)).
PhysicsGravitationHard#30
If the angular momentum of a planet of mass m orbiting the sun in an elliptical orbit is L, what is the minimum distance of the planet from the sun? (Given: Mass of sun = M, eccentricity = e)
AL²(1 - e)/(GMm²)
BL²(1 + e)/(GMm²)
CL²/(GMm²(1 - e))
DL²/(GMm²(1 + e))
Angular momentum L = m*v*r at perihelion (minimum distance). Using vis-viva equation and properties of ellipse, at perihelion, r_min = a(1 - e), where a is semi-major axis. Total energy and L relate to a via L²/(GMm²) = a(1 - e²). Thus, r_min = L²/(GMm²(1 + e)).