Question

Two bodies of masses 10 kg and 5 kg are moving in concentric orbits of radii r1 and r2 such that their periods are the same. The ratio of centripetal accelerations is:

(a) 1:2
(b) 2:1
(c) 1:1
(d) 4:1

Answer 1

The ratio of centripetal accelerations for two bodies moving in concentric orbits with the same period but different masses is 1:1. This is because the period being the same indicates that their linear speeds are equal, and hence centripetal acceleration depends solely on the radii, which in this case need to balance the gravitational force equally for both masses, leading to the ratios being equal.

Step-by-step explanation:

The question asks about the ratio of centripetal accelerations of two bodies with different masses moving in concentric orbits with the same period. To find the ratio, we can use the formula for centripetal acceleration, a = (v^2)/r, where v is the linear speed and r is the radius of the orbit.

Since the periods are equal, the linear speeds are also equal, because linear speed is given by the circumference of the orbit divided by the period (v = 2πr/T). Thus, since the speeds are equal, the ratio of the centripetal accelerations will depend solely on the ratio of the radii of the orbits.

Since the questions asks for the ratio and not the actual values, we don't need the details of the radii, only the fact that the masses would orbit at the same speed due to the equal period requirement. This means the centripetal acceleration ratio is c = (r2^2)/(r1^2).

Without further information, we can't directly calculate this; however, given that both bodies are constrained to have the same centripetal force to maintain the same orbital period, we conclude the ratio must be 1:1 (option c), as the force causing the acceleration (gravity) is balanced by the centripetal force which is provided by the circular motion, and is the same for both bodies due to their equal period of orbit.

Answer 2

The ratio of centripetal accelerations for two bodies moving in concentric orbits with the same period is 1:1, regardless of their masses and radii of orbits. Option (C) is correct.

Step-by-step explanation:

The question involves the concept of centripetal acceleration in circular motion within physics. According to the law of universal gravitation and circular motion dynamics, bodies moving in concentric orbits around a common center with the same period will have different centripetal accelerations depending on the radius of their orbits. The centripetal acceleration a is given by a = v^2 / r (where v is the linear velocity and r is the radius of the orbit). The period T of an object in circular motion is given by T = 2πr / v. If two objects have the same period, we can derive their velocities as proportional to their respective orbital radii.

For the bodies of masses 10 kg and 5 kg to have the same period, their velocities must be such that v1 / r1 = v2 / r2. Since a = v^2 / r, we can express the ratio of the velocities squared as (v1 / v2)^2 = (r1 / r2). Simplifying these equations and considering that T1 = T2, the ratio of their centripetal accelerations will be 1:1, which corresponds to option (c). Therefore, no matter the mass or radius, if two objects have the same orbital period, their centripetal accelerations will be the same.