Question

A block of mass \[M_1\] is attached to the end of a massless rope. The other end of the rope passes through a hole in the center of the table and is attached to a second mass \[M_2\] that is hanging motionless. The mass \[M_1\] is given a tangential velocity such that it moves in a circular path with a radius of \[R\]. The tension force between \[M_1\] and \[M_2\] results in the hanging mass \[M_2\] remaining at rest while \[M_1\] moves in a circular path. As \[M_1\] rotates around the circle, the rope is not impeded by the hole because the hole is coated with a frictionless surface. A student wants to change the hanging mass, \[M_2\], while keeping the radius of the path and the rotating mass, \[M_1\], the same. Which of the following changes will result in no change in the motion of the new hanging mass?

Answer 1

To change the hanging mass M2 with no impact on the system's motion, the new mass must have the same weight as M2, or adjust the velocity of M1 to achieve a new tension force that equals the weight of the new mass.

The scenario involves a system where mass M1 is rotating on a frictionless table while connected to mass M2, which is hanging and at rest. According to Newton's second law and the concept of centripetal force, the tension in the rope caused by the circular motion of M1 must equal the weight of the hanging mass M2 for it to remain at rest. Therefore, if you wish to change mass M2 without affecting the system's motion, you must ensure that the new mass either has the same weight as the original mass M2, so the tension in the rope and thus the centripetal force remains unchanged or adjust the tangential velocity of the mass M1 such that the new tension equals the weight of the new mass.

Remember, the formula for determining the tension in the rope due to a mass moving in a circular path (with radius R and tangential velocity v) is T = M1 * v² / R, where T is the tension, and this tension is balanced by the weight of the hanging mass M2, that is T = M2 * g, where g is the acceleration due to gravity.