Final answer:
The tension on both sides of a bead on a massless inextensible string is equal due to Newton's second law. In motion, the string's tension must be uniform to avoid infinite acceleration. This principle of uniform tension applies to both static and dynamic scenarios.
Step-by-step explanation:
To prove that the tension on both sides of a bead on a massless inextensible string is equal, we can use Newton's second law. When the string becomes taut and delivers an impulse to the bead, the forces acting on it are the gravitational force, which is the weight (mg), downward, and the tension in the string on either side of the bead.
Assuming the string is massless and the system is in motion, the tension throughout the string must be uniform because any difference in tension would result in an infinite acceleration which is not possible. This is illustrated by the scenario of a mass hanging from a string in static equilibrium, where the tension in the string supports the weight of the mass. The same principle applies when the bead is in motion - the tension on both sides of the bead must be equal to balance the forces and maintain the integrity of the massless inextensible string.
A similar situation is described in Figure 16.13, where a mass element of a string kept taut with tension Fr is in static equilibrium, and the force of tension acting on either side is equal in magnitude and opposite in direction.