Two bodies of masses m1 and m2 are connected by a light,...

Two bodies of masses $m_{1}$ and $m_{2}$ are connected by a light, inextensible string which passes over a frictionless pulley. If the pulley is moving upward with uniform acceleration $g$, then the tension in the string is (a) $\frac{4 m m_{2}}{m_{1}+m_{1}} g$ (b) $\frac{m_{1} m_{2}}{4 m_{1} m_{2}} g$ (c) $\frac{m_{1} m_{2}}{m_{1}+m_{1}} g$ (d) $\frac{m_{1}-m_{1}}{m_{1}+m_{1}} g^{2}$

Two bodies of masses $m_{1}$ and $m_{2}$ are connected by a light, inextensible string which passes over a frictionless pulley. If the pulley is moving upward with uniform acceleration $g$, then the tension in the string is
(a) $\frac{4 m m_{2}}{m_{1}+m_{1}} g$
(b) $\frac{m_{1} m_{2}}{4 m_{1} m_{2}} g$
(c) $\frac{m_{1} m_{2}}{m_{1}+m_{1}} g$
(d) $\frac{m_{1}-m_{1}}{m_{1}+m_{1}} g^{2}$

Key Concepts

Newton’s Second Law in Accelerating Systems

Newton’s second law, F = ma, must be applied carefully in non?inertial frames. The additional pseudo force is added to the real forces. For masses connected by a string over a pulley, the dynamics involve both the actual gravitational forces and the additional forces arising from the acceleration of the pulley, leading to modified equations that relate tensions and net accelerations.

Tension in a Pulley System with Connected Masses

In a system with two masses connected by a light, inextensible string over a frictionless pulley, the tension is determined by balancing the forces acting on each mass. In an accelerating system, the effective weight of each mass becomes m(g + a), where a is the acceleration of the pulley. This makes the expression for tension different from the standard case of a static pulley. Analyzing the forces with the effective gravitational acceleration allows one to derive the correct expression for the tension in the string.

Effective Gravitational Acceleration

When a system accelerates upward with acceleration equal to g, the effective gravitational acceleration perceived in the accelerating frame combines the actual gravitational pull and the extra pseudo force. For example, an upward acceleration of g makes the effective acceleration equal to g + g = 2g. This adjustment is crucial for correctly analyzing forces acting on masses in the system.

Inertial and Non?Inertial Reference Frames

In dynamics, a problem is set in either an inertial frame (one at rest or moving uniformly) or a non?inertial frame (accelerating). When dealing with an accelerating system, such as a pulley moving upward uniformly, one must account for additional apparent (or ‘pseudo’) forces that appear because the observer’s frame is accelerating relative to an inertial frame.

Pseudo Forces

A pseudo force (or fictitious force) is an apparent force that arises when observing motion from a non?inertial frame. In an upward accelerating frame, objects appear to experience an extra force downward. This additional force must be considered along with gravity when applying Newton’s laws in such frames.

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