A trolley is under the action of a constant force F. The...

A trolley is under the action of a constant force $F$. The sand contained by it is poured out through a hole in the floor at the rate of $m$ per second. If initial mass of sand and trolley was $M$ and initial speed was $u$, then acceleration of trolley is given by (Figure can't copy) (a) $\frac{F}{M-m t}$ (b) $\frac{F}{M+m t}$ (c) $\frac{F}{M-m}$ (d) $\frac{F}{M+m}$

A trolley is under the action of a constant force $F$. The sand contained by it is poured out through a hole in the floor at the rate of $m$ per second. If initial mass of sand and trolley was $M$ and initial speed was $u$, then acceleration of trolley is given by
(Figure can't copy)
(a) $\frac{F}{M-m t}$
(b) $\frac{F}{M+m t}$
(c) $\frac{F}{M-m}$
(d) $\frac{F}{M+m}$

Key Concepts

Variable Mass Systems

This concept refers to systems in which the mass changes over time. Such changes require a careful application of Newton's laws of motion because the standard form, F = ma, assumes constant mass. In variable mass systems, momentum must be treated as the product of a time-varying mass and velocity, and one must account for mass loss or gain when analyzing the dynamics.

Newton's Second Law in Systems with Changing Mass

Newton's second law is extended to account for changes in mass by considering the rate of change of momentum rather than strictly F = m(dv/dt). For systems losing or gaining mass, the derivative of the product of mass and velocity must be taken, which introduces additional terms related to the mass flow rate. This modification is critical in deriving accurate expressions for acceleration in such systems.

Rate of Mass Change

In problems involving systems with a time-dependent mass, the rate at which mass is being lost or gained directly influences the dynamics. When mass is lost at a constant rate, the effective mass of the system decreases linearly with time. This changing mass needs to be incorporated into equations of motion, affecting computed quantities like acceleration.

Constant Force Application

The application of a constant force to a system implies that the external driving force remains unchanged over time, regardless of the system's mass. However, when the mass of the system is not constant, the instantaneous acceleration will vary over time as it is defined by the ratio of the constant force to the instantaneous mass. This interplay is key to understanding the resulting motion.

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