Linear motion of a body of variable mass
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Linear Motion of a Body of Variable Mass. Newton's Second Law
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When an external force is applied to a body the rate of increase of momentum produced is directly proportional to the applied force. An instantaneous rate is given by the first derivative; thus this law is written as [Equation] The S.I. units of mass are kilograms (kg), of velocity metres/second (ms-1), and when mass and velocity are measured in these units the constant of proportionality is 1 and Newton's Second Law is: [Equation] Here m = m(t) is the mass of the object at time t and v = v(t) is the velocity of the object at time t. It is often possible to integrate this law directly to obtain an expression for v = v(t), the velocity of the particle at time t. This is the case in the following example. Example (1) The initial mass of a raindrop is M0. It falls from an initial velocity V0 through a cloud of still water vapour accumulating water from the cloud as it does so. The mass of the drop at time t is M and its velocity is V. It is assumed that the mass of the drop is governed by the equation: [Equation] (i) Ignoring air-resistance, find an expression for V. (ii) Sketch the graph of V against t. [Diagram] Newton's Second Law gives:- [Equation] Thus, we can solve problems by integrating Newton's 2nd law in the form: [Equation] However, we must issue a word of warning. This would only be applicable when the object by accumulating mass does not accumulate mass that imparts an impulse to it. This means that the mass increment must be at rest, from an external viewpoint so its momentum is zero. If the mass increment is at rest from an external viewpoint from the viewpoint of the object the mass increment is travelling toward the object with magnitude v and in the opposite direction. [Diagram] From an external viewpoint the raindrop is travelling towards the water vapour with velocity v. The water vapour is still and has no momentum. [Diagram] From the viewpoint of the raindrop the water vapour is travelling towards it with speed v. If the mass increment is not still when viewed from an external frame of reference then when the particle accumulates it there will be an impulse imparted. The impulse will alter the force acting on the particle and it will no longer be true that [Equation] because F will not take into account the effect of this impulse. Thus we can only integrate [Equation] directly when the mass increment has no momentum. Often this is assumed in questions but this would not apply in cases where a rocket, for instance, moves by propelling gas in the form of burnt fuel. Thus Newton's Second Law is [Equation] Likewise, it is tempting to differentiate this result to obtain: [Equation] However, this would be a mistake and only applies when the mass increment has no momentum. Hence, either the mass of the object is not constant or the velocity of the mass increment is 0. If the mass were constant dm/dt would equal 0 and the result would reduce to the familiar form of Newton's Second Law: F = ma. We are dealing with precisely the case where the mass is not constant and dm/dt ¹ 0, and we now consider the case where the velocity of the mass increment is not 0. The mass cannot be changing if it is not either gaining or losing mass - let the mass that is either joining or is being ejected have a velocity v' at the moment of impact or separation. [Diagram] The rocket ejects mass m' with velocity v'; the raindrop accumulates mass m' with velocity v'. A body gains mass by joining with particles that have a velocity of their own. A body loses mass by ejecting particles that likewise have a velocity of their own. We must, therefore, derive a "correct" formula from first principals. There are, in fact, two cases to consider - firstly, where the object gains mass as in the rain drop, secondly, where the object loses mass, as in the rocket.
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Contents of
Linear motion of a body of variable mass
1 Linear Motion of a Body of Variable Mass - Rockets and raindrops
2 Linear Motion of a Body of Variable Mass. Newton's Second Law
3 Linear Motion of a Body of Variable Mass. First Case (mass increment)
4 Linear Motion of a Body of Variable Mass. Second Case (mass decrement)
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