Irodov Problem 1.72

The force diagram of the problem is shown beside. We assume that that the masses are and, the tensions in the two strings are and and their accelerations are and respectively. This is indicated in the figure.

Now let us consider the relavent forces acting on each of the two masses first.

Forces acting on mass : For this problem we need to only consider forces acting on the body1 along a direction parallel to the inclined plane . There are two forces along this direction - i) the component of force of gravity and ii) the tension in the string , as shown in the figure. The mass accelerates up the inclined plane at a rate . Thus the equation of motion for body 1 can be written as,


Forces acting on mass : There are two forces acting on this mass, i) the tension in the second string and the force of gravity as shown in the figure. Since we have assumed that body 1 is moving up the plane, the reasonable thing is to assume that body 2 moves down. Suppose that it moves down with an acceleration of . Its dynamics are given as,


Forces on the Moving Pulley holding : There are three forces acting on the pulley, i) the tensions acting to pull the pulley down from each part of the thread that holds and ii) the tension in the thread holding mass . The mass of he pulley is 0 and it will accelerate at a rate downwards. Thus, we can write the dynamics of the pulley as,




How are and related ?: There is one more piece needed to solve the problem, the fact that the accelerations of the two bodies are related to each other. The basic principle that we use in figuring this relationship out is that the length of the thread is conserved. Please refer to the figure beside to understand the explanation. Suppose that the mass moves a length x along the inclined plane upwards. The moving pulley thus will descend a distance x. The length of the thread section AB now decreases by x. Thus, as seen in the figure A' B = AB-x. Since the length of the thread (AB + CD) has to be conserved, the length of the thread section CD must thus increases by x, C'D' = CD + x. In other words, will descend a distance x relative to the moving pulley. Since the pulley itself has descended x, the total distance that descends is x + x = 2x. To summarize, if moves a distance x units moves a distance 2x units. Thus, we have,



Now we are in a position to solve the problem. We can use (3) and (4) in (1) to get,

Irodov Problem 1.71

To a person in an accelerating elevator thats accelerating upwards at rate then the person of mass m will feel as if the gravitational is . The extra is some times referred to also as a pseudo force that a person in an accelerated frame experiences. In fact this also lead to Einstein's famous principle of equivalence. So we can solve the problem in the accelerated frame of reference of the elevator as if the effective gravitational constant were . However, here we shall solve with respect to a stationary frame.



Let T be the tension in the pulley string, F be the force between the pulley and the elevator car. Further let us assume that the acceleration of mass relative to the elevator car be w directed towards the elevator floor. Mass then accelerates at a rate w towards the elevator ceiling relative to the elevator car. The net accelerations of and relative to a stationary observer are thus and towards the elevator floor respectively.

Forces on mass : There are two forces acting on this mass i) gravity and ii) tension in the pulley string as shown in the figure. Thus, its dynamics are given by,



Forces on mass : There are two forces acting on this mass i) gravity and ii) tension in the string as shown in the figure. Thus, its dynamics are given by,



We can now solve of w from (1) and (2) by subtracting the two equations as,





From (3) now we can determine the acceleration of relative to the shaft in the direction towards the floor (stationary observer) as,









Forces acting on the Pulley: There are three force acting on the pulley i) tension T in the part of the string connecting mass , ii) tension T in the part of the string connecting mass , and iii) the force the elevator exerts on the pulley F. The pulley is accelerating with an acceleration of upwards but the pulley has no mass. Thus, we have,




Tension T can be calculated from (1) and (4), as,









From (6) the force exerted on the elevator on the pulley F is