9.3 Working with moving pulleys  (Page 3/3)

We assume a pseudo force on the body being studied in the accelerated frame. The magnitude of pseudo force is equal to the product of mass and acceleration of the frame of reference. It acts in the direction opposite to the acceleration of the frame of reference. While writing equation of motion, we also incorporate this force. But, we use acceleration of the body with respect to accelerated frame of reference - not with respect to inertial frame. This scheme is easily understood with an example. Considering the case as above, let us analyze the motion of block "1". Let us assume that moving pulley is accelerating upwards.

Here, pseudo force is :

$$F_s=m_1{a}_B$$

The pseudo force is acting in downward direction as the moving pulley "B" is accelerating upward. Now, applying Newton's second law of motion , we have :

$$T-m_{1}g-m_1{a}_B=a_{1B}$$

$$T-m_{1}g=m_1(a_{1B}+a_B)=m_1{a}_1$$

Thus, we see that analyzing motion in accelerated frame with pseudo force is equivalent to analyzing motion in inertial frame.

Advantages of analysis in moving frame of reference

The consideration of relative acceleration as against absolute acceleration for analysis using constraint relation (see Pulleys ) has many advantages. We enumerate these advantages here as :

1 : The relative accelerations of blocks with respect to moving pulley are equal in magnitude, but opposite in direction. This is based on the fact that blocks are attached with a single string that passes over moving pulley. This simplifies analysis a great deal.

On the other hand, if we refer accelerations to the ground, then we can not be sure of the directions of accelerations of the blocks as they depend on the acceleration of the pulley itself. It is for this reason that we generally assume same direction of absolute accelerations of the blocks. If the assumption is wrong, then we get negative value of acceleration after analysis, showing that our initial assumption about the direction was wrong and that the acceleration is actually opposite to that assumed. This technique was illustrated in the module on Pulleys .

2 : Sometimes observed values are given in terms of relative reference in the first place. In this situation, we have the easy option to carry out analysis in the accelerated reference itself. Otherwise, we would be required to convert given values to the ground reference, using concept of relative acceleration and carry out the analysis in the ground reference.

Transition between absolute and moving reference

We enumerated advantages of relative acceleration technique. It does not, however, mean that analysis of motion in ground reference has no virtue. As a matter of fact, we are interested in the values, which are referred to ground reference – measurement of accelerations as seen in the ground reference. Even though, relative acceleration technique allows us to simplify solution; the analysis in accelerated frame essentially yields values, which are referred to the accelerated frame of pulley. Ultimately, we are required to convert the solution or values of acceleration in the ground reference.

On the other hand, we have seen in earlier module (see Pulleys ) that the technique of constraint relation is extremely effective. It relates velocities and accelerations of the elements of the system in the ground reference very elegantly – notwithstanding the complexity of the system.

It emerges from above discussion that two frameworks of analysis have their relative strengths and weaknesses. It is, therefore, pragmatic to combine the strengths of two techniques in our analysis. We may transition between two analysis frameworks, using the concept of relative acceleration in one dimension.

The emphasis on the sequence of subscript almost makes the conversion mechanical. It is always helpful to read the relation : “the relative acceleration of “1” with respect to “2” is equal to the absolute acceleration of “1”, subtracted by the absolute acceleration of “2”.