# Pulley Problems: How to Solve Pulley Problems: CASE – 1: CASE – 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Pulley Problems

## How to Solve Pulley Problems?

For solving any pulley problem, the first step is to understand the given conditions and write down the constraint equations accordingly.

## CASE – 1

• Let, M1 & M2 be the mass attached to the pulley A.
• Now, consider that the mass M1 is moving down with acceleration a1 and mass M2 is moving up with acceleration a2.

From the Free Body Diagram (FBD) :

• The equation of force from FBD:
• Equation of force from FBD:

• From the above force equation, we have three unknowns but there are only 2 equations (Equation (1) & Equation (2)
• So we need a third equation relating the two unknowns. The relation is known as constraint equation because the motion of M1 and M2 is interconnected.
• The following assumptions must be considered before writing the equation:
• The string is taut and inextensible at each and every point of time.
• The string is massless and hence the tension is uniform throughout.
• Pulley is massless.
• The string is inextensible hence the total change in length of the string should be zero.
• Suppose mass goes down by distance and mass moves up by distance.
• Change in length (Taking elongation as positive and compression as negative)
• On equating it to zero we get, = 0 (x = displacement)
• On Differentiating equation (3) twice we get,
• = 0 … (3) (a = acceleration)
• Using Equation (1) , (2) & (3) we can calculate the values of T,
• This is an example of a Fixed Pulley System.

## CASE – 2

• Consider the following pulley system:
• First, we have to relate the tension between string 1 & string 2,

Consider F. B. D of pulley B:

(Since the pulley is massless)

• Now, Consider the accelerations as shown in the below figure:
• For String 2:
• Movable pulley B and Block M2 will have same acceleration otherwise the string will stretch or compress.

… (5)

• Increase in length to the left of fixed pulley A =
• Decrease in length to the right of fixed pulley A =
• Decrease in length to the right of movable pulley B
• And, .
• Therefore, the total change in length = (Since, )
• On differentiating the above equation twice we get,

… (6)

• Similarly, if we write force equation on both the blocks:

Using equations (5) , (6) , (7) and (8) ,

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