# 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)** ,