First slide

Questions using constraints

Question

In the figure shown a₃ = 6m/s² (downwards) and a₂ = 4m/s² (upwards). Find acceleration of 1.

Moderate

A

1m/sec² upwards
B

2m/sec² upwards
C

1m/sec² downwards
D

2m/sec² downwards

Solution

Since the points 1,2,3 and are movable, so let their displacements are x₁ , x₂ , x₃ and x₄ We observe that the length of the strings between 1 and 4 and 2 and 3 are constants.
$x_{1} + x_{4} = ℓ_{1}; (x_{2} - x_{4}) + (x_{3} - x_{4}) = ℓ_{2}$
Differentiating twice w.r.t time, we get
$\begin{array}{l} a_{1} + a_{4} = 0 \Rightarrow a_{1} = - a_{4} \to (1) \\ a_{2} - a_{4} + a_{3} - a_{4} = 0 \\ \Rightarrow a_{2} + a_{3} - 2 a_{4} = 0 \to (2) \\ a_{2} = - 4 {ms}^{- 2}, a_{3} = 6 {ms}^{- 2}; - 4 + 6 - 2 a_{4} = 0 \\ 2 a_{4} = 2 \Rightarrow a_{4} = 1 {ms}^{- 2}; From (1), \\ a_{1} = - a_{4}; a_{1} = - 1 {ms}^{- 2}; a_{1} = 1 {ms}^{- 2} upwards \end{array}$

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