Question

Find the constraint relation between ${{a}_{1}},{{a}_{2}}$ and ${{a}_{3}}$.

Answer 1

We will need to draw a diagram representing all three bodies represented for better understanding of the situation. Then we need to relate ${{a}_{1}}$, ${{a}_{2}}$ and ${{a}_{3}}$. For finding constraint relation, we must find the relation between displacements of the 3 bodies. Then we will double differentiate both the relations regarding displacement which will give us the constraint relation.

Complete step by step answer:
Firstly we will try to draw a detailed diagram which represents the displacement relations between the 3 bodies.

Here, the bodies marked as 1, 2, 3 and 4 are movable. Let us take the displacements from a fixed point which is extended as the dotted lines. Here, distance to the first body is taken as ${{X}_{1}}$ and similarly distances to the bodies 2, 3 and 4 are taken as ${{X}_{2}},{{X}_{3}}\text{ and }{{\text{X}}_{4}}$ respectively.
Now, we will derive the relation between ${{l}_{1}}\text{ and }{{l}_{2}}$ with the distances from the bodies.
We have ${{l}_{1}}$ given as,
${{l}_{1}}={{X}_{1}}+{{X}_{4}}$ - (1) (length of the first string)
Then we have ${{l}_{2}}$which is given as,

& {{l}_{2}}=\left( {{X}_{2}}-{{X}_{4}} \right)+\left( {{X}_{3}}-{{X}_{4}} \right) \\\ & {{l}_{2}}={{X}_{2}}-{{X}_{3}}-2{{X}_{4}} \\\ \end{aligned}$$ - (2) (length of second string) Now, we will double differentiate (1) and (2) w.r.t time which will give us the acceleration. Here $${{l}_{1}}\text{ and }{{l}_{2}}$$ are not changing, so the differentiation will be zero. $$\Rightarrow {{a}_{1}}+{{a}_{4}}=0$$ - (3) $$\Rightarrow {{a}_{2}}+{{a}_{3}}-2{{a}_{4}}=0$$ - (4) But, from (3) we can have, $$-{{a}_{1}}={{a}_{4}}$$. So, we will substitute this in (4). $$\Rightarrow {{a}_{2}}+{{a}_{3}}+2{{a}_{1}}=0$$ So, we have the constraint relation between $${{a}_{1}},{{a}_{2}}$$ and $${{a}_{3}}$$ as $$2{{a}_{1}}+{{a}_{2}}+{{a}_{3}}=0$$. **Note:** To solve these types of questions, we should make the relation between the given parameters and try to relate it with the one which is required to be found. Here, we are trying to find the length of the rope which is a constant quantity by using the distances of the body from a fixed point. The differentiation of $${{l}_{1}}\text{ and }{{l}_{2}}$$ are taken as zero because the length of the rope is constant.