# Constraint Relation

**Constraint Relation** is a beautiful and interesting concept which helps in solving questions related to pulleys and strings. It can be used to solve even the most complicated problems. Once understood it will be a very useful tool for you in solving problems in dynamics.

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## Definition

Constraint relationsays that the sum of products of all tensions in strings and velocities of respective blocks connected to the strings is equal to $0$. In other words it says that thetotal power by tension is zero. Mathematically it is represented by : $\displaystyle \sum T \cdot \overline{v} = 0$If the velocity vector is constant then differentiating the above equation with respect to time we get another relation. If the velocity vector is constant then the sum of products of all tensions in strings and accelerations of respective blocks connected to the strings is equal to $0$. It is mathematically represented by : $\displaystyle \sum T \cdot \overline{a} = 0$

**Note :** Constraint relation works only when **the strings are inextensible and taut.**

Before you learn on how to write the constraint relation there is one point that you have to remember :

When it is given that a specific pulley is mass less then the

tensions on both the sides of that pulley are equal.If you encounter with a situation as shown in the below picture and if the

pulley is mass lessthen the the tension on both sides are equal. That means : $\boxed{2T_p = T + T = 2T}$ This is a standard result and you will be going to use this continuously in solving questions.

**Technique for writing constraint relation :**

For step is to draw free body diagrams of all the pulleys and blocks showing tensions, velocities, etc clearly. If you encounter with a mass less pulley you can consider the block connected to that pulley and the pulley itself as one body.

Now, wherever you encounter with a tension in the string then multiply the tension and velocity (or acceleration) of the block connected to it. Sign convention is very important. If the tension in the string and velocity of the block are in same direction then $+ ve$ sign must be taken and $- ve$ sign will be taken if they are in opposite directions.

At last add all the products of tension's and velocities (or acceleration) and equate it to $0$.

And that's all you have written the required constraint relation.

## Problem Solving - Basic

In this section you are going to solve basic questions related to constraint relation.

Find acceleration of the two blocks and the tensions in $string~1$ and in $string~2$. Take $A = 7~kg$ and $B = 9~kg$. Assume that the pulley is mass-less and friction-less. Both the strings are in-extensible.

The first thing you have to do is to draw the free body diagram and to show all the forces acting on the two blocks. Now we will directly write the equations. $a$ is the acceleration of the blocks, $T_1,T_2$ are tensions in string $1,2$.$\begin{array}{c}~T_1 - 70 = 7a \\ 90 - T_1 = 9a \\ \hline 20 = 16a \implies \boxed{a = 1.25~m/s^2} \\ T_1 = 70 + 7(1.25) \implies \boxed{T_1 = 96.25~N} \\ \end{array}$

It is given that the pulley is mass-less so, $T_2 = 2T_1 \implies \boxed{T_2 = 192.5~N}$

Establish a relation between the velocities of blocks $A,B,C$. Assume that block $A$ moves with velocity $v_A \uparrow$ block $B$ moves with velocity $v_B \downarrow$ and block $C$ moves with velocity $v_C \downarrow$. The pulley is mass-less and the strings are in-extensible. Neglect friction between any two surfaces.

In the question the direction of velocities are already given. I assume that the tension connecting $B$ and $C$ is $T$ which is acting upwards. Also, the pulley is mass-less so $T_A = 2T$ in upward direction. For blocks $B$ and $C$ velocities and tensions are opposite in direction so you should take $-ve$ sign. But for block $A$ velocity and tension are in same direction so you should take $+ve$ sign. so, now write the relation $\displaystyle \sum T \cdot \overline{v} = 0$$\implies 2Tv_A - Tv_B - Tv_C = 0$

$\implies \boxed{2v_A = v_B + v_C}$

## Problem Solving - Intermediate

Given that:

- the velocity of $A$ with respect to $C$, $v_{AC} = \text{300 m/s}\uparrow$ (note the direction of the velocity)
- the velocity of $B$ with respect to $A$, $v_{BA} = \text{200 m/s}\downarrow$.

Find the absolute velocity of $A$, $v_A$.

**Assumptions:**

- The strings are massless and inextensible.
- The pulleys are massless and there is no friction in the pulley and between the string and pulley.

Try my World of Physics to solve many problems like this one.

## Problem Solving - Advanced

**Cite as:**Constraint Relation.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/constraint-relation/