## Monday, October 31, 2016

### Vectors Problems and Solutions One

A Physical quantity that has both magnitude and direction and satisfy  the laws of vector algebra is called a vector. Some physical quantities like displacement and velocity need both magnitude and direction to understand them properly and they were treated like vectors. Here we are solving problems based on this vector concepts. The first problem tells that the vector resultant of two vectors is given for us and we need to find the angular separation between the given vectors. The problem is as shown in the diagram below.

Solution

The resultant of two vectors can be found using the parallelogram law of vectors. According to it, if two vectors are represented as the two sides of a parallelogram from one common point, then the resultant of that two vectors is the diagonal of the parallelogram from the same point.

Applying that concept and making further simplification, we can solve the problem as shown below.

Problem

It is given in the problem that the two vectors are having the same magnitude and they are separated by known angle and they are in the same direction. We need to find the resultant of them. The problem is as shown in the diagram below.

Solution

Again we are using the Parallelogram law of the vectors. By applying the trigonometry rules, we can solve the problem as shown in the diagram below.

Problem

The next problem is also the same and we need to find the difference but not the sum in the given problem. It is as shown in the diagram below.

Solution

Even the solution is same. The only difference is,we need to find the difference but not the sum of the two vectors. Thus while we are applying the parallelogram law, we need to  apply the angle as the difference of one eighty degree and the given angular separation of the two vectors. Thus we are able to find the answer as shown in the diagram below.

Problem

It is given in the problem that eleven forces are acting on a point and each force makes an angle 30 degree with its neighbor. We need to find the resultant force acting on the particle. The problem is as shown below.

Solution

Force is a physical quantity that changes are try to change the state of a body. It is a vector that has both the magnitude and direction. In the given problem, out of eleven forces, ten forms five pairs such that each force has another force of equal magnitude but opposite direction. Thus they cancel each other. Hence only one force is remaining and that itself is the resultant. The solution is as shown below.

Problem

It is given in the problem that the sum of the two vectors is 16 newton. If the resultant force is eight newton and it is perpendicular to the smaller force, we need to measure two forces involved here.

Solution

By using the definition of the parallelogram law both in terms of magnitude and direction and by simplifying it further, we can solve the problem as  shown in the diagram below.

Problem

It is given in the problem that a weight has been suspended from the mid point of a rope connected between the two points of the same level. The rope lost its horizontalness and we need to know the minimum tension required to make to horizontal. The problem is as shown in the diagram below.

Solution

The weight appliyed makes the string non horizontal and it generates a tension in it. We can resolve it into components as shown. The sum of vertical components and it has to be equal to the weight. By solving the equation, we can get the answer as infinite.

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