## Saturday, November 5, 2016

### Vectors Problems and Solutions Four

In continuation to the problems with solutions in a topic called vectors, we are going to solve some more problems based on vector addition, parallelogram law, components of the vectors and Lami''s theorem. Lami' theorem is used to find comparison of three vectors and can be tried like a alternate option for triangle law of vectors.

Any vector can be resolved into components along X,Y and Z axis. To identify them with the particular direction, we can multiply the components with unit vectors along the respective axes like i,j and k. Adding all the components back in vector format gives back the original quantity and hence sanctity of the physical quantity is not disturbed.

It is given in the problem that a man of mass 80 kilogram is supported by two cables as shown and we need to know the ratio of the tensions in that cables.The problem is as shown in the diagram below.

Solution

The angles at the two corners of the triangle is given to us as 60 and 30 degree. As we know that the total angle in a triangle is 180 degree, the angle at the third corner is 90 degree.

According to Lami's theorem, the ratio of the force and its SIN component is constant. By applying that rule, we solve the problem as shown in the diagram below.

Problem

This problem is also a similar one to the earlier. It is given in the problem that two pegs are separated by 13 cm. A body of weight W is suspended using a thread of 17 cm. We need to know the tensions in the strings separated. The problem is as shown in the diagram below.

Solution

It is given in the problem that the total length of the string 17 cm is divided into two parts of 5 and 12 centimeter.The sides 5,12 and 13 are the combination of a right angled triangle. So we can find the angle at the remaining two corners as shown in the diagram below.

Once we know the angles, we can apply Lami's theorem and solve the problem as shown.

Problem

According to the given problem the vector sum of the given to vectors is the third vector and magnitude of the sum of the squares of the same two vectors is the square of the magnitude of the same third  vector and we need to find the angle between the two vectors. The problem is as shown in the diagram below.

Solution

We can write the resultant of  the two vectors using the parallelogram law of the vectors. By further substituting the condition given in the problem, we can solve the problem as shown below.

Problem

It is given in the problem that the sum of two unit vectors is also a unit vector in terms of magnitude. We need to find the difference of the two unit vectors and the problem is as shown in the diagram below.

Solution

Using the concept the parallelogram law of vectors, we can find the angle between them as shown below. We know that the vectors are unit in magnitude and we can apply this condition.

Problem

It is given that l,m and n are the directional cosines of a vector and we need to find the relation among them. The problem is as shown below.

Solution

We can define direction cosine as the ratio of the component of the vector and the magnitude of the vector. The same can be defined with different components of the vector. Their squares and sum of them as shown below.

Problem

A metal sphere is hung by a string fixed to the wall as shown in the diagram. It is pushed away from the wall and different forces acting on it are the weight, tension in the string and the applied force. We need to check which of the given relation is wrong.

Solution

At the instantanious position,  the sphere is in the equallibriem. Thus the three forces acting on it can be expressed as the three sides of the triangle and their vector sum is zero. We can solve the problem as shown below.

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