## Monday, November 21, 2016

### Work Power and Energy Problems with Solutions Six

Work is the way using some of our energy to perform a given task using some force. If this force or its component is successful in producing a displacement then work is said to be done. Work done is treated like a scalar and it gives complete meaning with out in need of any direction. Thus it is the dot product of force and displacement. If force is variable, we shall measure the work done with each force for a given time interval and to get the total work done, we shall add all that small works done. This can be done mathematically using a method called integration and it shall be used in this case.

Problem

It is given in the problem that a force on a particle of mass known and the displacement is given in terms of time. We need to know the work done in a specified time four seconds. The problem is as shown in the diagram below.

Solution

Displacement is given to us and that is variable with time. To  get velocity from displacement ,we shall differentiate displacement with time. We need to measure the work done between the time intervals zero and four. Thus we shall substitute that times and get initial and final velocity. So we can measure initial and final kinetic energy. Work done can be measured as the change in kinetic energy.

Problem

A position dependent force is given as shown in the diagram below. We need to measure the work done in between two locations.

Solution

Here in this problem force is variable with displacement and to get the total work  done, we shall first measure the work done for a small displacement and to get the total work done, we shall add all that small works done. That is mathematically called integration. By applying the rules of integration, we shall solve the problem as shown below.

Problem

It is given in the problem that a force is acting on a body and it is inversely proportional to the distance covered by the body. We need to find out how work done is dependent on displacement.

Solution

As the force is inversely proportional to displacement, the proportionality can be eliminated with a constant. That can be substituted and simplified like the previous problem as shown in the diagram below.

Problem

A ball of mass m at rest receives an impulse in the direction of south and after some time some other impulse in in the direction of south. We need to measure the final kinetic energy of the body and the problem is as shown in the diagram below.

Solution

Impulse is the large force acting on a body for a short interval of time and it is mathematically equal to momentum. It is a vector and its resultant can be measured using vector laws of addition. Thus we can measure the final kinetic energy and initial kinetic energy is zero. By using the concept of work energy theorem that work done is equal to change in kinetic energy, we can solve the problem as shown in the diagram below.

Problem

A body freely falls from a certain height on the ground in a time. During the first one third of the interval it gains a kinetic energy and in the last one third of the interval it gains a different kinetic energy. We need to measure the ratio of the kinetic energies and the problem is as shown in the diagram below.

Solution

The body is initially at test and its kinetic energy is zero. After one third of the time of journey, the body acquires some velocity and it can be measured using the equation of motion. We can also measure the respective kinetic energy as shown below.

We can also measure the final velocity after the other interval and ratio of kinetic energies as shown in the diagram below.

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Work Done by a Variable Force with Integration Process