Work is the phenomenon of using the available energy to produce the displacement along the direction of the applied force. If force is constant , we can measure the work done as the dot product of force and displacement. If the force is variable, we shall first measure the work done for each small force and to get the total work done, we shall add all that components of work done. This can be mathematically done using integration concept.
Problems
It is given in the problem that time is expressed in terms of the displacement. We need to measure the work done in first six seconds and the problem is as shown in the diagram below.
Solution
By simplifying the given equation of time, we can get for expression of displacement as shown below. His given intervals of time has to be applied in the equation and find the displacement in each case as the same value and it actually means there is no actual displacement between these two intervals. It means in that interval the body is coming back to the same position. Hence work done is zero.
Problem
A particle is moving along the x axis between the two given points and a force is applied that is given in terms of displacement. We need to measure the work done in this process. Problem is as shown in the diagram below.
Solution
As force is variable with displacement, we need to integrate to get the total work done as shown in the diagram below. By applying the rules of integration, it can be further simplified between the given limits of time and further simplified as shown.
Problem
Velocity and time graph is given to us when a particle of known mass is moving along a straight line. We need to measure the work done in this case and the problem is as shown in the diagram below.
Solution
When velocity is take on Y axis and time and X axis, the slope of the graph gives us acceleration. Area of velocity and time graph gives us displacement. My measuring them and substituting them in the work done definition, we can solve the problem as shown in the diagram below.
Problem
A truck can move up on a road with inclination with a known speed. Air resistance is taken into count and we need to measure the speed of the truck when it is coming down the hill with the same horse power. The problem is as shown in the diagram below.
Solution
It is given in the problem that the power while the truck moving in upward and downward direction is same. While he is moving up, he need to over come the air resistance also. By writing the power as the dot product of force and velocity, problem is solved as shown in the diagram below.
Problem
A human heart discharge 75 cc of blood through the arteries at each beat against an average pressure of 10 cm of mercury. If pulse frequency is known to us, we need to measure power of the heart. Problem is as shown in the diagram below.
Solution
Here in this case work done can be expressed as the product of applied pressure and change in the volume of the blood. It shall be multiplied with the number of beats. The solution is as shown below.
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