## Tuesday, November 22, 2016

### Work Power and Energy Problems with Solutions Seven

We are solving series of problems on the concept work, energy and power. When ever we apply a force on a body and if there is a displacement along the direction of force, work is said to be done. Even in the case of a spring compressed, we need to do some work to do that and that work done is stored in the form of spring energy or spring potential energy, Power is the ability of doing work in quick time. Efficiency is the productiveness of the energy used in doing the required work. If the system is ideal it will have hundred percent efficiency and it means all the energy used is doing productive work.

Problem

A man is running with a kinetic energy and his it is half of the kinetic energy of a boy who is having half of the mass of the man. If the man increases his speed by one meter per second both of them are having same kinetic energy. We need to measure the original speed of the man and the problem is as shown in the diagram below.

Solution

We know the formula for kinetic energy of the body and it is due to motion of the body and it is given in the problem that when the man increases his speed by one meter per second, both man and boy has same kinetic energy and that condition is applied as shown in the diagram below. By simplifying the equation further, we can find the velocity of the man as shown in the diagram below.

Problem

It is given in the problem that a motor is lifting water from a well of depth 20 meter and water has a further depth of 10 meter in the well. If the radius of the well is seven meter, we need to measure the work done in emptying the well. The problem is as shown below.

Solution

The scenario is as shown in the diagram below. Water starts at a depth of 20 meter from the surface. Water is further up to a depth of 10 meter and water is uniformly distributed over that depth. We need to consider a point where the mass appears t o be concentrated and that point is called center of mass. It is geometrically at the middle of the depth.

The system has potential energy and we need to measure the potential energy of both the cases as shown in the diagram below. We can further write mass as the product of volume and density. Volume can be further written as the product of area of cross section and depth of the water. The data is substituted and the problem is solved as shown below.

Problem

A motor has some electrical supply and it supplies energy of 30 kilo joule. It is used to lift a mass of 100 kilogram load to a height of 25 meter and we need to measure the efficiency of the system. Problem is as shown in the diagram below.

Solution

We can measure the work done in the form of potential energy and we can compare that with the input energy supplied and the ratio is called efficiency. We can solve the problem as shown in the diagram below.

Problem

A sphere of mass 16 kg is moving with a velocity 4 meter per second and it strikes a spring of spring constant known to us. We need to measure the compression if the spring and the problem is as shown in the diagram below.

Solution

The body is moving with some velocity and hence it has kinetic energy. This energy is transferred to the spring and it is stored in the form of potential energy of the spring. By equating this energies since the energy is always conserved, we can solve the problem as shown in the diagram below.

Problem

A particle of mass m is projected with a known velocity and known angle of projection from the horizontal. During the journey some where its velocity makes an angle as given in the problem shown and we need to measure the work done by the gravitational force.

Solution

Work done by the gravitational force is stored in the form of kinetic energy and we need to measure it in the form of initial kinetic energy. We know that at the maximum height of the projectile, it has only horizontal component velocity and vertical component velocity is zero. We can solve the problem as shown in the diagram below.

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