Showing posts with label Power. Show all posts
Showing posts with label Power. Show all posts

Work Power and Energy Problems with Solutions Ten

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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Work Power and Energy Problems with Solutions Six

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Work Power and Energy Problems with Solutions Nine

We are solving series of problems on the concept of work, energy and power. Here in this post we are going to solve few problems on power and it is the ability of doing the work in specified time. Though work is done or energy is used in each case how quick we are in doing the work is measured with the term power. It is measured with a unit watt and it is a scalar quantity. Power can also be expressed as the dot product of force and velocity.

Problem

It is given in the problem that input power of an electric motor is 200 kilowatt and its efficiency is eighty percent. It operates a crane of efficiency ninety percent. If the crane is used to lift a load of known mass, we need to measure the speed of that mass. Problem is as shown in the diagram below.


Solution

As there two efficiecies are involved in this case their effective value has to be measured as shown in the diagram below. Further using the concept of efficiency is the ratio of power output to the power input, we can measure the required output power in the problem as shown in the diagram below. Further this power can be defined as the dot product of force and velocity. Here force is nothing but the weight of the system.


Problem

It is given in the problem that a motor pump is used to deliver water at a certain rate in a water pipe. We need to increase the water to a certain times and we need to measure how much of force and power of the motor has to be increased to get this done. Problem is as shown in the diagram below.


Solution

We can express mass as the product of volume and density and volume can be further expressed as the product of area of cross section to the length of the pipe. Further rate of water flow is written by dividing that equation with time. It is further simplified and solved as shown below.


Problem

It is given in the problem that there is a friction less inclined surface and their location from the earth is given to us as shown below. The block stated from the state of rest and we need to know the velocity acquired by the body as it has reached the bottom part.


Solution

We know that both the points of the problem are at a certain height from the ground and there they possess potential energy due to their position. As it has reached the bottom there is a loss of potential energy and that energy is converted into kinetic energy. Thus by applying law of conservation of energy, we can solve the problem as shown in  the diagram below.


Problem

A particle of mass m is moving in a horizontal circle of known radius under the influence of centripetal force and its value is given to us. Then we need to measure the total energy of the system.


Solution

We know the concept of centripetal force as the effective force acting towards the center of a circular motion. Basing on that we can measure the kinetic energy of the body as shown in the diagram below.



We can also find its potential energy energy using integration concept as shown below. We can also find the total energy as the sum of potential and kinetic energies as shown in the diagram below.




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Work Power and Energy Problems with Solutions Six

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Work Power and Energy Problems with Solutions Eight

We are solving series of problems on work, power and energy. Here in this post we are solving problems based on kinetic energy and power in particular along a inclined plane. Kinetic energy is the energy  possessed by the body due to its motion and velocity and power is the rate of change of work done. When a body is on a inclined plane, entire acceleration due to gravity wont act normal to the plane and it can be resolved into components. It is because it is a vector quantity.

Problem

It is given in the problem that a proton is accelerated through along a straight line with a known acceleration. If its initial speed and distance covered by it is given to us, we need to measure the gain in the kinetic energy of the body. The problem is as shown in the diagram below.


Solution

Using the third equation of motion, we can measure the final velocity of the body as shown in the diagram below. We need to further solve the problem. Initial kinetic energy  and final kinetic energies can be found and the difference between them is the answer to the problem. We need to solve the data into electron volts as shown in the diagram below.


Problem

A body of mass half kilogram is on a inclined plane of known dimensions and it is allowed to slide down to the bottom again.Coefficient of friction is given to  us and we need to measure the work done by the frictional force over the round trip. Problem is as shown in the diagram below.


Solution

We can resolve weight of the body into components and we can find out the normal reaction. It is the reaction force applied by the lower surface when a component of the weight acts on in in the right angle. Frictional force is the product of coefficient of friction and normal reaction. We need to do the work in overcoming this frictional force while the body is moving up and coming down. We can solve the problem as shown in the diagram below.

Problem

An object of mass is tied to a string of known length and a variable horizontal force is applied on it so that the string makes some angle to the vertical. We need to measure the work done by this force and the problem is as shown in the diagram below.


Solution

When the pendulum is taken to a certain height from the mean position, some work is done and that work done is stored in the form of potential energy. We can express the height in terms of the length of the pendulum and the angle of inclination as shown in  the diagram below.


Problem

Water flows out horizontally from a pipe with a velocity and its area of cross section is given to us. We know the density of water and we need to measure the power needed to produce the required kinetic energy. The problem is as shown in the diagram below.


Solution

We know that as the water is  moving with a known velocity, its energy is in the form of kinetic energy. We can further express mass of the water as the product of volume and its density. Volume can be further expressed as the product of area of cross section of the pipe and the length of the pipe. Further length of the pipe divided by the time gives us velocity. It can be further simplified as shown in the diagram below.


Problem

A train of known mass has a constant speed is moving up along a inclined plane of known inclination and the power of the engine is given to us. We need to measure the resistance force acting on this system.


Solution

We know that a component of the weight and frictional force acts against the motion and they are down along the inclined plane when we are moving the train in the upward direction. We can express power as the dot product of force and velocity. Problem can be further solved as  shown in the diagram below.

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Work Power and Energy Problems with Solutions Six


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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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Work Power and Energy Problems with Solutions Six


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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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