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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