1.Work and kinetic Energy

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This video explains the definition of work and enery.(Youtube channel:Professor Dave explains)

Work is an action that is performed on an object or system and that transfers energy from one location to another or from one form to another. The basic mathematical formula is :

work = force x distance   or   W = FD

 kinetic energy (K) is the energy while the object or matter is in state of motion. If the object moves faster the greater the kinetic energy it posses.When the object is stationary or not moving the kinetic energy is zero. Now we all know that Work is equal to the energy we possess so we can say that W=FD=KI =(1/2)(mv２). Unit of Work and energy are the same .



Mathematical explanation why work is equal to the kinetic energy .
       
We all know that the common formula of force is :
                                     F=ma                        (equation 1 )
    where m is the mass of an object and a is the acceleration of an object . As the object moves through a displacement ,the force changes from an initial value Vo(Initial velocity) to some other value Vf(Final velocity) . But when the force is constant , We all know that the acceleration is also constant so using the formula of constant acceleration which is :

  equation 2

                       
       using equation 1 we get a=f/m ,subtituting a to equation 2 we get:

        Multiplying both sides by distance and dividing both sides by 2 we get:

equation 3   

            The first term is the Final kinetic energy or the kinetic energy at the end of displacement while the second term is the kinetic energy at the start these tells us that the kinetic energy  has been changed by the force . while on the right side of equation 2 tells us that the change in kinetic energy is equal to the Force multiply to distance(FD).
            take note: to calculate the force does on an object through some displacement we must use the force component along the objects displacement. the force perpendicular to the displacement does zero work.

Kinetic energy and work Theorem

    As we observe on our equation 3 we can tell that left side equation is the change in kinetic energy and right side equation is the work .For the particle-like object we can generazlize: Let △K be the change in kinetic energy of the object and let W be the net work done on it . Then

(equation 4)

which says that:

    

     (Change in the kinetic energy of a particle)=(net work done on a particle)

we can also write:

which says that kinetic energy after the net work is done is equal to kinetic energy before the net work plus the net work done. Since there is no speed or velocity in initial kinetic energy V=0 we can say that:

 

(Equation 5)

Example 1:

A factory worker pushes a Crate from building A to building B with a distance of 400m and a constant velocity of 1 meter per second he pushes the crate with a force of 50 N .A.) Compute for the work done in moving the crate B.) Compute the mass of the crate .

Solution :

A.) 

We all know that the formula of work is the force applied to the object multiply to the distance from starting point to some point .We say that:

                                                      

B.)

Computing for the mass We need equation 5

                                                              

 Where K= (1/2)mV^2 equating to mass we get:

                                                        

1.1 Work done by Gravitational Force

Gravity is defined as the force that attracts a body towards the earth or towards any other physical body having mass.

If a particular object is falling, the particle is bound to point in the direction of gravity. The magnitude of the falling body depends on the mass, gravitational constant and height from which it is falling.

Figure 1

The formula for work done by a gravity is:

(equation 6)

When the Particle like object is rising , the Gravity force is directed opposite to the displacement as indicated in figure 1.

when the particle like object reach the maximum height and falls down the angle between the displacement and the gravitational force becomes 0 because they are acting on same direction which is downward. this will give us :

 

Our equation becomes positive because gravitational force transfer energy to the kinetic energy of falling object Thus the speed of the object is now increasing .

Example :

A ball with a mass of 600g is being shot from a free throw line and luckily  shoots in the ring . Compute for the work done by the gravitational force on the ball starting from when it hits the ring until it drops on the floor .The height of the ring is 10ft from the ground. 

Work done in lowering and lifting an Object

Now imagine you are lifting a table by applying a vertical force (F) to it. 

During the upward displacement ,the vertical force applied to the table does positive work while the gravitational force does negative work .The applied force transfer energy to it while the gravitational force transfer energy from it thats why it is easier to lowered the table than to lift it because the gravitational force is always pointing towards the center of the earth.By using equation 4 ,the change in kinetic energy of the table due to the energy transfer from gravitational force and due to the energy applied to the force is :

(equation 7)

(equation of change in kinetic energy due to applied force and gravitaional force)

This equation can also apply on lowering an object but the gravitational force is now transferring the energy to the table while the applied force is now transferring energy from it.

If an object is stationary before and after a like picking up a bottle of water from the floor and put it inside the refrigerator this will give us zero in final kinetic energy and initial kinetic energy.We can rewrite equation 6 as:

(equation 8)

take note that we get the same equation if Kf and Ki is not zero but same in value .Eitherway equation 8 tells us that the work done by the applied force is equal to the negative work done by the gravitational force  .Using equation 6 we can rewrite equation 8 as :

(Work done in lifting and lowering the object;Kf=Ki)

Example:

A wall clocked having a mass of 400g is dropped on the floor and picked up to put it back on the wall with a height of 8ft compute for the applied force needed to put back the wall clock.

Solution:

Our formula for work is :

                              Work = Mass * Gravity * Height

Given Constant gravity force=9.8 m/s^{2 ,}mass=400g,height of 8Ft .

W=(400g)(9.8)(8Ft.)

W=31360J

1.2 Work done by a Spring Force

Figure 2

Figure 2 shows a blocked attached to the spring with the pring parrallel to the X axis and the relaxed state of the blocked attached to the spring is at the origin of x axis .

Figure 2(a) shows that the spring is on a relaxed state this means there is no force acting and there is no displacement yet .On figure 2(b) it shows that the blocked attached to the spring is being pulled so our displacement is pointing toward the positive side of X - axis .Figure 2(c) shows that the block attached to the spring is being pushed so the displacement of the block is going to the negative side of X-axis.

The formula for Force exerted by the spring is :

Where k is called the Spring constant or Constant force this is the measure of the stiffness of your Spring ."x" is the displacement of the block in the X-axis.

If and object is attached to the spring's  free end , the work Ws done on the onject by the spring force when the object moved from an initial position to final position is

If Xi=0 and Xf=X, then the equation becomes

Example 1: If the force stretch a spring given by (80N/m)(x) ,compute for the work needed to stretch the spring 7m from rest?

1.3 Work done by a Variable Force

If the force varies (e.g. compressing a spring) we need to use calculus to find the work done.

If the force is given by F(x) (a function of x) then the work done by the force along the x-axis from a to b is:

$\displaystyle{W}={\int_{{a}}^{{b}}}{F}{\left({x}\right)}{\left.{d}{x}\right.}$

The meaning of Variable Force is when an object needs or requires increasing force for it to moves from one point to another.Like the car shown below The car moves faster and faster because of the driver increasing engine force to increase the speed of the wheel.
If you observed the formula of Work define by variable force the integration from a to b with respect to function x is equal to the force varying in the displacement X or in a floor surface .

Watch this video to learn more about Work and kinetic energy

Watch this video to learn more about Work done by Gravitational Force

Watch this video to learn more about Work done by a spring

Watch this video to learn more about Work done by a Variable Force

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References:

1.)https://www.youtube.com/watch?v=B29lF9Jjm5Y

2.)https://www.youtube.com/watch?v=BowKiFJk3MM&t=143s

3.)Fundamental of physics 10th editiion by jearl walker

4.)https://www.introduction-to-physics.com/kinetic-energy-problems.html

                                                              

                                                       

                                               

                                     

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