Work Energy Theorem Of a Variable Force

Introduction 

The work done as characterized by physical sciences, is calculated by the net force (total summation of additions and subtractions according to both the direction and magnitude of the forces applied) acting on a body is defined as the equivalent of the total changes produced in kinetic energy of the body. 

To understand this fully, we must understand what Kinetic energy is. Kinetic Energy, in essence, is the total energy possessed by a body or object when it is in motion. This brings us to the Work energy theorem. Essentially, the work energy theorem equation boils down to the total work done. This can also be calculated as the change in Kinetic energy.

Alternatively, the work energy theorem also states that the work done can also be denoted as the transfer of energy, to and from an object. when a force is applied to the aforementioned body and the body has covered some distance that is, been displaced from its original location. 

Definition of Work and Energy

The basic units of energy and work are kg m²/s² . The Joule is assigned to this mixture as its own SI unit. However, work is commonly expressed in newton metres ( N m ). They are scalar quantities, which have only a magnitude; vector quantities, like F, a, v, and d, have both a magnitude and a direction. As per the work energy principle, energy can be kinetic (KE) or potential (PE), and it can take many different forms in each case. KE can involve observable motion and can be translational or rotational, but it can also encompass vibrational motion at the molecular level and below. The most common form of potential energy is gravitational, but it can also be found in springs, electrical fields, and other places in nature. The net work done is expressed by the equation Wnet = Fnet x Cosθ 

Where Fnet is the system’s net force, d is the object’s displacement, and θ is the angle between the displacement and force vectors as stated in the work energy theorem. Work is a scalar, despite the fact that both force and displacement are vector quantities. When the force and the displacement are in opposing directions (like when an item decelerates while continuing on the same path), cos is negative, and Wnet is negative. 

Work Energy Theorem Equation and how it was derived

Let the change in kinetic energy of a particle be equal to the work, denoted by ‘W’. 

This will be the work done by the net force (summation of all the forces applied accordingly to their directions) let this be denoted by KE. 

Following this, we look to Newton’s second law of motion which clearly states : 

The Force applied may be calculated using the product of the total mass of the body and its acceleration.

F = ma —–(1)

For the next step, we work in accordance with the work energy theorem where we must take into consideration the formula of work. Work can be defined as the by-product of force and displacement, encapsulated as the following formula :

W = Fs —–(2)

Upon Analysis, we find that the equation for Newton’s second law of motion can be incorporated into the equation for work through Newton’s definition of Force. This leads us to the third equation, where the work done can now be calculated using the mass, acceleration and the displacement of a body. 

W = mas —–(3)

Furthermore, Newton’s third equation of motion states that, If an object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A. Commonly referred to, as the action-reaction law, when expressed mathematically gives us the following expression: 

v2 = u2 + 2as 

2as = v2 – u2 

as = v2 – u2 / 2 —–(4)

Let this be (4) following the rearrangement of Newton’s third law of motion, we can combine the (3) equation of work done, to arrive at the mathematical expression of the Work energy theorem, where the change in Kinetic energy can be defined as the work done:

W = m (v2 – u2 /2) 

W = mv2/2 – mu2/2

W = KEf – KEi  ——-(5)                          

 ( As KE = ½ mv2 )

Or

W = Δ KE (Change in KE) 

Examples of work energy principle in real life 

Example 1

A 5,000 kg car comes to a complete halt from a speed of 50 m/s (112 miles per hour) across a distance of 100 metres. What is the force that the car is subjected to? (Hint: use Work Energy Equation)

ΔKE=0 – [(1/2)(5,000 kg)(50 m/s)2]

       = – 125,000 J

W = – 125,000 Nm=(F)(100 m)

F= – 1,250 N

-ve sign denotes that the work done is against the moving car.

Example 2

If the same car is stopped at a speed of 100 m/s (224 miles/h) and the same braking force is applied, how long will the car move before it stops?

ΔKE = 0 – [(1/2)(5,000 kg)(100 m/s)2]

         = -250,000 J

W = -250,000 Nm =(- 1250 N)(d)

d =200 m

Thus doubling speed causes the stopping distance to double, as well when all other conditions are kept the same.

Example 3 

Suppose you have the same momentum and there are two objects m1> m2 between v1 & v2. Do you need more work to stop bulky, slow, or light and fast objects?

Since we know that m1v1 = m2v2, we can express v2 in  other quantities: v2 = (m1 / m2) v1. Therefore, the KE for heavy objects is (1/2) m1v12 and the KE for light objects is (1/2) m2 [(m1 / m2) v1] 2. If you divide the equation of a light object by the equation of a heavy object, you can see that the light object (m2 / m1) has more features than the heavy object. This means less work is required for the bowling ball to stop when faced with the same swing bowling ball and marble.

Conclusion

From this we can conclude that how work and energy are linked together via the work energy theorem and we can also understand the application of the work energy principle. While also implementing the work energy theorem equation to solve real life problems also gave us a more clear idea as to how masses and speeds are interconnected to each other.