Work Done by Varying Force [PDF]

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Physics Work done by varying Force A: Learning Goals  



Work done by a varying force Work done by a spring



B : Prerequisites       



Work done by a constant force Vectors and their representation in the Cartesian coordinate system Scalar product of two vectors Free-body diagrams Basic knowledge of kinematics and laws of motion Differentiating simple polynomial and trigonometric functions Integration of simple polynomial and trigonometric functions



C: Work done by a Varying Force



If we express the resultant force in the x direction as



, then work done as the



particle moves from



to



will be:



Let us consider a particle being displaced along the x axis under the action of a varying force. The particle is displaced in the direction of increasing x from x = xi to x = xf. In such a situation, we cannot use W = (F cos  ).d to calculate the work done by the force because this relationship applies only when force F is constant in magnitude and direction. However, if we imagine that the particle undergoes a very small displacement , as shown in figure, then the x-component of force is approximately constant over this interval; for this small displacement, we can express the work done by the force as . This is just the area of the shaded rectangle in Figure shown here.



 Page 1



The work done by the force component 𝐹𝑥 for the small displacement 𝑥 is 𝑭𝒙 𝒙, which equals the area of the shaded rectangle. The total work done for the displacement from x = xi to x = xf is approximately equal to the sum of the areas of all the rectangles. 𝒙𝒇



𝑾



𝑭𝒙 𝒙 𝒙𝒊



If we imagine that versus curve is divided into a large number of such intervals, then the total work done for the displacement from x = xi to x = xf is approximately equal to the sum of a large number of such terms. That is: ∑



.



The work done by the component 𝐹𝑥 of the varying force as the particle moves from x = xi to x = xf is exactly equal to the area under this curve. We can express the work done by 𝑭𝒙 as the particle moves from 𝒙𝒊 to 𝒙𝒇 as: 𝑾



𝒙𝒇



𝒙𝒊



𝑭𝒙 𝒅𝒙



If more than one force acts on a particle, the total work done is just the work done by the resultant force. If we express the resultant force in the x- direction as ∑ 𝑭𝒙 ; then the total work, or net work done as the particle moves from 𝒙𝒊 to 𝒙𝒇 is: 𝑾



𝑾𝒏𝒆𝒕



𝒙𝒇 𝒙𝒊



𝑭𝒙 𝒅𝒙



 Page 2



If the displacements are allowed to approach zero, then the number of terms in the sum increases without limit but the value of the sum approaches a definite value equal to the area bounded by the curve and the x axis:



This definite integral is numerically equal to the area under the versus curve between and . Therefore, we can express the work done by as the particle moves from to as:



If more than one force acts on a particle, the total work done is just the work done by the resultant force. If we express the resultant force in the x- direction as ∑ ; then the total work, or net work done as the particle moves from to is:



Heading: Work done by spring



 Page 3



A common physical system for which the force varies with position is shown here. A block on a horizontal, frictionless surface is connected to a spring. If the spring is either elongated (stretched) or compressed a small distance from its unstretched (equilibrium) configuration, it exerts on the block a force of magnitude;



) position and Where x is the displacement of the block from its unstretched ( positive constant called the force constant or stiffness of the spring.



is a



The negative sign signifies that the force exerted by the spring is always directed opposite the displacement. In other words, the force required to stretch or compress a spring is proportional to the amount of elongation or compression; . This force law for springs is known as Hooke’s law. Stiff springs have large



values, and soft springs have small



values.



x max



 Page 4



- x max



If the spring is compressed until the block is at the point block moves from through zero to .



and is then released, the



And, if the spring is instead stretched until the block is at the point and is then released, the block moves from through zero to . It then reverses direction, returns to , and continues oscillating back and forth.



Heading: Spring is unstretched



When 𝑥 (natural length of the spring), the spring force is directed to the left and known as restoring force. In this position, force exerted by the spring 𝐹𝑠 .



When spring is in normal position or unstretched i.e. as shown here. In this position, force exerted by the spring ; because the spring force always acts toward the equilibrium position , this force is also known as the restoring force.  Page 5



Heading: Spring is compressed



When x is negative (compressed spring), the spring force is directed to the right, in the positive xdirection. Work done by the spring force as the block moves from 𝑥𝑖



𝑥𝑚𝑎𝑥 to 𝑥𝑓



is:



𝑥𝑓



𝑊𝑠



( 𝑘𝑥)𝑑𝑥



𝐹𝑠 𝑑𝑥 𝑥𝑖



𝑘𝑥𝑚𝑎𝑥



𝑥𝑚𝑎𝑥



Let us now suppose the block has been pushed to the left a distance xmax from equilibrium and is then released. Let us calculate the work, Ws done by the spring force as the block moves from to . Then, work done by the spring is: (



)



Heading: Spring is stretched



 Page 6



Now the block has been pushed to the right a distance xmax from equilibrium and is then released. Let us calculate the work, Ws done by the spring force as the block moves from to . Then, work done by the spring is: (



)



Therefore, the net work done by the spring force as the block moves from is zero.



Heading: Graph of



to



versus



 Page 7



If we plot a graph of versus for the block-spring system then the graph will be a straight line as shown here. The work done by the spring force as the block moves from to 0 (zero) is the area of the shaded triangle,



.



 Page 8