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Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,September 23,2010,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Physics 430:Lecture 8 Force and Potential Energy,Dale E.Gary,NJIT,Physics Department,September 23,2021,We have seen that the potential energy,U,(,r,),corresponding to a force,F,(,r,),can be expressed as an integral of,F,(,r,),.It should come as no surprise,then,that we can write,F,(,r,),as some kind of derivative of,U,(,r,),although we have to preserve the effect of the dot product in the integral,which turns the integral of,F,into a scalar.In other words,we need a derivative that turns the scalar,U,into a vector.A concept from,vector calculus,fills the billthe gradient.,Consider a particle acted on by a conservative force,F,(,r,),with corresponding potential energy,U,(,r,),.The work done by,F,(,r,),in a small displacement from,r,to,r,+,d,r,is:,On the other hand,from the definition of potential energy it is also,But from the definition of a derivative,4.3 Forces as Gradient of PE,September 23,2021,This suggests that we can write,where the derivatives of U are now partial derivatives with respect to x,y,z.For example,is the rate of change of U as x changes,keeping y and z fixed.,Equating the two alternative ways of writing,we have where the operator,is pronounced“grad.It takes a scalar“field such as U,and results in a vector pointing“uphill.Note that the force,being ,points“downhill.,Gradient of Potential Energy,September 23,2021,Scalar Fields,A scalar field is just one where a quantity in“space is represented by numbers,such as this temperature map.,Here is another scalar field,height of a mountain.,Contours,Side View,Contours close,together,steeper,Contours far apart,flatter,September 23,2021,Vector Fields,A vector field is one where a quantity in“space is represented by both magnitude and direction,i.e by vectors.,The vector field bears a close relationship to the contours(lines of constant potential energy).,The steeper the gradient,the larger the vectors.The gradient vectors point along the direction of steepest ascent.,The force vectors(negative of the gradient)point along the direction of steepest descent,which is also perpendicular to the lines of constant potential energy.,Imagine rain on the mountain.The vectors are also“streamlines.Water running down the mountain will follow these streamlines.,Side View,September 23,2021,Surface vs.Volume Vector Fields,In the example of the mountain,note that these force vectors are only correct when the object is ON the surface.,The actual force field anywhere other than the surface is everywhere downward(toward the center of the Earth.,The surface creates a“normal force everywhere normal(perpendicular)to the surface.,The vector sum of these two forces is what we are showing on the contour plot.,Side View,September 23,2021,Statement of the problem:,The potential energy of a certain particle is,U=Axy,2,+B,sin,Cz,where,A,B,and,C,are constants.What is the corresponding force?,Solution:,Formally,the force is,What we need,then,are the three partial derivatives,which we can write down by inspection:,Plugging back into the equation for force,we have:,Example 4.4:Finding,F,from,U,September 23,2021,Last time we saw that the two conditions for a force to be conservative are,It turns out that there is an easy way to check whether a force has the second property,using a concept from vector calculus.It can be shown via a theorem called Stokes Theorem(which you will have seen if you have had the vector calculus course)that a force has the desired property,that the work it does is independent of the path,if and only if everywhere.The quantity is called the curl of F,or just“curl F,or“del cross F.,It follows the usual rules for the cross product.,4.4 The Second Condition that F be Conservative,Conditions for a Force to be Conservative,A force,F,acting on a particle is conservative if and only if it satisfies two conditions:,1.F depends only on the particles position,r,(and not on the velocity,v,or the time,t,or any other variable);that is,F,=,F,(,r,).,2.For any two points 1 and 2,the work,W,(1,2),done by,F,is the same for all paths between 1 and 2.,September 23,2021,Two find the curl of a vector,you form the matrix and find its determinant:,It may not be obvious that this being zero is equivalent to the condition that,is path independent,but Stokes Theorem shows that it is.This gives a handy way to determine the path-independence property,as the following example shows.,Curl of,F,September 23,2021,Statement of the problem:,Consider the force,F,on a charge,q,due to a fixed charge,Q,at the origin.Show that it is conservative and find the corresponding potential energy,U,.Check that,Solution:,The Coulomb force is where we have substituted,g,for the constant,kqQ,.,Lets find the curl of,F,and see if it is zero.The,x,component is,But so,Th
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