Calculate the work done by a particle under the influence of a force #y^2 hat(i) - x^2hat(j)# along the curve #y=4x^2# from #(0,0)# to #(1,4)#?

Question

Charles Arocha · Accepted Answer

#1.2# #C=(x,4x^2)# and #F=((4x^2)^2,-x^2)# but #dC=(1,8x)# so #dW=<< F, dC >>dx = 16x^4-8x^3# then #W=int_0^1 << F, dC >>dx = 16/5-8/4=1.2#

Emily Ammerman · Answer

# int_C vec(F) * d vec(r) = 1.2#

The work done in moving a particle from the endpoints #A# to #B# along a curve #C# is. # int_C \ vec(F) * d vec(r) \ \ # where # \ \ {: (vec(F),=F_1 hat(i) + F_2 hat(j)),(d vec(r),=dx hat(i) +dy hat(j)) :} # The integral is known as a line integral. So we have: # vec(F) = y^2hat(i) -x^2hat(j) # and #C# is the arc of #y=4x^2# from #(0,0)# to #(1,4)# To evaluate the line integral we convert it to a standard integral by choosing an appropriate integration variable, In this case integrating wrt #x# would seem to make sense. On #C#, the variable #x# varies from #x=0# to #x=1#. Differentiating the equation for #C# wrt #x# gives: #dy/dx = 8x# And so we can express our vector fields in terms of #x# alone: # vec(F) = y^2hat(i) -x^2hat(j) # # \ \ \ \ = (4x^2)^2hat(i) -x^2hat(j) \ \ \ \ # (from the equation of #C#) # \ \ \ \ = 16x^4 hat(i) -x^2 hat(j) # And: # d vec(r)=dx hat(i) + dy hat(j) # # \ \ \ \ \ =dx hat(i) + 8x \ dx hat(j) \ \ \ \ # (from derivative of eqn for #C#) Hence, # int_C \ vec(F) * d vec(r) = int_C \ ( 16x^4 hat(i) -x^2hat(j) ) * (dx hat(i) + 8x \ dx hat(j))# # " "= int_0^1 \ 16x^4 \ dx -x^2(8x)dx # # " "= int_0^1 \ 16x^4 -8x^3 \ dx # # " "= [ 16/5 x^5 -2x^4 ]_0^1 # # " "= (16/5 -2) - 0# # " "= 6/5# # " "= 1.2#

Paisley Johnson · Answer

undefined To calculate the work done by the particle along the curve $ y = 4x^2 $ from (0,0) to (1,4), we first need to parameterize the curve. Let's denote $ x = t $ such that $ y = 4t^2 $, where $ t $ ranges from 0 to 1. Then, differentiate $ y $ with respect to $ t $ to find $ dx $ and $ dy $ in terms of $ dt $.

Next, express the given force as $ F = y^2 \mathbf{i} - x^2 \mathbf{j} $, and substitute the parameterized values of $ x $ and $ y $.

After that, integrate $ F \cdot dr $ along the curve from the initial point (0,0) to the final point (1,4), where $ dr $ is the differential displacement vector $ dx \mathbf{i} + dy \mathbf{j} $.

Finally, evaluate the integral to find the work done by the force along the given curve.