A force acting on a particle moving in the xy plane is given by

[AryRezvani]

Homework Statement

Homework Equations

W=∫ F_x dx

The Attempt at a Solution

I think you got to split this one in terms of axis.

∫(2yi + x²j) dx

Pretty lost on it.

 

[256bits]

Well, they do not say in which direction the vectors i and j point, so let us assume that vector i is parallel to the x-axis and vector j is parallel to the y-axis, which is the usual convention.

To start you off,
Path OAC:
From O to A, can you find the force on the particle along this path knowing that y = 0. In what direction does the force point from the equation given for force.? Does the force do any work along this path? Note that the particle is moving along the x-axis which is what vector direction, i or j. Can you find an equation for the path of the particle in terms of vectors i oj or both?

from A to C, same questions.

You might want to review dot and cross product as that that is what this problem seems to be designed for.

PS. Your picture shows up a bit large.

 

[AryRezvani]

Understood, so quick question, is the blue line the force? Then the other lines simply components of the force?

Still a little lost. We want to calculate the work required to mvoe the object to the right first. Force is constant right? So would you use fΔrCosθ?

 

[SammyS]

Understood, so quick question, is the blue line the force? Then the other lines simply components of the force?

Still a little lost. We want to calculate the work required to move the object to the right first. Force is constant right? So would you use fΔrCosθ?

No, the force is not along the blue line.

No, the force is not constant.

The force depends upon the location of the particle according to $\displaystyle \vec{F}=2y\hat{\text{i}}+x^2\hat{\text{j}}\ .$

You are to calculate the work done by that force in moving the particle from point, O, to point, C, along each of the colored paths.

 

[Nickie]

I would approach this problem by first identifying the forces acting on the particle. From the given equation, it appears that there are two forces acting on the particle, one in the y direction (2yi) and one in the x direction (x2j). To solve for the work done by these forces, we can use the equation W=∫ Fx dx, where Fx represents the force in the x direction and dx represents the displacement in the x direction. Similarly, we can use the equation W=∫ Fy dy to solve for the work done by the force in the y direction.

To find the total work done, we would need to integrate both equations over the appropriate displacement ranges. This may require splitting the integral in terms of the x and y axes to accurately represent the forces acting on the particle. Once we have the integrals set up, we can solve for the work done by plugging in the given values for the forces and displacement.

It is important to note that the work done by a force is equal to the change in kinetic energy of the particle. Therefore, solving for the work done can provide valuable information about the motion and energy of the particle.