Work done by a conservative force

In summary, the potential energy of a particle moving in the x-y plane is given by the equation U=3x+4y. If the particle is at rest at (6,8) at time t=0, the work done by the conservative force on the particle as it crosses the x-axis can be calculated by finding the difference in potential energy between (6,8) and (0,0). The force always points towards the y-axis and lines of constant potential energy are straight lines.
  • #1
Titan97
Gold Member
450
18

Homework Statement


The potential energy ##U## of a particle of mass 1kg moving in x-y plane obeys the law ##U=3x+4y##. x and y are in meters. If the particle is at rest at (6,8) at time t=0, then find the work done by conservative force on the particle from initial position to the instant when it crosses the x-axis.

Homework Equations


##\vec F=-\frac{\partial U}{\partial x}-\frac{\partial U}{\partial y}-\frac{\partial U}{\partial z}##
(This equation is not taken from any book. I thought the relation between F and U was just like the relation between electric field and potential)

3. The Attempt at a Solution

Using the equation: ##F=-3\hat i-4\hat j##
Since no other forces are acting, the particle will move in the direction of acceleration. I also have to find the x-coordinate when it crosses x-axis. Acceleration is at an angle ##\tan^{-1}\big(\frac{4}{3}\big)## with the horizontal towards the 3rd quadrant. Hence the particle moves along the line ##y=\frac{4}{3}(x-6)+8##.
So the x intercept is 0. Hence the total distance moved by the particle is 10m. And work done is 50J.
Is this correct?
 
Physics news on Phys.org
  • #2
It is. A lot of work done by you. The work done by the field can also be compared to the difference U(6,8) - U(0,0). Could that be a concidence or is there more to that ? :rolleyes:
 
  • #3
Work done by conservative force is path independent. But I still need to do "work" to find the final position :smile:. Or is there another way to solve the problem without having to find the final position?
 
  • #4
Force always points at (0,0) !
 
  • #5
Why should it always point at (0,0)? Is it because the force is conservative?
 
  • #6
No, but starting at (6,8) there is no component other than in the direction of (0,0)
 
  • #7
I don't understand that. Is it because the minimum magnitude of potential energy is at (0,0)? Since a particle tends to reach minimum potential energy. If that's the case, then is it true for ##U=2x^2+1##? Since force points at (0,1).
 
  • #8
Potential energy is a scalar. It has a value. Vectors have a magnitude. Unfortunately daily language mixes them up.

Lines of constant U are straight lines for ##
U=3x+4y##. Constant U means no force component along that line. The force points perpendicular to those equipotential lines, so once on a line through the origin means following a straight path through the origin if starting from x > 0 and away from the origin when starting from x ##\le## 0.

For ##
U=x^2+1## lines of constant U are straight lines also. The force always points towards the y axis, not at 0,1 !
 
  • Like
Likes Titan97

1. What is meant by "work done by a conservative force"?

Work done by a conservative force refers to the energy transferred to an object by a force that is independent of the path taken by the object. In other words, the work done by a conservative force only depends on the initial and final positions of the object, not the specific path it takes.

2. How is the work done by a conservative force calculated?

The work done by a conservative force is calculated by taking the negative of the change in potential energy of an object as it moves from its initial to final position. This can be represented by the equation W = -ΔU, where W is the work done and ΔU is the change in potential energy.

3. What are some examples of conservative forces?

Some examples of conservative forces include gravity, electric and magnetic forces, and elastic forces like the force exerted by a spring. These forces do not dissipate energy and the work done by them is dependent on the initial and final positions of the object.

4. How does the work done by a conservative force affect an object's kinetic energy?

The work done by a conservative force can either increase or decrease an object's kinetic energy. If the work done is positive, the object's kinetic energy will increase, and if the work done is negative, the object's kinetic energy will decrease. However, the total mechanical energy (kinetic + potential) remains constant.

5. How is the concept of "conservation of energy" related to work done by a conservative force?

The concept of conservation of energy is related to work done by a conservative force because in a conservative system, the total mechanical energy (kinetic + potential) is conserved. This means that the work done by a conservative force will result in a change in the object's potential energy, but the total energy of the system will remain constant.

Similar threads

  • Introductory Physics Homework Help
Replies
5
Views
523
  • Introductory Physics Homework Help
Replies
9
Views
856
  • Introductory Physics Homework Help
Replies
11
Views
991
  • Introductory Physics Homework Help
Replies
25
Views
933
  • Introductory Physics Homework Help
Replies
1
Views
264
  • Introductory Physics Homework Help
Replies
15
Views
300
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
477
  • Introductory Physics Homework Help
Replies
8
Views
855
  • Introductory Physics Homework Help
Replies
4
Views
1K
Back
Top