# Particle Motion with Forces

1. Feb 16, 2007

### brunie

Just need a bit of help with this question and to make sure im solving it properly. Thanks
______________

A single conservative force acts on a 5.00 kg particle.
The equation F_x = (2x + 4) N describes this force, where x is in meters.
As the particle moves along the x axis from x = 1.40 m to x = 6.20 m, calculate the following:

(a) work done by the force
(b) change in the potential energy of the system
(c) kinetic energy of the particle at x=6.20m
if its speed is is 3.00 m/s at x =1.40m

Ek = 0.5mv^2
Ep = mgh

***A)***

Ok I started by integrating the force as ∫ 2x +4
= x^2 + 4x
I then evaluated the difference in area between the two bounds, namely
((6.2)^2 + 4*6.2 - (1.4)^2 + 4*1.4) which gave 55.68
Since this was the force I multiplied by the change in distance 6.2 - 1.4 = 4.8
So 55.68 * 4.8 = 267.264

***B)***

I figured that since the particle is always on the x-axis, the potential energy difference should be zero.

***C)***

At 1.4m, speed is 3m/s.
Ek = 0.5(5)(3)^2
= 22.5

They want Ek at 6.2m.
And now I can't decide what to do. Because I think I need the speed at 6.2m in order to find the energy at 6.2m, but I don't kno why the Ek at 1.4m was given. Or I might need to use work and total energy. Im really not too sure.

Any help to verify this would be appreciated.
Thanks

Last edited: Feb 16, 2007
2. Feb 16, 2007

### drpizza

For part A, the integral finds the area under the curve. What are the units of the area? (look at the units for the width of the interval times the units for the height or F_x value). Since the F_x value is force, and you're multiplying that by a distance, the units don't agree with what you think you found (a force.) You're definitely on the right track to take the integral, but I think you might see what I'm talking about if you look back at your work.

I agree with your part B.

Therefore, for part C, you started with a certain amount of kinetic energy when x was 1.40 meters. What happened to all the work that you did, since none of it was "changed" into potential energy (and the force was conservative)?

Last edited: Feb 16, 2007
3. Feb 16, 2007

### Hootenanny

Staff Emeritus
Good
Not so good. What is the defintion of the work done by a force?

Last edited: Feb 16, 2007
4. Feb 16, 2007

### neutrino

What happened to the 4?

5. Feb 16, 2007

### Hootenanny

Staff Emeritus
Good catch neutrino, I missed that ...

6. Feb 16, 2007

### brunie

oops lol, let me fix that

7. Feb 16, 2007

### brunie

ok so for part A should I have multiplied again by the distance since it was integrated?

8. Feb 16, 2007

### drpizza

Hootenany, why does the work have to go to a change in potential energy for part B? (Vs. a change in mechanical energy, of which there's another possibility in this problem)

9. Feb 16, 2007

### Hootenanny

Staff Emeritus
Lets just pause a moment and think. Write down for my the defintion of work done by a variable force F(x).

10. Feb 16, 2007

### drpizza

1st, what do you multiply together to find work?
Then, draw a quick graph with Force on the vertical axis and displacement on the x-axis. Make a little rectangle somewhere on the graph. Now, if you were trying to find the area of that rectangle (using a formulas from geometry instead of an integral), what would you be multiplying together?

11. Feb 16, 2007

### Hootenanny

Staff Emeritus
Oops edit, my bad, misread the question, for some reason I thought we were in the vertical plane

12. Feb 16, 2007

### brunie

Isnt work just the force by the distance. And since the integrated area gave the total force in Nm shouldnt it just be that?

13. Feb 16, 2007

### Hootenanny

Staff Emeritus
No, not quite. The defintion of work done (W) by a variable force (F) is gived by;

$$W = \int^{x_1}_{x_0}F(x)dx$$

14. Feb 16, 2007

### brunie

By integrating that wouldnt it just be an extra x term, meaning that its divided by the distance to get rid of it?

15. Feb 16, 2007

### Hootenanny

Staff Emeritus

$$W = \int^{x_1}_{x_0} (2x+4)dx = \left[ x^2 + 4x \right]^{x_1}_{x_0}$$

$$W = \left(x_1^2+4x_1\right)-\left(x_0^2+4x_0\right)$$

16. Feb 16, 2007

### brunie

ok, i evaluated that already but im just confused about what to do with that result

17. Feb 16, 2007

### Hootenanny

Staff Emeritus
If you evaluated the integral correctly, it will have returned the value of the work done by the force as the particle moved between x0 and x1

18. Feb 16, 2007

### brunie

so by integrating the equation and evaluating w.r.t the endpoints i am finding the work done by the force?

19. Feb 16, 2007

### drpizza

Fixed.
Yep. Newton-meters are not a unit of force.

Maybe it'll help to go back to something simpler.

For a similar problem, lets do this:
On the vertical axis, put velocity. On the horizontal axis, let's put time. Now, here's an equation. v_t= 5.
Find the displacement from t=1 to t=7.
Note: if you graph this, it's just a constant function (a horizontal line).

If we integrate, we'd have $$displacement = \int^{7}_{1}5dt$$
This integral is 5t evaluated from 1 to 7, or 5*7 - 5*1 = 30 (units??)
Now instead, notice that it's just a rectangle. You can find the area in a rectangle by a simple length * width. The length is 6 seconds (from 1 second to 7 seconds) and the width is 5 m/s. Thus, you're multiplying meters/second by seconds, and the units cancel to give you an answer in meters.

The area under a velocity vs time curve is the displacement.
Likewise, the area under a Force vs. distance curve IS the work.
Thus, as Hootenanny said, Work = the integral...
(the integral is just a way to find the area under a curve.)

20. Feb 16, 2007

### drpizza

Yes
That *is* the work. There's no need to multiply by anything.

(Don't you wish we would have just told you that you went too far on your first step the first time you did it? :tongue2: ) Sometimes it's tough to help someone understand the process and what they're actually doing vs. just rote memorization of steps that you might end up confusing later on.