Calculate the work done by a force on a particle

[henrco]

Homework Statement

A force Fx acts on a particle that has a mass of 1.46 kg. The force is related to the position x of the particle by the formula Fx = Cx^3, where C = 0.550 if x is in meters and Fx is in Newtons. Calculate the work done by this force on the particle as the particle moves from x = 2.96 m to x = 1.47 m.

Homework Equations

W = F.x
Where F is the force and x is the displacement

The Attempt at a Solution

The displacement x = 1.47-2.96 = -1.49m
The force Fx = Cx^3 = (0.550)(-1.49)^3 = -1.82 (to 3 significant figures)
Input these values into the formula, W=F.x

W = (-1.82)(-1.49) = 2.71 Joules (to 3 significant figures)

This is the answer I get however it's not the expected answer. Could someone please point me in the right direction, so I can understand where I am going wrong.

 

[cnh1995]

W = F.x
Where F is the force and x is the displacement

This is true when the force is constant. What is the fundamental equation of work in terms of force and displacement? You need integral calculus for that.

 

[PeroK]

Homework Equations

W = F.x
Where F is the force and x is the displacement

Could someone please point me in the right direction, so I can understand where I am going wrong.

Here's how I interpret that equation:

The work done, ##W##, by a constant force ##F##, acting over a distance ##x## equals ##Fx##

Is that a relevant formula in this case?

 

[ChrisVer]

the work can be the area between your [itex]F(x)[/itex] and the x-axis from [itex]x= x_{in}[/itex] to [itex]x=x_{fin} [/itex] ...

 

[henrco]

Thank you gentlemen, that was very helpful. Here's a second attempt.

So because it's a variable force I need to integrate the area under the curve.

W = ∫ F(x) . dx (from x = 2.96 m to x = 1.47 m). Where F(x) and dx are both vectors - unable to write vector notation.

The dot product of two vectors:

F(x) . dx = F(x).dx.Cos(180). It's 180 as it's moving in the opposite direction. Is this correct?
F(x).dx.(-1)
-(F(x).dx)

Inputing the value for F(x) into the equation.
W = ∫ -(Cx^3) dx (from x = 2.96 m to x = 1.47 m)
W = - Cx^3 x (from x = 2.96 m to x = 1.47 m)
W = - C(1.47)^3(1.47) - (-C(2.96)^3(2.96))
W = - ((0.550)(1.47)^3)(1.47) - ((-0.550)(2.96)^3)(2.96)
W = -2.57 +42.2
W = 39.7 Joules

 

[PeroK]

Thank you gentlemen, that was very helpful. Here's a second attempt.

So because it's a variable force I need to integrate the area under the curve.

W = ∫ F(x) . dx (from x = 2.96 m to x = 1.47 m). Where F(x) and dx are both vectors - unable to write vector notation.

The dot product of two vectors:

F(x) . dx = F(x).dx.Cos(180). It's 180 as it's moving in the opposite direction. Is this correct?
F(x).dx.(-1)
-(F(x).dx)

That's one way to look at it. Another is that ##\int_a^b = -\int_b^a##

Inputing the value for F(x) into the equation.
W = ∫ -(Cx^3) dx (from x = 2.96 m to x = 1.47 m)
W = - Cx^3 x (from x = 2.96 m to x = 1.47 m)
W = - C(1.47)^3(1.47) - (-C(2.96)^3(2.96))
W = - ((0.550)(1.47)^3)(1.47) - ((-0.550)(2.96)^3)(2.96)
W = -2.57 +42.2
W = 39.7 Joules

Something's gone wrong with your integration there. First, you already reversed the integral value by taking the negative, but then you left the end points going high to low. It's one or the other. Second, you didn't integrate ##x^3## correctly.[/QUOTE]

 

[henrco]

PeroK, thank you for that.

I have now reversed the range, so it is from a= 1.47 to b= 2.96.
Also hopefully I've correctly integrated x^3.

W = - ∫ (Cx^3) dx (from a =1.47 to b = 2.96)
W = - ((C/4) x^4) + C (from a =1.47 to b = 2.96)

(I'm little confused by the constant of integration and the constant C provided in the question. I've discarded the constant of integration as I would normally when inputting a range).

W = - ((0.550)/4) (2.96)^4) - ((0.550)/4) (1.47)^4)
W = - 10.6 + 0.642
W = -9.96 Joules

 

[ChrisVer]

(I'm little confused by the constant of integration and the constant C provided in the question. I've discarded the constant of integration as I would normally when inputting a range).

For definite integrals, there is no need for the constant of integration... it dies out after the subtraction.
You got confused because you used the same symbol for both. Either use different symbols or even if you use the same, avoid them having the same capitalization/font etc.

 

[ChrisVer]

W = -9.96 Joules

are you sure you did the calculations right?

 

[henrco]

Thanks for feedback and for checking the calculations ChrisVer,

I redid the calculations more carefully and got.

W= -9.91 Joules

 

[henrco]

Hi,

Would anyone please check my last post, to see if I've calculated the final result correctly.
ChrisVer said I had made a mistake but I'm not sure if I've rectified it.

Thanks.

 

[cnh1995]

Hi,

Would anyone please check my last post, to see if I've calculated the final result correctly.
ChrisVer said I had made a mistake but I'm not sure if I've rectified it.

Thanks.

Looks correct to me!

 

[henrco]

I input the answer -9.91 Joules as the submission was electronic and it says my answer was wrong!

Could someone else please have a look at my calculation and let me know what I've done wrong.

 

[ChrisVer]

The right answer of your problem is -9.91322
you can always compare some of your results by double checking with wolframalpha:
https://www.wolframalpha.com/input/?i=integral+0.55*x^3+from+x=2.96+to+x=1.47

if all your input to the problem is correct:
Either it takes it wrong because you didn't keep the right number decimals (like 9.9 or 9.91 or 9.913?) or there was no need for the - (minus), since you are already told that work is being produced (however I wouldn't like this explanation since it's way too ambiguous).