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

Question

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

2 Answers

Impact of this question

8129 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-11-21T12:57:08

$1.2$ $1.2$

Explanation:

$C = (x, 4 x^{2})$ $C=(x,4x^2)$ and $F = ({(4 x^{2})}^{2}, - x^{2})$ $F=((4x^2)^2,-x^2)$ but $d C = (1, 8 x)$ $dC=(1,8x)$ so
$d W = ⟨ F, d C ⟩ d x = 16 x^{4} - 8 x^{3}$ $dW=<< F, dC >>dx = 16x^4-8x^3$ then

$W = \int_{0}^{1} ⟨ F, d C ⟩ d x = \frac{16}{5} - \frac{8}{4} = 1.2$ $W=int_0^1 << F, dC >>dx = 16/5-8/4=1.2$

Answer 2 · 2017-02-18T17:19:27

$\int_{C} \to F \cdot d \to r = 1.2$ $int_C vec(F) * d vec(r) = 1.2$

Explanation:

The work done in moving a particle from the endpoints $A$ $A$ to $B$ $B$ along a curve $C$ $C$ 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,

$int_C \ vec(F) * d vec(r) = int_C \ ( 16x^4 hat(i) -x^2hat(j) ) * (dx hat(i) + 8x \ dx hat(j))$
$" "= int_0^1 \ 16x^4 \ dx -x^2(8x)dx$
$" "= int_0^1 \ 16x^4 -8x^3 \ dx$
$" "= [ 16/5 x^5 -2x^4 ]_0^1$
$" "= (16/5 -2) - 0$
$" "= 6/5$
$" "= 1.2$

Science

Math

Humanities

... and beyond

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

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

Calculate the work done by a particle under the influence of a force y^2 hat(i) - x^2hat(j)y2ˆi−x2ˆjy^2 hat(i) - x^2hat(j) along the curve y=4x^2y=4x2y=4x^2 from (0,0)(0,0)(0,0) to (1,4)(1,4)(1,4)?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

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