Calculating Forces and Potential Energy Changes: Integrals, Curl, and Work

[joemama69]

Homework Statement

U(x,y) = 3x² - 7y

A) Calculate the force at the coordinate point (3,3)

B) Determine if the following forces are conservative and find the change in potential energy correspoinding to each for an interval 0 to x

i) F_x = ax + bs² a and b are constants

ii) F_x = Ae^Bx (A and B are constants)

c) a force F = 6i - 2j acts on a particle that undergoes a displacement of S = 3i + 5j

i)find the work done by the force on the particle
ii) find the angle between F and S

Homework Equations

The Attempt at a Solution

Part A)

by book says F(x,y) = -dU/dx i - dU/dy j those are partial derivatives

so F = -6xi + 7j

then F(3,3) = -18i + 7j and the magnitude is 19.31 N, the answer is given as 21.2 what's my mistake

Part B)

it says you need to take the curl and if it is 0, then it is conservative

i) curl F = (-a - 2bx)j + (a + 2bx)k = 0

ii) curl F = -ABe^Bx i + ABe^Bx k = 0

Part C)

i) W = F dot S = 18 - 10 = 8 N

ii)theta_F = arctan -2/6 = -18.4 degress
theta_S = artcan 5/3 = 59.0 degrees

59.0-18.4 = 77.5 degrees

 

[rock.freak667]

For part A

when taking the partial derivative w.r.t.x you hold y as a constant

so U= 3x² - 7y
∂U/∂x= ∂/∂x(3x² - 7y), so -7y is essentially treated like a constant.

 

[joemama69]

i did that dU/dx = 6x DU/dy = -7