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

In summary: F= -6xi + 7jIn summary, we calculated the force at a specific coordinate point and determined if two forces were conservative. We also found the work done and the angle between the force and displacement in a given situation.
  • #1
joemama69
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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) Fx = ax + bs2 a and b are constants

ii) Fx = AeBx (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 = -ABeBx i + ABeBx k = 0

Part C)

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

ii)thetaF = arctan -2/6 = -18.4 degress
thetaS = artcan 5/3 = 59.0 degrees

59.0-18.4 = 77.5 degrees
 
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  • #2
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.
 
  • #3
i did that dU/dx = 6x DU/dy = -7
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is a fundamental tool in calculus and is used to solve a variety of problems in physics, engineering, and other fields.

2. How is the integral of a function calculated?

The integral of a function is calculated using a process called integration. There are several methods of integration, including the fundamental theorem of calculus, integration by substitution, and integration by parts. The choice of method depends on the complexity of the function and the desired level of accuracy.

3. What is curl?

Curl is a mathematical operation that describes the rotation or "circulation" of a vector field in three-dimensional space. It is typically represented as a vector and is used in physics and engineering to describe the behavior of fluids, electromagnetism, and other physical phenomena.

4. How is curl calculated?

Curl is calculated by taking the partial derivatives of a vector field with respect to each of the three dimensions and then combining them in a specific way. This calculation can be done using various methods, including the cross product and the gradient operator.

5. What is the relationship between force and curl?

Force and curl are closely related, as force can be thought of as the physical manifestation of curl. In other words, the curl of a vector field represents the tendency of that field to rotate or "curl" around a point, and this rotation can produce a force on an object placed in that field. This relationship is fundamental in understanding the behavior of fluids, electromagnetism, and other physical systems.

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