Conservative Forces

# Conservative Forces - Potential energy Page 1 Contents 1...

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Unformatted text preview: Potential energy Page 1 Contents 1 Remember this from calculus class? 1 1.1 The gradient of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Total and exact differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4.1 The curl of the gradient of a function ( ∇ × ∇ g ) . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.5 Putting it all together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.6 Finding g when we know ∇ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.6.1 An example with an exact differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6.2 An example with an inexact differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 How does this boring math stuff relate to forces and work? 9 2.1 The work done by F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 How do you tell if a force is conservative? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Example - a constant force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Example - a force linearly proportional to distance . . . . . . . . . . . . . . . . . . . . . . . . 11 A Other coordinate systems 14 A.1 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A.2 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 Remember this from calculus class? 1.1 The gradient of a function If you have a scalar function, g , you can find the rate at which the function changes in the different coordinate directions of a multi-dimensional coordinate space by evaluating the gradient of the function. The gradient of g is written mathematically as grad g or ∇ g (1) In a 3-dimensional space, grad ≡ ∇ is the usual vector-like operator 1 which, for Cartesian coordinates, is: ∇ = ˆ ı ∂ ∂x + ˆ ∂ ∂y + ˆ k ∂ ∂z (2) Since we’re all familiar with them, we’ll stick with Cartesian components for the rest of this discussion. 1.2 Total and exact differentials As you may remember from calculus, the total differential, dg , of a function, g , is defined in terms of its partial derivatives to be (again in Cartesian components): dg = ∂g ∂x dx + ∂g ∂y dy + ∂g ∂z dz (3) You may further remember that an expression of the form g x ( x, y, z ) dx + g y ( x, y, z ) dy + g z ( x, y, z ) dz (4) is called an exact differential if and only if a function, h ( x, y, z ), exists such that the total differential of...
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## This note was uploaded on 04/28/2008 for the course AME 301 taught by Professor Shiflett during the Fall '06 term at USC.

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Conservative Forces - Potential energy Page 1 Contents 1...

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