Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Introductory Physics Homework Help
Conservative Forces - Maths of Force & Energy
Reply to thread
Message
[QUOTE="geoffrey159, post: 4994876, member: 532398"] [h2]Homework Statement [/h2] A particle of mass m moves in a horizontal plane along the parabola ##y = x^2##. At t=0, it is at the point (1,1) with speed v0. Aside from the force of constraint holding it to the path, it is acted upon by the following external forces: A radial force: ##\vec F_a = -A r^3\hat r## A force given by : ##\vec F_b = B (y^2\hat \imath - x^2 \hat \jmath)## where A,B are constants. a- Are the forces conservative? b- What is the speed ##v_f## of the particle when it arrives at the origin ? [h2]Homework Equations[/h2] curl, integration on a path [h2]The Attempt at a Solution[/h2] Hello, I've just started a new chapter about mathematical aspects of force and energy. It's a little hard to digest at the beginning so maybe you can check my work. Thanks ! a - To test whether a force is conservative, I must check that ##\vec \nabla \times \vec F = \vec 0 ##, but I'm going to need an expression of the gradient in polar coordinates for the (x,y) plane. I believe that it is ##\vec \nabla = \frac{\partial .}{\partial r} \hat r + \frac{1}{r} \frac{\partial .}{\partial \theta} \hat\theta + \frac{\partial .}{\partial z} \hat k## because if ##g = g(r,\theta,z)##, then its differential is ## \begin{array}{ccr} dg := (\nabla_{\hat r} g)\ dr + (\nabla_{\hat \theta} g)\ ds + (\nabla_{\hat k} g)\ dz & \text{and} & \begin{align} dg =& \frac{\partial g}{\partial r}\ dr + \frac{\partial g}{\partial \theta} \ d\theta + \frac{\partial g}{\partial z}\ dz \\ =& \frac{\partial g}{\partial r}\ dr + \frac{\partial g}{\partial \theta} \ (\frac{ds}{r}) + \frac{\partial g}{\partial z}\ dz \end{align} \end{array} ## So if you confirm this is right, ## \begin{array}{cc} \vec \nabla \times \vec F_a = \begin{vmatrix} \hat r & \hat \theta & \hat k \\ \frac{\partial .}{\partial r} & \frac{1}{r} \frac{\partial .}{\partial \theta}& \frac{\partial .}{\partial z} \\ -A r^3 & 0 & 0 \end{vmatrix} = \vec 0 & \vec \nabla \times \vec F_b = \begin{vmatrix} \hat \imath & \hat \jmath & \hat k \\ \frac{\partial .}{\partial x} & \frac{\partial .}{\partial y}& \frac{\partial .}{\partial z} \\ B y^2 & -Bx^2 & 0 \end{vmatrix} = -2B(x+y) \hat k\neq \vec 0 \end{array} ## So that only the radial force is conservative.b- ##\vec F_b## does a non-conservative work on the path ##y = x^2## from x=1 to x=0, so its work is : ## \begin{align} W^{(nc)} =& B \int_{(1,1)}^{(0,0)}(y^2 dx - x^2 \ dy) \\ =& - B\int_{0}^{1} (x^4\ dx - x^2 (2x\ dx)) \\ =& \frac{3B}{10} \end{align}## The potential fonction of the radial force is ##U_a(\vec r) = \frac{A}{4} r^4##. By conservation of total energy, ##E_f - E_i =W^{(nc)} \Rightarrow v_f^2 = v_0^2 + \frac{A}{2m} +\frac{3B}{5m}## [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Introductory Physics Homework Help
Conservative Forces - Maths of Force & Energy
Back
Top