- #71
Thank you for your reply @haruspex!haruspex said:So r is a constant, but you integrate wrt it?
Right.Callumnc1 said:Thank you for your reply @haruspex!
I think take it out of the work integral that will wrt to theta
Many thanks!
Thank you for your reply @haruspex ! That is good advice! I have tried to include my reasoning below.haruspex said:Right.
Don't just throw integrals together that seem to have the right ingredients. Think what the integral is saying.
In the present case, you are considering spiderman moving as the rope swings, so the independent variable is the angle. As it swings through a small angle ##d\theta##, Spiderman moves a distance (what?) at an angle (what?) to the vertical force (what?) thereby doing work (what?).
When you wrote S=rθ, weren’t you taking θ to be that variable?Callumnc1 said:the angle the rope makes with the vertical ϕ (I just defined that variable)
Thank you for your reply @haruspex !haruspex said:When you wrote S=rθ, weren’t you taking θ to be that variable?
Callumnc1 said:I think I changed the variables. I now take ##\phi## as the angle between the vertical and the rope and ##d\theta## as the angle between the force of gravity and differential displacement ##\vec {ds}##
Please update those equations using your new definitions.Callumnc1 said:Given that the arc length of the path is ##S = r\theta##, we take the derivative of arc length with respect to theta giving ## ds = r\theta d\theta##
## dW = Fr\cos\theta d\theta ## Where ##d\theta## is the angle between the weight and displacement vector ##\vec ds## tangent to the circular path at each point
Thank you for your reply @haruspex!haruspex said:Then this is wrong:
Please update those equations using your new definitions.
Edit: just noticed your step from ##S = r\theta## to ## ds = r\theta d\theta## is also wrong.
If ϕ is the angle between the vertical and the rope then S=rφ.Callumnc1 said:Here are the new equations @haruspex ,
We start with ## dW = Fr\cos\theta d\theta ##
And we must integrate from ##\phi_1## to ##\phi_2## so ## dW = \int_{\phi_1}^{\phi_2} Fr\cos\theta d\theta ##
Therefore, since ##\theta## depends on ##\phi## we must get ##\theta## in terms of ##\phi##. I did this geometrically and proved that ##\theta = \phi##.
Am I correct?
Many thanks!
Thank you for your reply @haruspex !haruspex said:If ϕ is the angle between the vertical and the rope then S=rφ.
When ϕ=0, what is the angle between ##\vec ds## and the gravitational force?
Of course: draw the right diagram. The special cases are just an easy way to check your answer.Callumnc1 said:Is there a way to find the relationship between ϕ and θ without considering intuitive cases along the circular path?
Thank you for your reply @haruspex !haruspex said:Of course: draw the right diagram. The special cases are just an easy way to check your answer.
If you thought you had proved a different relationship, try to see where that proof went wrong.
You have drawn ##\theta## as the angle the final displacement makes to the vertical. You defined it as the angle the displacement element ##\vec ds## makes to the vertical at an intermediate point.Callumnc1 said:Here is are my diagrams @haruspex
View attachment 322556
View attachment 322558
Could you please give me some more guidance, I am still getting ##\theta = \phi##
But I guess I'm meant to be looking at differential displacement so I can see here that ##\theta = 180##
View attachment 322560
many thanks!
Thank you for your reply @haruspex !haruspex said:You have drawn ##\theta## as the angle the final displacement makes to the vertical. You defined it as the angle the displacement element ##\vec ds## makes to the vertical at an intermediate point.
Draw a diagram with the rope at two angles to the vertical, ##\phi## and ##\phi+d\phi##.
The displacement element is length ##r.d\phi##. Note that it is at right angles to the rope.
##ds## is (or behaves like) an infinitesimal. The incremental displacement is infinitesimally close to being at right angles to both the initial and final angles. Its angle differs from the perpendicular by only ##d \phi## at most.Callumnc1 said:I'm not sure if the displacement vector makes at right angle with the dark orange line (finial rope position)
Thank you for your reply @jbriggs444 !jbriggs444 said:##ds## is (or behaves like) an infinitesimal. The incremental displacement is infinitesimally close to being at right angles to both the initial and final angles. Its angle differs from the perpendicular by only ##d \phi## at most.
But that is window dressing. Surely you are after the incremental work done. This should be the vector dot product of the force of gravity, ##\vec{mg}## and the incremental displacement, ##\vec{ds}##.
Yes. The concern with differentials is distracting and the drawings could be done more clearly -- in particular, labelling the angle you are calling ##\theta##. But I agree with the conclusion.Callumnc1 said:Is my proof for θ=90+ϕ geometrically correct?
Thank you so much for your reply @jbriggs444 !jbriggs444 said:Yes. The concern with differentials is distracting and the drawings could be done more clearly -- in particular, labelling the angle you are calling ##\theta##. But I agree with the conclusion.
It might make it easier to evaluate the resulting integral if you would write ##\cos ( 90 + \phi )## as ##-\sin \phi##.
Personally, I would have looked at the formula for the projection of a vector normal to a [oriented] surface. That involves ##\sin \theta## where ##\theta## is the angle the vector makes with respect to the surface. In this case, the displacement vector makes an angle of ##- \phi## with respect to a horizontal surface. The projection thus involves ##\sin -\phi = - \sin \phi##.
You can get the dot product of two vectors by taking the projection of one in the direction of the other and multiplying the magnitude of the one by the [signed] magnitude of the projection of the other.
Actually, I would have been sloppy with the sign convention, ignored the orientation of the surface, looked at ##\sin \phi## and then reasoned that the displacement and gravity are pointing generally away from each other and inverted the sign of the resulting path integral.
Actually, I would have recognized a conservative field, reasoned that all path integrals between a particular pair of endpoints across a conservative field will yield the same result and either picked a trivial path or computed the potential difference.
But we can definitely get to work evaluating the integral.
Well, a drawing of an isosceles triangle with two right angles is a bit distracting.Callumnc1 said:Yeah true, there is a much easier method if I had recognized the conservative field! Sorry, what did you find distracting about my concern with differentials?
Oh true! Thank for letting me know @jbriggs444 !jbriggs444 said:Well, a drawing of an isosceles triangle with two right angles is a bit distracting.