What is the velocity vector of (rsin(phi), rcos(phi), 1) ?

  • Thread starter Thread starter physicss
  • Start date Start date
  • Tags Tags
    Velocity
AI Thread Summary
The discussion centers on finding the velocity vector of a pendulum's bob represented by the position vector (rsin(phi), rcos(phi), 1). The initial attempt yielded a velocity vector of (rcos(phi), -rsin(phi), 0), which was later identified as incorrect. Participants emphasized the importance of applying the chain rule correctly, noting that both phi and the position vector are functions of time. Clarifications were made regarding the proper notation and the need to differentiate with respect to time rather than phi. The conversation highlights the need for careful calculus application in determining the correct velocity vector.
physicss
Messages
25
Reaction score
4
Homework Statement
Hello,

Given is the following location vector: (of a string-pendulum)
x= (rsin(phi), rcos(phi), 1)
My task was to determine the velocity vector of the location vector.
Quote: "Let the location of a pendulum be given by the vector x= (rsin(phi), rcos(phi), 1), where r is the length of the filament and φ is the angle to the y-axis.

Calculate the velocity vector v for the case where only the angle φ is time-dependent."
Relevant Equations
x= (rsin(phi), rcos(phi),1)
I did try to solve the problem by forming the derivative and my result was: v=(rcos(phi), -rsin(phi),0). My solution is wrong, the tutor corrected the task but he didn’t give us the results. My question is what the solution is. Thanks in advance.
 
Physics news on Phys.org
physicss said:
my result was: v=(rcos(phi), -rsin(phi),0)
Does your result have the dimensions of velocity?

Suppose you have a function ##f(\phi##), where ##\phi## is a function of time ##t##.
Recall the chain rule of calculus: $$\frac{d f}{dt} = \frac {df}{d \phi} \cdot \frac{d \phi}{dt}$$.
 
  • Like
Likes vanhees71, jbriggs444, Steve4Physics and 1 other person
Hi @physicss. Welcome to PF!

physicss said:
My task was to determine the velocity vector of the location vector.
Being picky, I think you mean ".. . to determine the velocity vector of the pendulum 'bob' (the mass at the end of the string)".

physicss said:
Quote: "Let the location of a pendulum be given by the vector x= (rsin(phi), rcos(phi), 1), where r is the length of the filament and φ is the angle to the y-axis.

Calculate the velocity vector v for the case where only the angle φ is time-dependent."
Relevant Equations: x= (rsin(phi), rcos(phi),1)

I did try to solve the problem by forming the derivative and my result was: v=(rcos(phi), -rsin(phi),0). My solution is wrong ...
Are you familiar with the chain rule? ##\vec x## is a function of ##\phi##. And ##\phi## is a function of ##t##.

Aha! Beaten to it by @TSny.
 
  • Like
Likes vanhees71, TSny and MatinSAR
Steve4Physics said:
Aha! Beaten to it by @TSny.
Very close :oldsmile:
 
  • Like
Likes vanhees71 and Steve4Physics
physicss said:
I did try to solve the problem by forming the derivative and my result was: v=(rcos(phi), -rsin(phi),0). My solution is wrong, the tutor corrected the task but he didn’t give us the results. My question is what the solution is. Thanks in advance.
If I understand the question properly, your work looks good. However, we could clean things up a bit.

First, let us clean up the format of the question. Using ##x## for the name of the position vector is poor form. It invites confusion with the ##x## component of the position vector. Let me put that in ##\LaTeX## format as well:$$\vec{S} = (r \sin \phi, r \cos \phi, 1)$$Now let us turn to the calculus part of the exercise. We want the derivative of position with respect to time:$$\vec{v} = \frac{d\vec{S}}{dt}$$If we take the derivative of ##r \sin \phi##, you say that we get ##r \cos \phi##. But that is not the derivative with respect to time (##t##). That is the derivative with respect to phi (##\phi##):$$\vec{v} \ne \frac{d\vec{S}}{d\phi}$$Drat -- beaten to it by both of you.
 
  • Like
Likes MatinSAR and Steve4Physics
Thread 'Struggling to make relation between elastic force and height'
Hello guys this is what I tried so far. I used the UTS to calculate the force it needs when the rope tears. My idea was to make a relationship/ function that would give me the force depending on height. Yeah i couldnt find a way to solve it. I also thought about how I could use hooks law (how it was given to me in my script) with the thought of instead of having two part of a rope id have one singular rope from the middle to the top where I could find the difference in height. But the...
Thread 'Voltmeter readings for this circuit with switches'
TL;DR Summary: I would like to know the voltmeter readings on the two resistors separately in the picture in the following cases , When one of the keys is closed When both of them are opened (Knowing that the battery has negligible internal resistance) My thoughts for the first case , one of them must be 12 volt while the other is 0 The second case we'll I think both voltmeter readings should be 12 volt since they are both parallel to the battery and they involve the key within what the...
Back
Top