- #1
RicardoMP
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Homework Statement
Hi! I'm trying to solve a simple problem of mechanics, but I'm getting the wrong results and I suppose I don't yet grasp the concept of instantaneous axis of rotation very well.
So, a cone (see attached picture) is rolling without slipping on a plane. Vp is point P linear velocity on the laboratory frame with its origin on the cone's apex. I must choose "z" as the vertical axis and, instantaneously, "x" along the line of contact.
I have to find the instantaneous angular velocity vector [tex] \omega [/tex].
Homework Equations
[tex]\omega = \Omega + \omega _s[/tex]
\Omega = \frac{v_p}{hcos(\alpha)}
\omega _s = \frac{v_p}{R}
The Attempt at a Solution
My first thought was that there are 2 contributions for the angular velocity: [tex]\Omega[/tex] pointing upwards and relates to the rotation of the cone around the vertical axis going through the origin and [tex]\omega _s[/tex] which is the spin angular velocity of the cone. I simply decomposed the spin ang.vel in its x and z components and got
[tex]\omega = (\omega_s cos(\alpha))e_x+(\omega_s sin(\alpha) + \Omega)e_z[/tex]
And I thought that was it.
The solution states that the angular velocity only has component along x: \omega = (\omega,0,0) and I'm failing to see how. Is the line of contact an instantaneous axis of rotation? And what does that mean for this problem?
Thank you in advance
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