kinematic questions(velocity of 2 contact points)

hihiip201

in the picture below



http://tinypic.com/view.php?pic=9riq9l&s=6


if we know the angular velocity of the gear on the left, as well as their current distance from contact point, what can we say about the velocity of the right rod at the contact point at anytime t?


we know it is not just the velocity of the contact point of the left rod because the right rod can be sliding along the left rod. but how do we find its exact velocity?

Andrew Mason

If you know the angular velocity of the left rod and the distance of the contact point from the centre of the left gear (r_l), you can determine the velocity of the contact point (ie. in terms of ω_l and r_l). You then just have to relate that to the velocity of the contact point on the right rod. That relationship changes at the point where the two rods are parallel (contact point is the whole rod). So you would have to break it down into two parts (ie. one where the end of the right rod contacts the left and the other where the end of the left rod contacts the right).

AM

hihiip201

how exactly do we relate the velocity of the contact point of left rod and right rod?

we know the velocity of left rod at contact point, but certainly right rod's velocity at the contact point wouldn't be the same since it is sliding along the left rod right?

Andrew Mason

You have to work it out from the geometry.

Consider the situation shown with the tip of the right rod is contacting the body of the left rod. You have to relate the velocity of the tip of the right rod to the velocity of the point that it is in contact with:

ω_rL_r = ω_lr_l + v where v is the speed of the end of the right rod along the length of the left rod (ie. dr_l/dt)/

I haven't worked it out but I think that is how you have to do it. The trick is to express v in terms of the angular velocities of the two rods.

AM

hihiip201

my first guess is also the equation you have above, but what i have trouble is that, should vector v in your equation be based on the geometry of the two rods at current time t? or t = t +dt? to be it makes more sense to use the time at t+dt or else the left rod will end up in a awkward position.

Andrew Mason

v is just the component of the tangential velocity of the end of the right rod that is in the direction of the radius vector r_l. So it is based on the angle between the two rods.

AM

hihiip201

right but the angle between two rods at the current time? or some dt later??


if I break the velocity into two component, the component that is along the contact point of left rod , and the velocity tangential to the left rod.

After some dt later, the position of the contact point of my right rod would be a the position of the contact point of the left rod, minus some distance away from the center of the left rod.


I don't know if my question make sense, sorry about the confusion.

Andrew Mason

It does not matter. dt is arbitrarily small. If you set v(t) at t then at t+dt it will be v(t+dt) = v + dv. If you set θ(t) at t then at t+dt it will be θ(t+dt) = θ+dθ = θ+ω(dt)

What you want to find is the velocity of the contact point on the left rod. Can you express that in terms of the velocity of the end of the right rod? ie. break down the tangential velocity of the right rod ω_rr_r into components that are in the direction of the r_l and perpendicular to that direction (ie tangential to r_l).

AM