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Fall 2008 -- Purdue University -- West Lafayette, IN
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10 comments:
Do take the whole device as your system.
T1 is zero for all three components (the two ball pendulums and the hanging mass) because the system is released from rest.
You can assign a datum line for each component so that V1 is equal to zero for each component. But because the system moves V2 will not be zero.
There are no outside forces acting on the system, so U(1 to 2)NC is zero.
So your end equation will only consist of V2 and T2.
do you assume that all the velocities are equal to each other?
You can't assume that. But you can relate the velocity of the two pendulums to the origin of the pulley they are attached to using rigid body kinematics. Since the pulley isn't moving the velocity at its center is zero.
The mass falling is what causes the pulley to move. The velocity of the falling mass determines the omega of the pulley. Knowing omega, you can express the pendulum velocity's in terms of the velocity of the mass using rigid body kinematics.
How do you relate the angular velocity of the pendulum to the velocity of the 1.5 kg block?
Why would V1 be zero? I placed my datum at the center of the pulleys, which eliminates any potential energies from the left 2 kg mass, but what about the other 2 kg mass and the 1.5 kg one? They are both lower than the datum at the start, wouldn't that create potential energy? (hence V1)
Let's call the velocity of the mass Vc.
Vc acts on the pulley. The resultant angular velocity can be found using the equation:
w(pulley)=Vc/r
Where r is the radius of the pulley, which is given.
Now you have to find the velocity of each pendulum. I'm just going to use a general equation and not focus on a specific pulley. To find the velocity of a pendulum you treat the pendulum arm as a rigid body with one end being the mass of the pulley and the other being the center of the pulley, which I called O. So now you can use kinematics to find the velocity of the pendulum Vp.
Vp=Vo+w(pulley)Xr(pulley to pendulum)
Vo is zero because the center of the pulley is not moving.
Because V0 is zero you can just reuse the w=V/r equation so it looks like this
w(pulley)=Vp/(length of pulley arm)
You know w(pulley) so to solve for Vp:
Vp=(length of pulley arm)*(Vc/(radius of pulley))
And now you can find the velocity of both pendulums in terms of Vc.
Jstrode:
You can use separate datum points for each thing. The datum point you pick for the mass does not have to be the datum point you use for the pulleys. Using one for each thing makes it easier.
If you define separate datum line for different objects, are you still treating the mass and the pendulum as a single system? How can there be to different datum line for a whole system? Thank you.
You can use separate datum points for each component because you just want to know the change in potential energy the component experiences. Check out this equation:
T1+V1+U=V2+T2
Where T is potential energy. If you rearrange the equation like this:
V1+U=V2+(T2-T1)
You can see that(T2-T1)is the change in potential energy. So you really only need to know how much each component changes. You could use the same datum line for all three components, but you will get the same answer.
It's like if you wanted to know the changed in potential energy of a ball 2 feet off the ground when dropped from five feet. It doesn't matter where you define your datum line for that ball, it's still going to have the same change in potential energy from 5 feet to 2 feet.
Sorry, for the convention we've been using T is not potential energy, V is. Sorry for any confusion.
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