How do I calculate the mass acting on the cart in a pendulum impacting device?

[frob]

I am designing a pendulum impacting device for an experiment. The pendulum will impact a cart on a track. I'd like to calculate theoretical kinetic energy values for the pendulum at the point of impact. The pendulum is made of a hollow aluminum bar, with a steel plate bolted on at the bottom. I cannot neglect the weight of the bar since it has so much mass. How much of this mass is acting on the cart at the bottom? How do I calculate this? The lever arm is 45" long and there's 3" of metal above the pivot point.

 

[Simon Bridge]

You use conservation of energy ... do calculations with the center of mass.

 

[frob]

You use conservation of energy ... do calculations with the center of mass.

So I should find the velocity at the center of mass and use the entire mass of the arm, despite the fact that not all of the mass is in the lever arm and bob?

 

[mfb]

Determine the momentum p of the whole object and an "effective velocity" of v=p/m, use this for the collision - it should give a good approximation.

 

[Simon Bridge]

Track the changes in energy ... loss of potential energy in the pendulum falling becomes kinetic energy in the object struck.

The loss in PE depends on the distance the center of mass falls.

 

[sophiecentaur]

If the arm has a non-negligible mass / moment of inertia, then you need to include this in the overall calculation - both to find the momentum on impact and in the momentum transfer during the collision.

 

[russ_watters]

OP: please google "charpy impact test". That's what you are describing, with only very minor variation.

 

[Simon Bridge]

<curious>
How come everyone wants to use momentum arguments?
The device converts gravitational potential energy in a raised hammer into kinetic energy in the cart (or whatever gets hit). Isn't this a conservation of energy problem? If steps are taken to minimize losses then we can treat it as elastic even!

Of course, I am imagining that the swing stops when it hits the cart (or whatever).

Usually when we do this sort of thing, it is to predict some design parameter ... say, how high we need to start the hammer to get a particular speed on the object struck ... or how long to make the hammer haft... otherwise you can just build it and see. Oh OK - maybe to get funding: "our calculations predict that the device will deliver a bit less than x lbs of impact so ti is worth paying for!"

 

[russ_watters]

Yes, a charpy tester measures the height of a pendulum on the downswing and upswing to calculate change in GPE needed to fracture a sample. For a collision with motion though it gets more complicated if the collision isn't fully elastic: some fraction of energy will be dissipated. Momentum, however, is conserved either way.

 

[Simon Bridge]

It seems to me that when you do the conservation of momentum at the impact, you are saying that the momentum of the hammer before the impact becomes the momentum of the cart after the impact - since the hammer is stationary ... or else, how do you figure working out other ways momentum can transfer in advance? OTOH: if the mass of the hammer is much different from the mass of the cart, then you'll need conservation of momentum as well to help work out the recoil... but if it is a LOT heavier, then the recoil will be small.

Is this approximation more accurate than treating the collision as elastic?

Sure, no collision is elastic - there will at least be the noise of the strike ... but that means the air gets some momentum doesn't it?

On top of which - care has been used, in this case, to choose very hard and elastic materials. Put a hard bumper on the cart and you have a highly elastic setup ... unless this is much bigger than a bench-top rig. (Recall the wrecking-ball Newton's cradle from Mythbusters?) But then, isn't it still just as hard to predict, in advance, what the momentum will do after the impact?