How much a spring is compressed

Question

On the ground, a cart is placed against a compressed spring. The spring launches the cart toward a vertical loop (like a roller coaster loop). The cart then rolls around the loop. Assume that the cart has a mass of 522.4kg and it’s speed at the top of the loop is 21.7m/s. Assume that the loop has a radius of 8.6m and the spring constant is 69,697.2 N/m. Ignore friction on the cart. How much was the spring compressed ?

Answer 1

Hi Kayla C.,

Not so bad a problem, approached in the correct way of course!

The cart has LOST kinetic energy = gravitational potential energy, in the vertical direction. That's mgh. But it still HAS 21.7 m/s of kinetic energy equivalent, which you have to calculate as (1/2)mv^2. So the INITIAL kinetic energy was the sum of those two quantities (there's no other place for the initial kinetic energy to go, is there!). So figure the initial velocity from that, if you wish. But the kinetic energy is all you really need, since the spring initially had it as (1/2)kx^2, right? So that will give you x (the initial spring compression).

I suspect you really didn't want to be on that cart -- you could figure your initial acceleration, from F=kx and F=ma (neglecting the mass of the spring, of course). I estimate it at about 30 g. That's why you calculate these things, rather than just hopping in and hoping for the best (though you might survive in a neutral bouyancy suit, perhaps)!

-- Cheers, -- Mr. d.