Question

Ap Physics C 2002 M2?

Answer 1

a) They nicely give us the rotational inertia of a disk, so all we have to do is substitute the given values into the formula. We know that each tire has a mass of `m/4` and a radius of `r`. Thus

`I_"tire" = 1/2(m/4)r^2 = 1/8r^2`

There is 4 tires on the cart, so to find the total rotational inertia we must multiply by `4`.

`4(1/8mr^2) = 1/2mr^2`

b) As soon as I see "find the height of the incline", I think of conservation of energy. Since there is rotation in the tires there will be translational AND rotational kinetic energies.

`mgh = 1/2mv^2 + 1/2Iomega^2`

We know the value of `I` from part `a`. The total mass of the cart is `m + 4(m/4) = 2m`

`2mgh = 1/2(2m)v^2 + 1/2(1/2mr^2)omega^2`

`2mgh = mv^2 + 1/4mr^2omega^2`

`2gh = v^2 + 1/4r^2w^2`

We know that `omega = v/r`.

`2gh = v^2 + 1/4r^2(v/r)^2`

`2gh = v^2 + 1/4v^2`

`2gh = 5/4v^2`

`8/5gh = v^2`

`v = sqrt(8/5gh)`

c) Once again conservation of energy is required here. Recall the potential energy of a spring is

`1/2kx^2`

If we equate this to the kinetic energy of the cart, we can get the value of `x`.

`1/2kx^2 = 5/4mv^2`

`1/2kx^2 = 5/4m(8/5gh)`

`kx^2 = 4mgh`

`x = 2sqrt((m gh)/k)`

d) An example response would be "mechanical energy was lost in the collision and as a result less was transferred to the spring, reducing the compression of the spring" .

Hopefully this helps!