Ap Physics C 2002 M2?

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Ap Physics C 2002 M2?

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Answer 1 · 2018-06-11T21:29:49

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 $\frac{m}{4}$ $m/4$ and a radius of $r$ $r$ . Thus

$I_{tire} = \frac{1}{2} (\frac{m}{4}) r^{2} = \frac{1}{8} r^{2}$ $I_"tire" = 1/2(m/4)r^2 = 1/8r^2$

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

$4 (\frac{1}{8} m r^{2}) = \frac{1}{2} m r^{2}$ $4(1/8mr^2) = 1/2mr^2$

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.

$m g h = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}$ $mgh = 1/2mv^2 + 1/2Iomega^2$

We know the value of $I$ $I$ from part $a$ $a$ . The total mass of the cart is $m + 4 (\frac{m}{4}) = 2 m$ $m + 4(m/4) = 2m$

$2 m g h = \frac{1}{2} (2 m) v^{2} + \frac{1}{2} (\frac{1}{2} m r^{2}) ω^{2}$ $2mgh = 1/2(2m)v^2 + 1/2(1/2mr^2)omega^2$

$2 m g h = m v^{2} + \frac{1}{4} m r^{2} ω^{2}$ $2mgh = mv^2 + 1/4mr^2omega^2$

$2 g h = v^{2} + \frac{1}{4} r^{2} w^{2}$ $2gh = v^2 + 1/4r^2w^2$

We know that $ω = \frac{v}{r}$ $omega = v/r$ .

$2 g h = v^{2} + \frac{1}{4} r^{2} {(\frac{v}{r})}^{2}$ $2gh = v^2 + 1/4r^2(v/r)^2$

$2 g h = v^{2} + \frac{1}{4} v^{2}$ $2gh = v^2 + 1/4v^2$

$2 g h = \frac{5}{4} v^{2}$ $2gh = 5/4v^2$

$\frac{8}{5} g h = v^{2}$ $8/5gh = v^2$

$v = \sqrt{\frac{8}{5} g h}$ $v = sqrt(8/5gh)$

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

$\frac{1}{2} k x^{2}$ $1/2kx^2$

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

$\frac{1}{2} k x^{2} = \frac{5}{4} m v^{2}$ $1/2kx^2 = 5/4mv^2$

$\frac{1}{2} k x^{2} = \frac{5}{4} m (\frac{8}{5} g h)$ $1/2kx^2 = 5/4m(8/5gh)$

$k x^{2} = 4 m g h$ $kx^2 = 4mgh$

$x = 2 \sqrt{\frac{m g h}{k}}$ $x = 2sqrt((m gh)/k)$

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!

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Ap Physics C 2002 M2?

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