Rank the four objects from fastest to slowest

[JessicaHelena]

Homework Statement

Rank the four objects (1kg solid sphere, 1kg hollow sphere, 2kg solid sphere and 1kg hoop) from fastest down the ramp to slowest. (Please see the attached screenshot for more details.)

Homework Equations

KE_rot = 1/2Iw^2 (where omega = w)

The Attempt at a Solution

Since we are given the values of I for each object, I was able to calculate the KE's of each:
the solid spheres had KE of 1/5mv^2; the hollow sphere had a KE of 1/5 mv^2, and the hoop had 1/2mv^2 (for KE).

Since KE = W = Fd = mad, a = KE/(md). Since all KE's had m's in their equations, the m's can be canceled out. This will give us a, and since from v^2 = v_0^2 + 2ax where x is the same for all and v_0 = 0 for all (and thus v depends solely on the a). Then I had a_D = v^2/(2d), a_B = v^2/(3d), and a_A = a_C = v^2/(5d).

That tells me that then the order from fastest to slowest should be
D > B > A = C.

The answer, however, is the exact opposite: A=C > B > D.

Could someone please help me see why asap?

 

[stockzahn]

Homework Statement

Rank the four objects (1kg solid sphere, 1kg hollow sphere, 2kg solid sphere and 1kg hoop) from fastest down the ramp to slowest. (Please see the attached screenshot for more details.)

Homework Equations

KE_rot = 1/2Iw^2 (where omega = w)

The Attempt at a Solution

Since we are given the values of I for each object, I was able to calculate the KE's of each:
the solid spheres had KE of 1/5mv^2; the hollow sphere had a KE of 1/5 mv^2, and the hoop had 1/2mv^2 (for KE).

Since KE = W = Fd = mad, a = KE/(md). Since all KE's had m's in their equations, the m's can be canceled out. This will give us a, and since from v^2 = v_0^2 + 2ax where x is the same for all and v_0 = 0 for all (and thus v depends solely on the a). Then I had a_D = v^2/(2d), a_B = v^2/(3d), and a_A = a_C = v^2/(5d).

That tells me that then the order from fastest to slowest should be
D > B > A = C.

The answer, however, is the exact opposite: A=C > B > D.

Could someone please help me see why asap?

The kinetic energy of the objects increases, when loosing potential energy: $mgh =\frac{mv^2}{2} + I \omega^2/2$. The value of $I$ is proportional to the distance between the mass and the rotational axis (to the power of 2), therefore the larger $I$, the lower the rotational speed (i.e. the slower the object). Just imagine two spheres of the same mass, but different size and compare the sum of their translational and roatational energies ($\frac{mv^2}{2}+I \omega^2/2 = \frac{m(r\omega)^2}{2}+I \omega^2/2$).

 

[JessicaHelena]

@stockzahn

Okay, so I just calculated the I's for each. (Assuming R's are all the same)
A: 2/5 R^2
B: 2/3 R^2
C: 4/5 R^2
D: R^2

Now I get why D is last. But A != C here... Could you help me out again?

 

[haruspex]

@stockzahn

Okay, so I just calculated the I's for each. (Assuming R's are all the same)
A: 2/5 R^2
B: 2/3 R^2
C: 4/5 R^2
D: R^2

Now I get why D is last. But A != C here... Could you help me out again?

The moment of inertia is not the time taken. What equation can you write relating I, M and R to the linear acceleration?