# Problem: You have a pumpkin of mass M and radius R. The pumpkin has the shape of a sphere, but it is not uniform inside so you do not know its moment of inertia. In order to determine the moment of inertia, you decide to roll the pumpkin down an incline that makes an angle θ with the horizontal. The pumpkin starts from rest and rolls without slipping. When it has descended a vertical height H, it has acquired a speed of v = √(5gH/4). Find the moment of inertia I of the pumpkin in terms of M and R.

###### FREE Expert Solution

Angular velocity:

$\overline{){\mathbf{\omega }}{\mathbf{=}}\frac{\mathbf{v}}{\mathbf{r}}}$

The total kinetic energy of a rolling object:

$\overline{){\mathbf{KE}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbit{m}}{{\mathbit{v}}}^{{\mathbf{2}}}{\mathbf{+}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbit{I}}{{\mathbit{\omega }}}^{{\mathbf{2}}}}$

Energy conservation:

$\overline{){{\mathbf{U}}}_{{\mathbf{0}}}{\mathbf{+}}{{\mathbf{KE}}}_{{\mathbf{0}}}{\mathbf{=}}{{\mathbf{U}}}_{{\mathbf{f}}}{\mathbf{+}}{{\mathbf{KE}}}_{{\mathbf{f}}}}$

Potential energy:

$\overline{){\mathbf{U}}{\mathbf{=}}{\mathbf{mgh}}}$

U0 = MgH

Uf = 0 since h = 0 m

KE0 = 0 - since V = 0 m/s

KEf = (1/2)Mv2 + (1/2)Iω2 = (1/2)Mv2 + (1/2)Iv2/R2

###### Problem Details

You have a pumpkin of mass M and radius R. The pumpkin has the shape of a sphere, but it is not uniform inside so you do not know its moment of inertia. In order to determine the moment of inertia, you decide to roll the pumpkin down an incline that makes an angle θ with the horizontal. The pumpkin starts from rest and rolls without slipping. When it has descended a vertical height H, it has acquired a speed of v = √(5gH/4).

Find the moment of inertia I of the pumpkin in terms of M and R.

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Conservation of Energy with Rotation concept. You can view video lessons to learn Conservation of Energy with Rotation. Or if you need more Conservation of Energy with Rotation practice, you can also practice Conservation of Energy with Rotation practice problems.