Grigoriy S. answered • 12/03/21

AP Physics / Math Expert Teacher With 40 Years of Proven Success

**A. **The moment of inertia of the sphere *I = ⅖MR*^{2}.

To get the numeric value you put in this formula M = 4.30 kg and R = .200 m

**B. **To find linear velocity of the sphere ** v**, we will apply the law of conservation of energy.

On top of incline the sphere has only potential energy E_{1} = Mgh, where h = 5.30 m.

On the bottom it has translational and rotational kinetic energy, hence we can write

E_{2 } = E_{t} + E_{r}

Translational energy E_{t} = ½ Mv^{2} and rotational E_{r} = ½ I∙ω^{2} (here ω is angular velocity of the sphere)

We know that linear and angular velocities connected to each other by the formula

v = ω∙R ,

So

ω = v/R.

The total energy of the sphere on the bottom can be written as follows:

E_{2} = ½ Mv^{2 } + ½ ∙ ⅖MR^{2}∙(v/R)^{2}

After simplification we obtain

E_{2} = ½ Mv^{2 } + ⅕Mv^{2} = .7Mv^{2}

But E_{1 }=_{ }E_{2}

Or Mgh = .7Mv^{2}

Then for velocity we have

* v = √ (gh/0.7)*

To get an answer you need to substitute known values in the formula above.

**C. **At the bottom the angular speed

*ω = v/R *

Just put the known numbers.

**D. **The angular acceleration α and linear acceleration a related to each other by the formula

*α = a/R*

To find acceleration of the body on incline with the angle θ when friction is negligible, you need to use the formula

a = g∙sin θ

Finally you get

*α = g∙sin θ / R*

After plugging in the numbers, you can get the numerical result.

Hope you understood the problem!