Please help (correct answer only)

Question

Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.35 m high. Assume it starts from rest and rolls without slipping. Express your answer with the appropriate units.

Answer 1

We will use the law of conservation of mechanical energy. At the top the energy of the cylinder is E_1, and at the bottom of incline - it is E₂. Using the law, we can write

E₁ = E₂ ,

where

E₁ = mgh - is a potential energy of the cylinder,

E₂ = E_tr + E_rot - the total kinetic energy of the cylinder, that includes

the translational energy E_tr = 1/2 mv² and

rotational energy E_rot = 1/2 Iω^2.

Here

I = 1/2mR² - moment of inertia of the cylinder,

ω = v / R - angular velocity of the cylinder.

Now we have

mgh = 1/2 mv² + 1/4 mR²⁽ v²/ R²) = (3/4)mv²

From this equation translational velocity

v = √ 4gh / 3 = 2√gh / 3.

Plugin in the values of g and h, we finally have

v = 2√9.81 m/s² ·7.35 m / 3 = 9.80 m/s