Question

II) A uniform solid sphere of radius r0 = 24.5 cm and mass m = 1.60 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. Calculate its translational and rotational speeds when it reaches the bottom. Avoid putting in numbers until the end so you can answer.

Accepted Answer

Start by identifying the forces acting on the sphere. The sphere rolls without slipping, so both translational and rotational motion are involved. The gravitational force causes the sphere to accelerate down the incline, and the frictional force ensures rolling without slipping. Apply the principle of conservation of energy. The total mechanical energy at the top of the incline (potential energy) is converted into both translational kinetic energy and rotational kinetic energy at the bottom. Write the energy conservation equation: $$ m g h = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 $$, where $$ h  is $$the height of the incline, $$ v  is $$the translational speed, $$ I  is $$the moment of inertia of the sphere, and $$ \omega  is $$the angular speed. Relate the height $$ h  to $$the length of the incline $$ L $$ and the angle $$ 	heta $$ using trigonometry: $$ h = L \sin(	heta) . $$Substitute this expression for $$ h $$ into the energy equation. For a solid sphere, the moment of inertia about its center is $$ I = \frac{2}{5} m r_0^2 . $$Also, since the sphere rolls without slipping, the condition $$ v = r_0 \omega $$ holds. Use this relationship to eliminate $$ \omega  in $$the energy equation by substituting $$ \omega = \frac{v}{r_0} . $$Simplify the energy equation to solve for the translational speed $$ v . $$Once $$ v  is $$determined, use the rolling condition $$ \omega = \frac{v}{r_0}  to $$find the rotational speed $$ \omega . $$Finally, substitute the given values ($$ r_0 = 0.245  m$$, $$ m = 1.60  kg$$, $$ L = 10.0  m$$, $$ 	heta = 30.0° ) to $$calculate the numerical results for $$ v $$ and $$ \omega .$$

II) A uniform solid sphere of radius r₀ = 24.5 cm and mass m = 1.60 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. Calculate its translational and rotational speeds when it reaches the bottom. Avoid putting in numbers until the end so you can answer.

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Key Concepts

Conservation of Energy

Moment of Inertia

Rolling Without Slipping