Question

A solid sphere of radius R is placed at a height of 30 cm on a 15° slope. It is released and rolls, without slipping, to the bottom. From what height should a circular hoop of radius R be released on the same slope in order to equal the sphere's speed at the bottom?

Accepted Answer

Step 1: Begin by analyzing the energy conservation principle for both the solid sphere and the circular hoop. The total mechanical energy at the top of the slope is purely potential energy, and at the bottom, it is converted into both translational kinetic energy and rotational kinetic energy. Step 2: Write the energy conservation equation for the solid sphere. The potential energy at the top is $$ mgh $$, and the kinetic energy at the bottom is $$ \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 $$, where $$ I  is $$the moment of inertia of the sphere and $$ \omega  is $$the angular velocity. For a solid sphere, $$ I = \frac{2}{5}mR^2 $$ and $$ \omega = \frac{v}{R} . $$Substitute these values into the equation. Step 3: Write the energy conservation equation for the circular hoop. The potential energy at the top is $$ mgh $$, and the kinetic energy at the bottom is $$ \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 . $$For a hoop, $$ I = mR^2 $$ and $$ \omega = \frac{v}{R} . $$Substitute these values into the equation. Step 4: Compare the final speeds of the sphere and the hoop at the bottom of the slope. The speed of the sphere is determined by its energy conservation equation, and the speed of the hoop is determined by its own energy conservation equation. Set the speeds equal to each other to find the height $$ h $$ from which the hoop must be released. Step 5: Solve the resulting equation for $$ h  in $$terms of the given height of the sphere (30 cm). This involves algebraic manipulation to isolate $$ h $$ for the hoop, ensuring that the speeds match at the bottom of the slope.

A solid sphere of radius R is placed at a height of 30 cm on a 15° slope. It is released and rolls, without slipping, to the bottom. From what height should a circular hoop of radius R be released on the same slope in order to equal the sphere's speed at the bottom?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Conservation of Energy

Moment of Inertia

Rolling Without Slipping