in a pickup game of dorm shuffleboard, students crazed by final exams use a broom to propel a calculus book along the dorm hallw
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Answer:
vcyl / vsph = 1.05
Explanation:
- The kinetic energy of a rolling object can be expressed as the sum of a translational kinetic energy plus a rotational kinetic energy.
- The traslational part can be written as follows:
- The rotational part can be expressed as follows:
- where I = moment of Inertia regarding the axis of rotation.
- ω = angular speed of the rotating object.
- If the object has a radius R, and it rolls without slipping, there is a fixed relationship between the linear and angular speed, as follows:
- For a solid cylinder, I = M*R²/2 (4)
- Replacing (3) and (4) in (2), we get:
- Adding (5) and (1), we get the total kinetic energy for the solid cylinder, as follows:
- Repeating the same steps for the spherical shell:
- Since we know that both masses are equal each other, we can simplify (6) and (9), cancelling both masses out.
- And since we also know that both objects have the same kinetic energy, this means that (6) are (9) are equal each other.
- Rearranging, and taking square roots on both sides, we get:
- This means that the solid cylinder is 5% faster than the spherical shell, which is due to the larger moment of inertia for the shell.