Minimum height that the marble must start from to make it around the loop.

Answer to Problem 69QAP
Explanation of Solution
Given info:
Spherical marble of,
Radius
Mass
Loop-the-loop track,
Radius
Spherical marble rolls without slipping down the track.
Marble starts from rest and just barely clears the loop to emerge on the other side of the track.
Formula used:
Let's name the minimum height that the marble must start from to make it around the loop
As
Let's name the
Let's name the linear speed of spherical marble at the top of the track as
Let's name the moment of inertia of spherical marble as
Conservation of mechanical energy:
Kinetic energy for an object that undergoes both translation and rotation:
Condition for rolling without slipping:
Equilibrium of vertical forces of spherical marble at the top of the track,
Centrifugal power
Calculation:
Let's consider the motion of sphere,
Initially the spherical marble is at rest with zero kinetic energy, so
The initial gravitational potential energy is
Final gravitational potential energy is
Conservation of mechanical energy:
Let's consider the kinetic energy
Kinetic energy is part translational and part rotational. We can use
In terms of
Using
Kinetic energy for an object that undergoes both translation and rotation:
Condition for rolling without slipping:
Substitute into kinetic energy equation:
From the general knowledge we know that moment of inertia of a sphere is
So, let's substitute the
Since
Marble starts from rest and just barely clears the loop to emerge on the other side of the track.
So,
Equilibrium of vertical forces of spherical marble at the top of the track,
Centrifugal power
Substituting
Let's substitute the
Conclusion:
Thus, minimum height that the marble must start from to make it around the loop is
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