Mr.p: Good morning. Let's determine which of these three objects reaches the bottom of this incline first.
Bo, please read the problem and Billy, please translate.
"Flipping Physics" Intro
Bo: A hollow sphere, solid sphere, and thin hoop
are simultaneously released from rest at the top of an incline. Which will reach the bottom first?
Assume all objects are of uniform density.
Billy: Okay. There are three objects, a hollow sphere, a solid sphere and a thin hoop.
Their initial velocities all equal zero. Each object has a constant density.
And we are trying to determine which one reaches the bottom of the incline first. That's it, right?
Mr.p: Billy, that is it. Bobby, please begin solving the problem.
Bobby: Well, we know mechanical energy is conserved when an object is rolling without slipping.
Billy: Actually, the problem does not say the objects will not slide while rolling down the incline.
Mr.p: Oh, sorry. You may assume they roll without slipping.
Bobby: Right. So, for conservation of mechanical energy, set the initial point at the top of the incline,
the final point at the bottom of the incline,
and the zero line at the height of the center of mass of the objects at the final point.
Initially, the only type of mechanical energy the objects have is gravitational potential energy
and the only type they have at the end is kinetic energy.
Although they have both translational and rotational final kinetic energies.
Bo: Did we not show last time that the acceleration of those objects rolling without slipping down the incline
only depends on acceleration due to gravity, incline angle and rotational inertia?
Billy: That's close. Actually, it does not. The acceleration does not have to do with the rotational inertia.
We showed that the acceleration down the incline
depends on the fraction in front of the mass times radius squared equation.
Bo: Close enough. Billy: Not for physics.
Right. We have to be careful here.
The acceleration down the incline depends on the acceleration due to gravity, the incline angle, and the
fraction, X, in front of the mass times radius squared equation for the rotational inertia of each object.
The acceleration due to gravity and incline angle are the same for all three objects,
so the only variable which is different for each object is the fraction, X.
Bo, what are the equations for translational and rotational kinetic energies?
Translational kinetic energy equals one half mass times velocity squared
and rotational kinetic energy equals one half rotational inertia times angular velocity squared.
Okay. So, the lower the fraction for the rotational inertia equation,
the lower the percentage of the initial gravitational potential energy
which will be converted to rotational kinetic energy final.
The more energy will be left for translational kinetic energy final,
the faster the object will be going so it will get there first.
Bobby: No. Sorry. That was too fast. Billy: Yeah. I'm sorry, what?
Bo: Okay, look at the conservation of mechanical energy equation.
Initially each object has a finite amount of mechanical energy
in the form of initial gravitational potential energy
As each object rolls down the incline, the gravitational potential energy decreases and is converted to both
translational and rotational kinetic energies.
The smaller the amount that needs to be converted to rotational kinetic energy,
the more that is left over for translational kinetic energy.
Bobby: Okay. So,
a smaller rotational inertia means a smaller percentage necessary for rotational kinetic energy
and more left over for translational kinetic energy and a larger final velocity,
which means it takes less time to roll without slipping down the incline.
Billy: Oh, I guess we need to look up the rotational inertia equations for those three objects then.
Mr.p: Actually Billy, rather than looking up those rotational inertia equations or having them memorized,
I think it is better to use the rotational inertia equation of a system of particles
to estimate their relative fractions in the rotational inertia equations. Billy could you please do that?
Billy: Well, the rotational inertia of a system of particles equals
the sum of mass times the square of the distance from the axis of rotation
for each individual piece which makes up the object.
In other words, if the mass of the object is concentrated near the axis of rotation,
its rotational inertia fraction will be smaller.
If the mass of the object is concentrated farther from its axis of rotation,
it's rotational inertia fraction will be larger.
So, the largest fraction will be the thin hoop because all of its mass is far from its axis of rotation.
The smallest fraction of the three will be the solid sphere because
more of its mass is close to its axis of rotation than the other two.
and the hollow sphere's fraction is somewhere in the middle.
That means the solid sphere will have the most translational kinetic energy left over
and have the fastest final velocity followed by the hollow sphere and then the thin hoop.
So, the solid sphere is the first to the bottom of the ramp, then the hollow sphere, then the thin ring.
Mr.p: And there you have it.
The object with the smallest fraction for the rotational inertia equation, the solid sphere,
has the largest percentage amount of translational kinetic energy
and therefore is moving the fastest and therefore reaches the bottom first.
The reverse is also true, the object with the largest fraction for the rotational inertia equation,
the thin hoop, has the smallest percentage of translational kinetic energy
and therefore is moving the slowest and reaches the bottom last.
Thank you very much for learning with me today. I enjoyed learning with you.