Why disc moves faster than a hoop while moving down an inclined plane?
Answers
Answer:
The hollow cylinder or ring or hoop has all its mass a a distance r away from its axis of rotation. The solid cylinder, however, has its mass distributed throughout. ... With its smaller "rotational mass", the solid sphere is easier to rotate so the solid sphere will roll down a hill faster.
Answer:
Explanation:
Without answering your question directly, let’s look at what factors apply to it. And the surprise is that neither the masses nor the radii of either object affect the result. What? How can that be, you ask? Read on.
Actually, the mass issue is no different than the question of which falls faster a heavy object or a lighter one in the absence of any other forces (buoyancy, air resistance, etc.). Most people know the answer without knowing why. That is, the mass of an object affects both the gravitational force on it and the objects inertia - so cancels out of the calculation and the acceleration is constant and independent of mass.
So let’s look at what goes into determining what determines how fast an object rolls without slipping down an incline. It turn out to be an easier problem to think in terms of energy rather than forces (although the result will be the same). That is, consider how the potential energy at the top of the incline becomes kinetic energy at the bottom if there are no frictional losses along the way.
Then all you have to do is determine how to write the kinetic energy of an object that is rolling on an incline. And whichever object has the higher speed at the bottom of the incline will also win the race.
So how do you write the kinetic energy of a rolling object whose speed is v? And what you need to remember is that the kinetic energy has two components because the object is both moving at speed v, and rotating about its axis.
The kinetic energy associated with the motion of the center of mass is just:
The kinetic energy associated with the rotation about its axis as it rolls down the incline is expressed similarly, but in terms of its moment of inertia and its angular velocity, that is:
And the total kinetic energy is the sum of the two.
But how does the angular velocity relate to the speed it has as it travels down the ramp? That depends on the radius of the rotating object, call it r. That is:
And how does the moment of inertia I relate to the mass of the object and the radius? And that is the part that most people might not remember. Why? Because it depends on how the mass is distributed about the axis. That is, the moment of inertia of a rotating object of mass M and radius r about the axis through its center is given by
where the parameter b depends on how the mass is distributed. That is, if all of the mass is at the same radius (as in a hoop), b=1; if the mass is uniformly distributed (as in a disk), b=1/2; if the object is a solid sphere, b=2/5; if it is a hollow sphere, b=3/5. That is, the moment of inertia of a rolling object of mass M and radius r depends on that parameter as well as M and r. Where did those values come from? It’s a calculus problem - so we’ll just accept the result here.
Now you have all of the pieces to answer the question.
Set the potential energy at the top of the incline equal to the total kinetic energy at the bottom - and solve for the speed at the bottom of the incline.
And here’s the surprise: Even though each piece depends on the mass M and the the rotational kinetic energy depends on the mass as well as the radius, when you put everything in terms of the total kinetic energy in terms of the final speed, both the mass and the radius cancel out! All you need to do is assemble the above pieces.
And the answer to the original question only depends on that parameter b - which depends on how the mass is distributed about the axis of rotation.