Which reaches the bottom of the incline first? Which has the greater speed there? And greater Kinetic Energy?

Question

Two solid spheres simultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which sphere reaches the bottom of the incline first? Which has the greater speed there? Which has the greater kinetic energy at the bottom?

Answer 1

If we ignore air resistence, then both the spheres will reach the bottom of the incline at the same time.

Both will have the same velocity when they reach the bottom of the incline.

But the sphere with greater mass will have more kinetic energy since it has more mass.

Answer 2

Neglecting air friction and surface friction, the smaller diameter will reach the bottom faster because of a beautiful physics principle called rotational intertia.

As the balls proceed down the incline, the same potential energy is being converted into kinetic energy for both, but the larger sphere converts a larger proportion of the initial potential energy into rotational energy than the smaller one.

At the bottom, if both had the same mass, then both would have the same kinetic energy with the larger sphere having more energy in rotational motion. A collision with a frictionless wall would deliver the same impulse and energy, but while the larger would be spinning slower on our frictionless surfaces, it would have more rotational energy.

With twice the mass, the larger sphere would have begun with twice the potential energy and would have converted it at the bottom to twice the kinetic energy (much of it being in the form of rotational energy). With greater rotational intertia, it would clearly accelerate slower down the incline.

If there were no rolling in a frictionless environment, there would be no difference in speed, and the larger sphere would still have twice the kinetic energy by virtue of its having twice the mass, but this time it would all be in forward motion, as there would be no rotation.

I would like to note that doubling the radius does not double the volume (it's actually a factor of 8 times larger), therefore if the larger sphere has twice the mass, its density should be much less than the smaller one. Volumes of spheres increase by the cube of the relative increase in radius. Density is mass divided by volume, therefore m/r^3>(2*m)/(2*r)^3 thus making the larger sphere 1/4 the density. It would also be twice the cross-sectional area of the smaller one. Both of these factors would create significantly more air friction and rolling friction for the larger sphere.

Answer 3

All the above answers were given with the best intentions, but I actually sat down and did the math. The speed at the bottom of the ramp is a function of neither mass OR radius. They both arrive simultaneously at the same speed, so the larger one has KE equal to twice the smaller one. The expression I came up with is v^2 = gh/.7

The only thing of note in the answer here is that a non-rolling sphere will have 18.3% more speed than either of the rolling ones.

Answer 4

Since they are rolling and not sliding or falling, some of the energy that would have gone into the velocity of the spheres down the ramp instead goes into rotation of the spheres. Since the larger sphere has a higher moment of inertia, it would take more energy to get it rotating. Since it is larger in diameter, it does not have to rotate as fast to roll at the same speed as the small one, so it isn't clear to me yet if it accelerates at the same rate as the smaller sphere or not, but it might. In any case, the larger sphere has more kinetic energy because it is heavier.

OK, I looked up moment of inertia on wikipedia and it is 2/5MR^2 for a sphere. So a sphere twice the mass and twice the radius has 16 times the moment of inertia. But it has twice the circumference, so it has to rotate at rate only half that of the smaller sphere to roll at the same speed down the ramp. Half the speed with 16 times the moment of inertia means it will use up 8 times as much of its energy in rotation as the small sphere does, so the larger sphere will reach the bottom of the ramp after the small one.

Answer 5

since kinetic energy is 1\2mv2 where m=mass of an object,,,v=velocity of the object

now solid whose radius is twice that will reach the bottom at first.... because it mass is greater 2m than that of the object whose mass is 1m so increasing the kinetic energy ......

and kinetic energy are connecting to one an other

Answer 6

RE: Rotational, Translational Kinetic ability? A sphere of radius 20.0cm and mass one million.80kg starts off from relax and rolls without slipping down a 30.0(ranges) incline it extremely is 10.0m long. (a) Calculate its translational and rotational speeds whilst it reaches the backside. (b) what's the ratio of translational to rotational KE on the backside? (c) Do your...