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All other things being equal, if a heavier object will roll at a higher speed down hill than a lighter one, because it's heavier with more mass, then how is it that dropped objects fall at the same speed regardless of their weight?

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  • $\begingroup$ A reminder that comments are meant for requests for clarification and suggestions for improvement. I deleted some off-topic comments. P.S. I also edited the title and deleted comments calling for its improvement (although I would have hoped someone else would have fixed it already). $\endgroup$
    – David Z
    Apr 9, 2016 at 5:19
  • $\begingroup$ Concerning free fall, see physics.stackexchange.com/q/11321/2451 and links therein. $\endgroup$
    – Qmechanic
    Apr 9, 2016 at 12:34

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if a heavier object will roll at a higher speed down hill

Free fall and rolling are two different behaviors of objects. It is correct that for free fall all objects get the same acceleration ( minus friction and drag) but free fall is not the same as rolling. For going down a hill free fall can be compared to sliding, as was pointed out in the comments to the question. Rolling is another story because angular momentum comes in, and the moment of inertia:

Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion. It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be specified with respect to a chosen axis of rotation. For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis, I = mr2. That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point masses.

The shape of a body enter into the problem.

rolling downhill

So it is not a problem of mass, but of center of mass, when rolling under gravity. Of course friction, which depends on the weight, and drag will play a role making the outcome more complicated in real life.

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All other things being equal, if a heavier object will roll at a higher speed down hill than a lighter one

With the qualification "All other things being equal" your statement is not correct.

The falling acceleration is the same because doubling the mass of an object doubles the force causing the acceleration (the object's weight) which means that when Newton's second law id used $F=ma \Rightarrow mg = ma \Rightarrow a = g$ the acceleration is independent of the mass.
Another way of looking at this is that the loss in gravitational potential energy $mg\Delta h$ is equal to the gain in kinetic energy $\frac 1 2 m v^2$ and again the masses cancel out.

The same is true for rolling motion where you have torques and forces which depend linearly on the mass of the object and moments of inertia (and masses) which also depend linear on the mass of the object (remember "all other things being equal") and so the accelerations do not depend on the mass as mass is a common factor on both sides of the force/torque = mass/moment of inertia $\times$ linear/angular acceleration equations.

Once other factors are changed introduced then there will might well be a difference.

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The reason for the two behavior of the same objects accounts for the forces acting on it. When you drop a 5 kg body and another 25 tonne body vertically downwards, they will fall under the influence of gravity alone and both will fall with the same acceleration g (free fall). In such a case the weight of the body is zero. You know that fact, which you may have studied from the free fall of a lift illustration. The force acting on the body that pulls the body down is the force of gravity. We know that for sure. The force of gravity is the weight of the body. Then why the weight of the body is zero under free fall? It's because under free fall, the force of gravity and the normal force are acting equal and opposite, which makes the resultant force zero.
Now when you consider a slope, a body with greater mass will win the game. In such a case, for example, take the case of a 5kg body and a 20kg body. Since the objects are placed at a slope, the force of gravity (weight) of the body acting vertically downwards from the centre of gravity, and the normal force which acts in the direction perpendicular to the body, away from it are in different directions. So they will not cancel each other. There will be a resultant force which will be proportional to the mass of the object. Hence an object with greater mass feels greater force than the other one. So even if the slope is same for both objects, a massive object moves faster through the slope than a less mass object.

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In physics we recognize two different kinds of mass: inertial and gravitational. Inertial mass tells us how much an object resists a change in motion - or how much force is needed to effect an acceleration.

Gravitational mass describes how much attraction (due to gravity) an object experiences as a result of gravity.

Now despite very careful experiments, it has not been possible to show there is a difference between the two - which is another way of saying that when an object is twice as heavy (twice the force of gravity), then it requires twice the force to get the same acceleration.

And that means that, absent effects of friction or drag, objects of different weight fall with the same acceleration.

Note that if you have object that ROLLS, then other factors come into play - specifically the moment of inertia which depends on the size of the object and the weight distribution.

Specifically, the acceleration for a solid sphere of radius $R$ will be smaller when $R$ is larger (so a "big" ball will roll more slowly than a "little" ball - see for example this earlier answer) but there is no effect of the mass ("all other things being the same"). And again, if a heavy object is sitting on a "real" hill (for example a sandy dune) it might sink into the sand and not move, while a light object (think beach ball) would actually roll down the hill.

But I don't believe your question was about that...

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  • $\begingroup$ the question is asking about rolling objects $\endgroup$
    – anna v
    Apr 9, 2016 at 4:29
  • $\begingroup$ Even if the types of mass were different it would not matter in this example because in both cases one type of mass occurs on one side of the equation of motion and the other type of mass occurs on the other side of the equation of motion. $\endgroup$
    – Farcher
    Apr 9, 2016 at 7:49
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No, we don't know that for sure about gravity. One object cannot have 2 different masses: The force that makes things roll, and fall, is the same force; it is gravity. This is what my question is about. We don't know for sure that Newton was correct: The fact that something heavier rolls at a higher speed shows us that Newton's theory of gravity is wrong. Because aside from friction and drag and things like that, the main force is gravity. Lets say that we have 2 soapbox derby cars at the top of a hill-with all other things being equal, on a smooth street, and lets make it very steep- almost like falling, and we put an elephant-assuming that it will fit, in one car, and a mouse in the other car, and let them go - what will happen? I think the elephant's car will reach the bottom first. Doesn't this prove Newton wrong?

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