# Two Rolling logs

Suppose we have two logs rolling down a hill, one of gold and the other of wood; the acceleration for both will be equal, something which is unclear to me; I get that this may be due to their form, which is the same, but how come the mass of the objects doesn't matter?

• If you tied the two logs together with a string, making one even more massive object, would you expect the acceleration to be even greater? – WillO Jul 7 '16 at 17:33
• it would be the same – Bak1139 Jul 7 '16 at 17:37
• do you know why? – Bak1139 Jul 7 '16 at 17:37
• They are tied to one another so they make one object otherwise the string would rip off – Oussama Boussif Jul 7 '16 at 17:41
• i dont know Osama, what you say doesnt sound really right or on the subject to me – Bak1139 Jul 7 '16 at 17:45

All objects are accelerated by the same value in a gravitational potential in accordance with newton's laws of gravitation and motion

$F = mGM/r^2 = mg$ where G is newtons constant and M is mass of the earth.

Applying the equivalence of gravitational and inertial mass we have

$F = mg = ma => g = a$ (independent of mass)

where $a=9.8 m/s^2$ is the acceleration due to the earth.

For an inclined place the acceleration is reduced by the sin of the angle ($\theta$) the plane makes with the ground. I.e. $a = g*sin(\theta)$

Since mass cancels out of this equation then you will find that the 2 logs accelerate equally.

The deep concept here is that inertial and gravitational mass are equivalent.

Astronauts performed this type of experiment on the moon by dropping a feather and a er at roughly the same time and found they both landed at roughly the same time

Hope that helps!

:D

• although I do concur I get $a=2gsin(\theta)$ instead of $a=gsin(\theta)$ when taking into account the moment inertia of cylinders. – Bak1139 Jul 7 '16 at 18:43

$$F = M A$$

Simply, more mass doesn't mean more acceleration, but more force. The logs will accelerate at equal speeds, but the heavier one will carry more force with it.

So, the gold one would have much more force behind it, and would take more force to stop it. If you put up bowling pins down the hill to attempt to stop the logs (Not the best method of halting I know), the gold one would plow through them more than the wooden one.

I'll avoid using any formula in my answer, so you can forget about the mathematical side - maybe it will give you a deeper understanding.

Suppose you are holding two objects. One is a bowling ball and the other is a tennis ball. Now you release both from the same height - which ball do you think will hit the ground first?

If you haven't already seen such an experiment perhaps you might have guessed by intuition that the bowling ball would have hit first because it weighs more and therefore it must experience a greater force and a greater acceleration. The only wrong part of this statement is acceleration part. Yes, the bowling ball weighs more than the tennis ball, but both balls have the same acceleration if you do the experiment.. how can that be?

This is all about inertia. In layman terms, an objects inertia determines how hard it is to accelerate that object - to get it moving - when you apply a force on it. The bigger the inertia, the bigger force needed to get it moving.

The bowling ball has more inertia, which means it has more mass/inertial mass, so even though it experiences a greater gravitational force the ball is harder to accelerate.

If you look at the tennis ball has a relatively small mass meaning it is easy to accelerate it, but it experiences a smaller gravitational force because it is light. It is this relationship that gives equal accelerations. The inertial mass and the gravitational force "cancel" each other so the acceleration stays the same for all objects independent of their mass.

In your example with the logs, the golden log is heavier and so it experiences a greater gravitational force, but because of its large inertia it will be harder to move and the accelerations of the logs are equal.