# Why is only one end at rest when a rope is attached to a cube? [duplicate]

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I know there is something wrong in the way I think about physics generally, but I don't know what it is.

Suppose there is a cube with a hook on one face and a rope attached to that hook (see diagram). Now let's say there is some force $F$ pulling the cube from the face opposite the hook & rope. The hook will force the rope to accelerate in the direction of $F$. The rope in reaction would exert the force $-F$ on the cube through the hook. Now there are two forces acting on the cube: $F$ and $-F$, which would cancel, and the cube should be at rest. But what happens really is that the block is accelerating in the direction of $F$.

Now, where am I going wrong in this problem? Or in thinking about forces, in general?

## marked as duplicate by garyp, ACuriousMind♦, Gert, John Rennie newtonian-mechanics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 31 '15 at 8:38

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• How does the rope exert a force -F? – Jahan Claes Oct 30 '15 at 14:38
• If the end of the rope is tied to an immovable rigid object then the cube will not accelerate but if the rope is able to slip around the hooked point then it would accelerate. – Sathyaram Oct 30 '15 at 14:42
• @SathyaramGanapathy I don't see how It addresses the original question. – user66452 Nov 2 '15 at 7:02

## 2 Answers

What you are confusing here is that the force is given by $F=ma$, the mass times the acceleration. Both the cube and the rope will have the same acceleration if they are attached, but they will not have the same force because the masses are different. What you'll end up with is something like this for the acceleration of both objects:

$$a_{sys} = \frac{F_{app}}{m_{cube}+m_{rope}}$$

You can now use this acceleration to calculate the force on each object

$$F_{cube} = m_{cube}a_{sys}$$

$$F_{rope} = m_{rope}a_{sys}$$

From Newtons Laws you can deduce that

$$F_{cube} = F_{app} - F_{rope}$$

• Why the cube and the rope have to have the same acceleration if they are attached? – Samama Fahim Oct 31 '15 at 15:34
• Note that: We have considered the whole system $$m_{cube}+m_{rope}$$. When you are running you count acceleration of your hand same as that of your head and throughout the body. – user73555 Nov 2 '15 at 7:16

"But what happens really is that the block is accelerating in the direction of F": that is only before the hook starts to act. Once they are in equilibrium, and as you said before, the cube will be at rest and the acceleration will be zero. It is easy to see this: $ma=F-F=0$ so the acceleration (or the mass, which I assume is not the case) must be zero.