Three weights and two pulley problem

[rudransh verma]

Homework Statement

Three masses of 2 kg , 5kg,3 kg are connected to each other with threads and are placed on a frictionless table as shown in fig. What is the acceleration of the system? (g=10m/s^2)

Relevant Equations

$$F=ma$$

$Net force=50-T2+T2-T1+T1-20=10a$
$a=3 m/s^2$
I want to ask when we do this way rather than taking individual masses we can’t decide the direction of the system as we can when they are taken individually. So is it correct to just leave $a$ as it is and solve ?

 

[Orodruin]

Net force on what? What is your free body diagram?

 

[Lnewqban]

Homework Statement:: Three masses of 2 kg , 5kg,3 kg are connected to each other with threads and are placed on a frictionless table as shown in fig. What is the acceleration of the system? (g=10m/s^2)
Relevant Equations:: $$F=ma$$

$Net force=50-T2+T2-T1+T1-20=10a$
$a=3 m/s^2$
I want to ask when we do this way rather than taking individual masses we can’t decide the direction of the system as we can when they are taken individually. So is it correct to just leave $a$ as it is and solve ?

For the same reasons that we have previously discussed, you can leave all those T's out of any calculation, because those are simple internal forces that exist to link or connect the three bodies as a flexible one of total mass equal to m1+m2+m3.
That complication is not necessary in this case, it would only use time.

Just analice what is the direction in which each body is free to move under the action of gravity, which is the only external source of energy in this case.
Then, what is the magnitude of the external forces acting on that direction for each body.
Add all those forces up and consider that all the inter-linked bodies must move together (same speed and acceleration), if they move.

 

[rudransh verma]

Then, what is the magnitude of the external forces acting on that direction for each body.
Add all those forces up and consider that all the inter-linked bodies must move together (same speed and acceleration), if they move.

Can we determine the direction of acceleration of the system somehow?

 

[Lnewqban]

Can we determine the direction of acceleration of the system somehow?

Not only the direction, but the magnitude.
One resultant force in the direction of free movement induces an acceleration in the same direction, how much depends on the total mass to be accelerated.

 

[rudransh verma]

Not only the direction, but the magnitude.
One resultant force in the direction of free movement induces an acceleration in the same direction, how much depends on the total mass to be accelerated.

What is the direction of acceleration of the system here in this case?

 

[Steve4Physics]

I want to ask when we do this way rather than taking individual masses we can’t decide the direction of the system

Yes we can decide. That's exactly what you have done!

You have chosen to take the weight of the 5kg mass as being in the positive sense. That's why your equation contains +50 and -20 rather than -50 and +20.

If you had used -50 and +20 you would have obtained a=-3m/s².

In my opinion, for a simple problem like this, common sense should prevail. It should be obvious that the difference in weights will cause acceleration and that the sense of acceleration is such that the heavier weight moves downwards.

In more complicated problems (e.g. with slopes) the sense of acceleration may not be clear. Consequently if friction is present, you may not know in which direction it acts. (E.g. a recent problem you posted.)

If you don't have friction you can probably get away with guessing. The acceleration will be negative if you guessed wrongly.

With friction present a preliminary calculation may be needed to determine the sense of acceleration.

If you are looking for a general method, covering all possible scenarios, where you don't have to think about the sense of acceleration, you may be disappointed!

 

[Lnewqban]

What is the direction of acceleration of the system here in this case?

After reading the above post of @Steve4Physics, what do you think?

 

[rudransh verma]

After reading the above post of @Steve4Physics, what do you think?

Well it is not so obvious in which direction body accelerates always. Sometimes we need calculation. Sometimes like here we cannot say whether the system accelerates up or down, I think

 

[Lnewqban]

Well it is not so obvious in which direction body accelerates always. Sometimes we need calculation. Sometimes like here we cannot say whether the system accelerates up or down, I think

Not in this case.
At a glance, we can see that there is no balance of forces here; therefore, there is a net or resultant force (weight of 5 kg minus weight of 2 kg).
Once we have a net mass (m1+m2+m3) and a net force, acceleration naturally happens.
Being acceleration the way in which we describe how velocities change respect to time, it must happen in the same direction in which the system is moving while its velocity is increasing.

The direction of gravity acceleration is straight down.
If your body free falls, its downwards velocity increases each second (it accelerates).
If you vertically jump up, your upwards velocity decreases each second until you reach the highest point and stop (your body accelerates in a negative way).

 

[rudransh verma]

happen in the same direction in which the system is moving while its velocity is increasing.

So in which direction is the velocity of the system increasing and moving. Is it downwards ? Is it upwards? Is it sideways?
We cannot say here.

 

[Lnewqban]

So in which direction is the velocity of the system increasing and moving. Is it downwards ? Is it upwards? Is it sideways?
We cannot say here.

It does not matter much, really.
Up and down are involved in this problem only because they are using gravity as source of constant force doing the work of increasing the velocity of the three masses.
The source of force can be any other one in other problems; then, up and down would not mean anything.
We could say that the mass of 5 kg is leading (movement-wise), and that the 3 kg and 2 kg masses are following at the same rates of speed and acceleration (as those are all linked together).

 

[Steve4Physics]

So in which direction is the velocity of the system increasing and moving. Is it downwards ? Is it upwards? Is it sideways?
We cannot say here.

Agreed.

I'll add to what @Lnewqban has said. The original question is 'informally' written. The question should ask for the magnitude of the acceleration. (It's not an uncommon mistake.)

But from the list of possible answers, this should be clear.

 

[kuruman]

So in which direction is the velocity of the system increasing and moving. Is it downwards ? Is it upwards? Is it sideways?
We cannot say here.

Sure we can just by looking at the drawing. Suppose you remove the 3 kg mass from the middle and you have just the 5 kg mass on the right and the the 2 kg mass on the left. Can the smaller 2 kg mass pull the 5 kg mass up whilst it accelerates moving down? Of course not. It's the other way around.

Now put the 3 kg mass back on the table and reconnect it with the string to the other two. Is the acceleration going to reverse direction considering that the weight of the added mass is entirely supported by the table? Of course not.