Friction Problem involving 3 blocks

[zorro]

Homework Statement

Find the accelerations of the blocks shown in the figure when a force of 10N acts on the lower block.

The Attempt at a Solution

For the 3kg block, by drawing F.B.D. we get
3a = (0.3)x5x10
a=5m/s2

Similarly for the block of mass 7 kg, we get
7a = 10 - 5x0.3x10 = -5!

I don't understand what is my mistake.
When 10N force acts on the lower block, its motion is opposed by the friction due to the 3 kg block. The maximum value of this friction is 15N. So I don't think that the lower block or any other block will move.

But the answer is different.

 

[ehild]

It is possible that two blocks do not move relatively to each other, or all blocks move together. But there is a net external force: The CM of the three blocks will move anyway.

ehild

 

[zorro]

It is possible that two blocks do not move relatively to each other, or all blocks move together. But there is a net external force: The CM of the three blocks will move anyway.

ehild

How will you prove that the two blocks (rather 3) move together?
Actually this is a part of the problem...in the first 2 parts we have to find accelerations when the same force acts on 2 kg and 3 kg blocks respectively.

 

[ehild]

The interaction between to joint blocks need not to be equal to the coefficient of friction times the normal force. If two blocks do not move relative to each other then the friction is static. But there is motion with respect to the ground because the centre of mass of the whole system has to move with the acceleration F/12. Investigate all possibilities, if the resultant of the forces on a block is enough to accelerate the block in the forward direction, but with less or equal acceleration as the block just below it. It is clear that the 7 kg block can not accelerate backwards. So the force of friction is less than the kinetic friction, that is the friction is static, the 7 kg mass and the 3 kg one move together. Find out what happens with the 2 kg mass.

 

[zorro]

The interaction between to joint blocks need not to be equal to the coefficient of friction times the normal force. If two blocks do not move relative to each other then the friction is static. But there is motion with respect to the ground because the centre of mass of the whole system has to move with the acceleration F/12. Investigate all possibilities, if the resultant of the forces on a block is enough to accelerate the block in the forward direction, but with less or equal acceleration as the block just below it. It is clear that the 7 kg block can not accelerate backwards. So the force of friction is less than the kinetic friction, that is the friction is static, the 7 kg mass and the 3 kg one move together. Find out what happens with the 2 kg mass.

Well then there must be some static frictional force acting on the 3 kg block which causes it to move with the 7 kg block. What is its value then?
If 3 kg block moves with the 7 kg block, the 2kg block ought to move together.

 

[ehild]

Well then there must be some static frictional force acting on the 3 kg block which causes it to move with the 7 kg block. What is its value then?

Just the proper value to make the 3 kg block move with the same acceleration as the 7kg block.

If 3 kg block moves with the 7 kg block, the 2kg block ought to move together.

It is not evident, you have to prove.

ehild

 

[zorro]

It is not evident, you have to prove.

ehild

Let me try then.

Writing equations of motion for each block-
7a3 = 10 - f
3a2 = f - f1
2a1 = f1

Now the maximum static frictional force between 3kg and 7 kg block is 15N. But applied force is only 10N. So the friction adjusts in such a way that 3kg and 7 kg blocks move together i.e. a2=a3 ( f<10)

We can reduce the first 2 equations in to
10a=10-f1 or f1=10-10a

Now the max. frictional force that can act on 2kg block is 4N and its max acceleration is 2m/s². But it will not move with this acceleration as this will violate the last equation.
Hence they all move together with an acceleration 10/12=5/6

Is everything correct?

 

[ehild]

Hence they all move together with an acceleration 10/12=5/6

Is everything correct?

Yes, well done.

ehild