# Spring and blocks

#### zorro

I am stuck up in a situation created by me.

Consider a block A resting on a smooth horizontal surface. There is another block B of the same size/mass resting over it. There is some friction present in between them, with coefficient of friction μ. A spring of spring constant K is attached to block B (the other side attached to a wall off course) and the blocks are displaced through a distance 'x' together and released. The block B oscillates without slipping over the block A.

At the max. displacement, there should be a max. value of the friction force. Now is this value of friction force equal to μmg? If we draw a F.B.D. of block B at max. displacement, we find that Kx should be greater than friction force for the block to oscillate. If the max. value is μmg, and Kx>μmg, then why doesn't the block B slip over the block A?

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#### androidx219

if kx exceeds μmg, there will be slipping and relative motion between the blocks. The frictional force at max displacement has to be less than μmg, only then the blocks will oscillate together without slipping.
Reason: The frictional force is STATIC and not DYNAMIC. Static friction adjusts itself with the amount of force applied on the body. So μmg (where μ is coefficient of static friction) can be a quantity which is greater than Kx (where x is the maximum displacement). If even the extreme value of Kx is less than μmg, then the two bodies never slip against each other.

suppose frictional force is f between the blocks and at maximum displacement,
f(max) = kx (x is maximum at maximum displacement)

f(max) < μmg (for the bodies to never slip)

#### zorro

f(max) = kx (x is maximum at maximum displacement)
If f(max)=kx, how do you think will the block return to its mean position and perform oscillations?

#### androidx219

I am sorry, I was grossly wrong at the statement I made there. Thanks for pointing that out and correcting me. f(max) can never be equal to kx and will always be less than kx, otherwise the blocks wouldn't move at all as you rightly said.

Lets consider the FBDs of A and B separately,

suppose at anypoint say, the combined acceleration is a, then
K.x - f = m.a (for B)
f= m.a (for A)

so k.x = 2m.a

so when considering A and B as a combined body together, f comes out to be a mere inernal force for this system. This should make things a bit clearer to both of us.

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