Find Min Horizontal Force to Keep Small Cube from Falling

[PhunWithPhysics]

Question? There is a Large Cube, with a mass of 25 kg, it is being accelerated across a horizontal FRICTIONLESS surface by a horizontal force P. There is a small cube with a mass of 4 kg in contact with the front surface and will slide down and fall off unless P is sufficiently large. The coefficient of static friction between the cubes is 0.71 , what is the smallest magnitude P can have in order to keep th small cube from falling down? Any help would be great, thanks.

 

[El Hombre Invisible]

The normal reaction of the smaller cube is, in this case, equal and opposite to the contact force applied to it by the larger cube. So long as the friction force is equal to the weight of the smaller cube it will not fall. You know the friction coefficient and the weight of the smaller cube and g. Solving P should be a doddle.

 

[PhunWithPhysics]

is normal force?

Is the normal force = P? should the answer be in Newtons?

 

[El Hombre Invisible]

The contact force is P. The normal force is the other force in the Newton pair, so is equal and opposite: -P. Since the question is 'how small can P be' and P is a force acting on the large cube, yes: the answer should be in Newtons.

What this is asking here is how much should the contact force P be such that the normal reaction to it causes a static friction force that overcomes the weight of the smaller cube.

You know the weight of the smaller cube, so you know what the friction force must be to counteract against the weight.

You also know what the static friction coefficient is, so you can find what the normal reaction needs to be to give the required friction force. This is a reaction to the contact force, so you can find what the contact force should be. That's P.

 

[PhunWithPhysics]

Tell me if i am wrong?

Static Force or P? is equal to (4.0kg)(9.8)=the normal force 39.2
times that by the coefficient 0.71 and it is 27.31 Newtons?

 

[El Hombre Invisible]

No, the normal force N must be equal and opposite to the contact force: N = -P.
The friction force F has to overcome the weight of the cube: F = -W = -mg.
If the coefficient of static friction is 0.71, then F = 0.71 * N = -0.71P = -mg.
Solve that for P and you're done.

 

[PhunWithPhysics]

so it should be

So it should be N= -0.71*P=-(4.0)(9.8), then just divide both sides by -0.71 to solve for P and it is P = -(4.0)(9.8)/-0.71 = 55.2 N?

 

[El Hombre Invisible]

That sounds about right. Do you dig why, though? That's the important question. This basically comes down to Newton's 1st and 3rd laws. For the cube to not fall, it must have no unbalanced force acting on it in the vertical direction, so the friction force must be equal and opposite to its weight. The friction force is given by some coefficient multiplied by the normal reaction. The normal reaction is a consequence of Newton's 3rd law, by which if the larger cube exerts some force P on the smaller one, the smaller cube must exert an equal and opposite force, the normal reaction, on the larger one.