Force and Friction problem help

[crcowboyfan]

Homework Statement

The coefficient of static friction between the 3 kg crate and the 35* incline is 0.3. What is the magnitude of the minimum force, F, that must be applied to the crate perpendicular to the incline to prevent the crate from sliding down?

 

[hotcommodity]

What are the forces acting on the crate parallel to the incline? We're told that we don't want the crate to slip, in other words, it does not accelerate from rest once it's set on the incline. This tells us that the net force parallel to the incline is zero, this is our constraint. Does this help?

 

[crcowboyfan]

Yeah I understand that the net force on the incline is zero, but I'm not sure where to go from there.

Thanks

 

[hotcommodity]

Can you identify the forces acting parallel to the incline? This will help you set up the equation (or in this case, the inequality) and solve for the minimum force. Hint: Use Newton's Second Law.

 

[crcowboyfan]

Wouldn't the forces acting parallel to the incline be the force of static friction and the x component of gravity?

 

[hotcommodity]

The static friction and the x component of the weight, you've got it. Now how can we plug this information into Newton's Second Law?

 

[crcowboyfan]

That's where I'm confused :)

F=ma

I'm not sure how to plug it in

 

[hotcommodity]

Ok, so we're going to say that the sum of the forces on an object along the incline is equal to the objects mass times its acceleration. For example, if I was pushing a box up the incline, and there was no friction, then I would know that the net force of the box along the incline is equal to the force that I push with (F) minus the x component of the weight (W_x), which is equal to the objects mass (m) times its acceleration (a).

So if I take the positive x direction to be up the incline, I would have:

$$\Sigma F_{x} = F - W_x = ma$$

So knowing this, how would you set up your equation? In which directions do the two forces act?

 

[crcowboyfan]

Opposite each other, so wouldn't friction just equal Wx and the acceleration would be 0?

 

[hotcommodity]

Yes, you have the right idea. Let's jut be careful about what we're saying about the static friction. We cannot say that it is equal to some value, because the static friction is given to us by an inequality:

$$f_s \leq \mu_s N$$

or in this case, with zero acceleration we have:

$$f_s \leq W_x$$

Now you can find the minimum value of the force. Does this help?