Minimum force required to prevent sliding down

[Pcmath]

Recently I've come across a question that seems very simple, but had puzzled me for a while.

Suppose a point object with mass M is placed on a rough plane inclined at 30 degree to the horizontal and is subjected to the force of gravity acting down vertically (to make it simple, assume g = 10 ms^-2). The inclined plane is rough and thus friction exists to oppose the motion of the object and given that the coefficient of static friction between 2 surfaces is 0.2. Now it can easily be shown that the object will slide down because the component of gravity parallel to the plane is greater than the max friction opposing the motion. Now suppose a new horizontal(perpendicular to gravity) force F acts on the mass M to prevent it from sliding down, and the force F is increasing gradually until it can stop the object completely. I want to find the formula for the minimum force F needed to stop the object.

The problem I get is that the horizontal force F can be resolved to 2 forces, parallel and perpendicular to the inclined plane. I notice that the force perpendicular to the inclined plane will increase the reaction force and thus friction. So when deriving the formula also need to account for the increasing friction force as F increases. But I make it very complicated and unable to solve it.

Can anyone help me?

MENTOR Note: Moved here from another forum hence no HW template.

 

[PeroK]

Recently I've come across a question that seems very simple, but had puzzled me for a while.

Suppose a point object with mass M is placed on a rough plane inclined at 30 degree to the horizontal and is subjected to the force of gravity acting down vertically (to make it simple, assume g = 10 ms^-2). The inclined plane is rough and thus friction exists to oppose the motion of the object and given that the coefficient of static friction between 2 surfaces is 0.2. Now it can easily be shown that the object will slide down because the component of gravity parallel to the plane is greater than the max friction opposing the motion. Now suppose a new horizontal(perpendicular to gravity) force F acts on the mass M to prevent it from sliding down, and the force F is increasing gradually until it can stop the object completely. I want to find the formula for the minimum force F needed to stop the object.

The problem I get is that the horizontal force F can be resolved to 2 forces, parallel and perpendicular to the inclined plane. I notice that the force perpendicular to the inclined plane will increase the reaction force and thus friction. So when deriving the formula also need to account for the increasing friction force as F increases. But I make it very complicated and unable to solve it.

Can anyone help me?

This would be better in the homework section.

In any case, the policy here at PF is for you to show us your working first and we can help you finish the problem. Your analysis is correct about the force $F$ increasing the normal force, hence the friction force. Can you find the equations using trigonometry?