Box on an inclined surface with Force of Friction and angled applied Force.

AI Thread Summary
The discussion focuses on analyzing a box on an inclined surface subjected to friction and an angled applied force. The force of friction is defined as Ff = -μN, where μ is the coefficient of kinetic friction and N is the normal force. The applied force's y-component reduces friction, leading to the equation Fy = N = sin(30)F. Participants derive equations for the net force and acceleration, incorporating the effects of gravity and friction. The final formula for the applied force F is expressed in terms of mass, acceleration, and angle components.
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Incline Surface force

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In that case:

\vec{F}f=-μ.N\hat{x'}

Being μ the coefficient of kinetic friction and N the normal reaction of the ramp.
As the applied force makes 30º with the ramp, it's y component is going to contribute to a decrease in friction, and so being Fy=N=sin(30)F, you have:

\vec{F}f=-μ.sin(30)F\hat{x'}
 
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You are given the acceleration up the incline so you can calculate F_x. Set up your force equations for the x and y components of all forces. Knowing the horizontal force will allow you to solve for the magnitude of F_a and you can get F_a from that.
 
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Fnet=Fcos(30)-mgsen(20)-μN , with N=mgcos(20)-Fsen(30)
ma=Fcos(30)-mgsen(20)-μmgcos(20)+μFsen(30)
F=mg(sen(20)+μcos(20)+(a/g))/(cos(30)+μsen(30))
 
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