Inclined plane, physics question. Help is very much appreciated.

Question

A 100N Box (initially at rest) is pushed 10.0m, up a rough ramp by a horizontal applied force of 150N. The ramp is inclined at an angle of 30 degreees and the coefficient of kinetic friction between the box and the ramp is 0.10. Compute the net work done on the box.

Answer 1

   There are four forces acting on the box: gravity, the normal force, friction and the "push".  Each force has its own magnitude and direction.  The simple definition for work is W = F•d  (W = Fdcosθ ) where F and d are constant  and θ is the angle between their directions.

 

   The problem can be simplified by choosing a coordinate system with the x-axis parallel to the incline.  This way d is along the x-axis and only those forces, or their components, parallel to the incline (i.e. x-axis) will end up doing any work.  (cos 90° = 0 for those components perpendicular to the incline)  

 

   The net force, in components, is 

 

          x:   P(cos 30°) - mg(sin30°) - f = ma  (the box might be accelerating up the ramp)

          y:   n - mg(cos30°) - P(sin30°)  = 0     (the box stays on the ramp) (the push increases the normal force)  

 

where P = 150 N, mg = 100 N, and f = µn = 0.10(100*0,866 + 150*0.500) = 16.16 N  

 

      The net force in the x-direction, up the incline, is F_x = 150*0.866 - 100*0.500 - 16.16 = 63.7 N and therefore the net work is w = Fd = 63.7N*10.00m = 637 J

 

Comment:  Note that the push is doing positive work,  i.e.adding energy to the system, and gravity and friction are doing negative work, i.e. taking energy out of the system.  The net work done here goes into increasing the box's kinetic energy  (it is slowly accelerating up the ramp, going faster and faster).