Which requires more work?

Question

Two inclines have the same height and same coefficients of friction. However, incline A is much steeper than incline B. If two identical blocks were pushed up the ramp, which block requires more work to get to the top of the inclines?

Answer 1

The incline where more energy is lost from friction will require more work.

 

E = energy lost from friction

L = length of incline (longer for B, since B is less steep)

θ = angle above horizontal (higher for A, since A is more steep)

h = height (same for both inclines)

b = length of horizontal base of incline (longer for B, since B is less steep)

 

E = μmg cosθ L

 

Note that 

 

L cosθ = b ⇒

 

E = μmgb

 

Since b is greater for incline B, the energy lost is greater on B than on A, so incline B requires more work under the conditions of this problem.