a crate of mass m = 180 kg is pushed at constant speed up the frictionless ramp ( = 26.0°)?

Question

by a horizontal force F.
What is the magnitude of F?

What is the magnitude of the force exerted by the ramp on the crate?

Answer 1

First, separate F into two orthogonal components: one parallel to the ramp and the other perpendicular to the ramp:
// part: F*cos26.0°
T part: F*sin26.0°
Also, separate gravity on the mass m into two orthogonal components: one parallel to the ramp and the other perpendicular to the ramp:
// part: mg*sin26.0°
T part: mg*cos26.0°
These two // parts must be the same for the crate to be pushed at constant speed up the frictionless ramp:
F*cos26.0° = mg*sin26.0°
F = mg*tan26.0° = 180*9.8*tan26.0° = 860(N)
The magnitude of the force exerted by the ramp on the crate, is the sumation of both "T" parts, since the T part of F also push the crate normally against the ramp:
F*sin26.0° + mg*cos26.0°
= 860*sin26.0° + 180*9.8*cos26.0°
= 1.96(kN)