A man pushes on a piano with mass 180 kg; it slides at constant velocity down a ramp that is inclined at 19.0° above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes (a) parallel to the incline and (b) parallel to the floor.

(a) F = 574.3 N

(b) F = 607.4 N

Explanation:

We apply Newton's second law:

∑F = m*a (Formula 1)

∑F : algebraic sum of the forces in Newton (N)

m : mass in kilograms (kg)

a : acceleration in meters over second square (m/s²)

Data

β = 19° : Angle of inclination of the ramp

μk = 0 : Coefficient of kinetic friction

m = 180 kg : piano mass

g = 9.8 m/s² : acceleration due to gravity

W = m*g : Piano Weight

W = 180*9.8 = 1764 N

X-Y axes in the inclined plane

We define the x-axis in the direction of the inclined plane, 19° to the horizontal.

We define the y-axis and in the direction of the plane perpendicular to the inclined plane.

We calculate the weight component parallel to the displacement of the piano:

Wx = W*sin19° = 1764*sin19° = 574.3 N

Problem development

a) The man pushes the piano with a force (F) parallel to the inclined plane.

We apply formula (1):

∑F = m*a, a=0 : Because the velocity is constant

F-Wx = 0

F = Wx

F = 574.3 N

b) The man pushes the piano with a force (F) parallel to the floor.

We apply formula (1):

We define the F force component parallel to the displacement of the piano (Fx):

Fx = F*cos19°

∑F = m*a, a=0 : Because the velocity is constant

Fx-Wx = 0

F*cos 19° - 574.3 N = 0

F = ((574.3) / (cos 19°)) N

F = 607.4 N