What is the coefficient of friction between the piano and the ramp?

Hi  ,


Here's the full solution.

m = mass

g = gravitational acceleration = -9.81 m/s^2

a = acceleration = 4.62 m/s^2

A = angle = 38.5 degrees

u = coefficient of friction

N = normal force = mass*gravity = m*g*cos(A)

F(fric) = force of friction = u*N = m*g*cos(A)



In the direction parallel to the surface of the incline there are two forces acting on the mass: 1) the fore of friction opposing the motion (directed "back up" the incline) and 2) the force of gravity accelerating the mass down the incline.  By Newton's 2nd Law, the sum of these force must equal the mass times acceleration:

F(total) = m*a = F(grav) - F(fric)

*F(fric) is negative because it acts OPPOSITE to the motion

We know that:

F(fric) = -u*N = -u*m*g*cos(A)

*we must multiply by the cos(A) term because only one of the components of gravity acts down the slope of the incline

We also know:

F(grav) = m*g*sin(A)

So now we have:

F(total) = m*a = m*g*sin(A) - u*m*g*cos(A)

Notice the "m" term cancels on both side of the equation.  Now we solve for u:

u = [ a - g*sin(A) ] / g*cos(A)

= [ 4.62 - 9.81*sin(38.5) ] / -9.81*cos(38.5)

= 0.19


So the coefficient of friction is 0.19.

That's your answer!







aNAGAH.