Penny is adjusting the position of a stand up piano of mass mp = 150 kg in her living room. The piano is lp = 1.35 m in length. The piano is currently at an angle of θp = 36 degrees to the wall. Penny wants to rotate the piano across the carpeted floor so that it is flat up against the wall. To move the piano, Penny pushes on it at the point furthest from the wall. This piano does not have wheels, so you can assume that the friction between the piano and the rug acts at the center of mass of the piano. Required:
a. Write an expression for the minimum magnitude of the force FS in N Penny needs to exert on the piano to get it moving. Assume the corner of the piano on the wall doesn't slide and the static friction between the rug and the piano is µs.
b. The coefficient of kinetic friction between the carpet and the piano is uk = 0.27. Once the piano starts moving, calculate the torque τ in N·m that Penny needs to apply to keep moving the piano at a constant angular velocity.
c. Calculate the amount of work Wp, in J Penny does on the piano as she rotates it.

a)  F = ½ μ m g / cos  θ,   b)  F = 908.5 N, c)   W = 720.9 J

Explanation:

a and b)  To rotate the piano you need the rotational equilibriu.   For this we fix the reference system in the corner of the piano that does not move; we will assume that counterclockwise rotation is positive

               Fₓ L sin θ  - fr L / 2 = 0

where we use trigonometry for the force components

            cos 36 = Fₓ / F

            Fₓ = F cos 36

            sin 36 = / F

the component F_{y} does not torque  any because because its distance to the pivot point of zero

the equation for the transactional equilibrium on the y axis is

             N -W = 0

             N = W

the friction force is shaped

             fr = μ N

             fr = μ mg

substitutes

           Fₓ L sin θ = μ mg L / 2

           Fₓ = ½ μ m g / sin θ

we substitute the values

           F = ½ 150 9.8 / sin θ

           Fₓ = 735 / sin θ

The angle between the force F and the direction of the piano is what determines the different values of F, in this case we apply the derivative to find the minimum

           dF / dθ = 0

           735 cos θ / sin² θ = 0

           cos θ = 0

           θ = 90º

therefore the angle between the force and the piano must be perpendicular, therefore the angle of the force with respect to the wall is 36º

The expretion is

      F = ½ μ m g / cos  θ

let's look for the value of strength

            F cos 36 = 735

            F = 735 / cos 36

            F = 908.5 N

c) Work is defined by

           W = F. d

           W = F d cos θ

the force is applied in the direction of movement so that   θ = 0

           W = F d

let's use trigonometry to find the distance the piano must move to touch the wall

           sin 36 = y / L

           y = L  sin 36

           y = 1.35 sin 36

           y = 0.7935 m

         W = 908.5 * 0.7935

         W = 720.9 J