Calculate the force and coefficient of static friction

Question

A 90 kg Elton John is trying to keep a 310 kg piano from accelerating down a ramp inclined at 35°. The kinetic co-efficient of friction between the piano and the ramp is 0.25.

a) Calculate the force that Elton has to apply to keep the piano from accelerating. Assume the force he applies is parallel to the incline.

b) Calculate the minimum static coefficient of friction between his glittery platform boots and the ramp in order for him to apply the force you calculated in part (a).

Answer 1

Let's first find the horizontal force of gravity applied by the piano:

a)

m_(piano) x g x sin35 = 310 x 9.8 x sin35 =~ 1742.52521363 N (to be rounded)

Since the piano 'wants' to accelerate down, friction opposed it thereby helping Elton.

Let's find the friction force

N = 310 kg * 9.8 m/s² * cos(35°) = 2,524 N

F_friction = 0.25 * 2,524 N = 631 N

So total force to be applied by Elton to hold the piano still is 1742.5 - 631 = ~ 1112 N

b) Due to N3 Law, the same force is also applied on Elton by piano. Besides, Elton has weight as well.

The forces that push Elton down are F(piano_on_Elton) + m_(Elton)g x sin35

The force that prevents him slide down, Friction_max = m_(Elton) g x cos35 x μ(_max)

So let's calculate

1112 + 90 x 9.8 x sin35 = ~1618N

1618 = Friction_(max) = 90 x 9.8 x cos35 x μ(_max)

μ(_max) =~ 2.24