Physics Question, please help?

Question

A 17 kg block sits on a plane inclined at 47 degrees. On one side of the block, a
string is attached and goes up the inclined plane and over a pulley, after which it attaches to a
dangling mass. On the other side of the block, a string stretches down the inclined plane and
attaches to another block. Determine the following:

b. [Assume the plane is frictionless] If the dangling block has a mass of 131 kg, determine
the mass of the other block on the inclined plane needed to keep the system in
equilibrium, i.e. to prevent the blocks from accelerating.
c. [Assume there is friction] If both strings are cut simultaneously, find the single
coefficient of static friction needed to prevent both blocks from sliding down the incline?
d. What is the net acceleration of m3?

Answer 1

Since the blocks are in equilibrium then the force exerted by the dangling block up the inclined plane is equal to the sum of the parallel forces of the two blocks down the inclined plane. In an equation: mg = F_||17 + F_||x , where 17 is for the parallel force of the 17 kg block and x is for the parallel force of the block with the unknown mass.

The parallel component of the weight of each block on the inclined plane is given by F_|| = sin47 (mg).

Thus our equation to solve for the mass of the unknown block becomes:

mg = sin47 (mg)₁₇ + sin47 (mg)_x substituting our given values,

(131 x 9.8) = sin47 (17 x 9.8)₁₇ + sin47 (m x 9.8)_x , we can simplify by dividing out g, or 9.8 from both sides,

131 = sin47 (17) + sin47 (m), multiply it out to simplify,

131 = 12.433 + 0.73135 (m)_x , more math,

(131 - 12.433)/0.73135 = m_x

162.12 kg = m_x answer for part b.

 

The coefficient of friction is given by the equation: µ = F_f / F_N where f is for friction and N is for normal. On the inclined plane, the force of friction will be equal in size to the parallel force. The normal force will be equal in size to the perpendicular component of force from the weight of each block. Thus our equation for the coefficient of friction becomes: µ = F_|| / F_⊥  .

The components of the weight of each block are given by: F_|| = sin47 (mg) and F_⊥ = cos47 (mg) and so we can solve for the coefficient of friction for each block by using: µ = sin47 (mg) / cos47 (mg) 

µ = 1.072  answer for part c.

 

I am not sure about part d, since m3 is not labeled. If it was the dangling block then its acceleration is 9.8 m/s².