PLEASE SHOW ALL WORK AND STEPS INCLUDE PICTURE AND FREE BODY

PLEASE SHOW ALL WORK AND STEPS. INCLUDE PICTURE AND FREE BODY DIAGRAM

Figure shows a block of mass m resting on a 20\" slope. The block has coefficients of friction/is = 0.76 and = 0.50 with the surface. It is connected via a massless string over a massless. frictionless pulley to a hanging block of mass 2.0 kg. What is the minimum mass m that will stick and not slip? Express your answer to three significant figures and include the appropriate units.

Solution

(a) Pulling the mass upslope we have T = 2kg * 9.8m/s² = 19.6 N
Resisting are the friction and gravity forces of mass m:
gravity Fg = mgsin = m * 9.8m/s² * sin20º = m*3.35 m/s²
friction Ff = µmgcos = 0.76 * m * 9.8m/s² * cos20º = m * 7 m/s²
m * (3.35 + 7) m/s² = 19.6 N
m = 1.894 kg

(b) now Ff = 0.50 * 1.894 kg * 9.8m/s² * cos20 = 8.72 N
and Fg = 1.894 kg * 3.35 m/s² = 6.34 N
so the net force accelerating the system is Fnet = 19.6N - 8.72N - 6.34N = 4.54 N
total mass = 3.894 kg
acceleration a = Fnet / m = 4.54N / 3.894kg = 1.166 m/s²