Block A has mass of mA = 58kg and rests on a flat surface. The coefficient of static friction between the block and the surface is 0.300. The coefficient of static friction between the rop and the fixed peg B is 0.310. The width of the block is d = 0.290m. Find the greatest mass, mC, that weight C can have such that block A does not move.

The greatest mass, that the weight C can have such that block A does not move is approximately 23.259 kg

Explanation:

The given parameters are;

The mass of the block A= 58 kg

The coefficient of static friction between the block and the surface, = 0.300

The coefficient of static friction between the rope and the fixed peg, B = 0.310

Let T represent the tension in the rope

Therefore, when the rope is static, T = The normal reaction at the peg, B,

The angle of inclination of the rope holding the block A = arctan(3/4) ≈ 36.87°

The length of the rope = √(0.4² + 0.3²) = 0.5

∴ sin(θ) = 3/5 = 0.6

cos(θ) = 4/5 = 0.8

The vertical component of the tension in the rope = T × sin(θ) = 0.6·T

The horizontal component of the tension in the rope = T × cos(θ) = 0.8·T

The friction force = μ×(W - 0.6·T) = 0.300×(58×9.8 - 0.6·T) = 170.52 - 0.18·T

The block will start to move when we have;

The horizontal component of the tension in the rope = The friction force

∴ 0.8·T = 170.52 - 0.18·T

0.8·T + 0.18·T = 170.52

0.98·T = 170.52

T = 170.52/0.98 = 174

Therefore, the tension in the rope = T = 174 N = The normal reaction at the peg, B

The frictional force at the peg, = × = 0.310 × 174 N = 53.94 N

The weight of the mass, , = The frictional force at the peg,  + The tension in the rope

∴ The weight of the mass, , = 53.94 N + 174 N = 227.94 N

Weight, W = Mass, m × The acceleration due to gravity, g, from which we have;

m = W/g

Where;

g = 9.8 m/s²

∴ = /g = 227.94 N/(9.8 m/s²) ≈ 23.259 kg.

The greatest mass, that the weight C can have such that block A does not move = ≈ 23.259 kg.