4/109 The ramp is used as passengers board a small commuter airline. The total mass of the ramp and six passengers is 750 kg with mass center at G. Determine the force in the hydraulic cylinder AB and the magnitude of the pin reaction at C.

the force in the hydraulic cylinder AB is 5311.9 N

the magnitude of the pin reaction at C is 4672.9 N

Explanation:

first we calculate the angle made by the triangle ACD from the Image

tan∅ = AB/605

tan∅ = 740 / 605

tan∅ = 1.2231

∅ = tan⁻¹ ( 1.2231 )

∅ = 50.73°

so the second image now shows the diagram of the ramp and the angle, the total weight of all the passengers and the ramp act at point G.

Now we equate the moment at point A to 0

∑MA = 0

Fcy (4115) - Mg(1815) = 0

our M is 750 kg and g is 9.81 m/s²

Fcy (4115) - ((750 × 9.81)(1815)) = 0

Fcy (4115) = 13353862.5

Fcy = 13353862.5 / 4115

Fcy = 3245.2 N

Now force in the hydraulic cylinder will be minimum only when the force Fab is perpendicular to the ramp

We equate the moment about the point C to 0

∑Mc = 0

Fab (4115sin50.73) - Mg(2300) = 0

our M is 750 kg and g is 9.81 m/s²

Fab (4115sin50.73) - ((750 × 9.81)(2300)) = 0

Fab(3185.7167) = 16922250

Fab = 16922250 / 3185.7167

Fab = 5311.9 N

Therefore the force in the hydraulic cylinder AB is 5311.9 N

We equate the forces in the horizontal direction to 0

Fcx - Fab cos50.73°  = 0

we substitute

Fcx -  5311.9 × cos50.73° = 0

Fcx = 3362.3 N

Next we calculate the resultant pin reaction force at C from the equ

Fc = √( F²cx + F²cy)

we substitute our values

Fc = √( (3362.3)² + (3245.2)² )

Fc = √ 21836384.33

Fc = 4672.9 N

Therefore the magnitude of the pin reaction at C is 4672.9 N