A hydraulic crane supports a mass of 500 kg with its piston-cylinder actuator, inclined at 60° with horizontal, as shown in the figure below. The diameter of the piston is 30 cm and gravitational acceleration is 10 m/s\(^2\). Neglecting mass of the piston, the reading of the pressure gage in kPa (gage) is
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Resolve the weight of the load along the direction of the inclined piston rod before dividing by the piston area.
Treat the piston rod as a two-force member: the oil pressure below it can only push along the rod's own line, so gravity has to be resolved along that line to find the pressure needed.
The load's weight is $W = mg = 500 \times 10 = 5000$ N, acting vertically. The rod sits at $60$ degrees to the horizontal, which is $30$ degrees off the vertical, so the part of the weight that lines up with the rod is $W\cos 30^{\circ} = 5000 \times 0.8660 = 4330.1$ N. The piston face has diameter $0.30$ m, giving an area of $A = \pi (0.15)^2 = 0.07069$ m$^2$.
Dividing the axial force by this area gives the gage pressure: $P = 4330.1 / 0.07069 = 61257$ Pa, which is $61.26$ kPa. This matches option (A); the other values come from mixing up the angle (using $\sin 30^{\circ}$ or $\cos 60^{\circ}$ instead of $\sin 60^{\circ}$) or using diameter in place of radius in the area formula.
