A hydraulic press has a small piston with a diameter of 4\text{ cm} and a large piston wit… — JAMB UTME

A hydraulic press has a small piston with a diameter of $4\text{ cm}$ and a large piston with a diameter of $20\text{ cm}$ . If an effort force of $80\text{ N}$ applied to the small piston raises a load of $1500\text{ N}$ placed on the large piston, what is the efficiency of the machine?

A
$133.3\%$
$75.0\%$ Answer
C
$375.0\%$
D
$26.7\%$

Answer

The efficiency of the hydraulic press is

75.0\%

The mechanical advantage is

MA = \frac{1500}{80} = 18.75

. The velocity ratio for pistons of diameters

20\text{ cm}

and

4\text{ cm}

VR = \left(\frac{20}{4}\right)^2 = 25

. Efficiency is

\frac{MA}{VR} \times 100\% = \frac{18.75}{25} \times 100\% = 75.0\%

Step-by-Step Solution

Calculate Mechanical Advantage (MA)

MA = \frac{\text{Load}}{\text{Effort}} = \frac{1500\text{ N}}{80\text{ N}} = 18.75

Mechanical advantage is defined as the ratio of load force to effort force.

Calculate Velocity Ratio (VR) for the hydraulic press

VR = \frac{A_2}{A_1} = \left(\frac{d_2}{d_1}\right)^2 = \left(\frac{20\text{ cm}}{4\text{ cm}}\right)^2 = 5^2 = 25

The velocity ratio of a hydraulic press equals the ratio of the cross-sectional areas of the pistons, which simplifies to the square of the ratio of their diameters.

Calculate Efficiency (\eta)

\eta = \frac{MA}{VR} \times 100\% = \frac{18.75}{25} \times 100\% = 75.0\%

Efficiency is the ratio of mechanical advantage to velocity ratio expressed as a percentage.

Key Concept

Hydraulic Press Efficiency and Velocity Ratio

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