To find the force exerted by the ram piston in a hydraulic press, we can utilize the concepts of mechanical advantage (MA), efficiency, and the relation between the areas of the pistons involved.
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Calculate the force exerted by the lever (Input Force): \[ F_{\text{input}} = \text{effort} = 200 , \text{N} \]
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Calculate the output force using mechanical advantage: The mechanical advantage is defined as the ratio of output force (\(F_{\text{output}}\)) to input force (\(F_{\text{input}}\)): \[ \text{MA} = \frac{F_{\text{output}}}{F_{\text{input}}} \]
Rearranging gives: \[ F_{\text{output}} = \text{MA} \times F_{\text{input}} \] Given that the mechanical advantage of the lever is 15: \[ F_{\text{output, ideal}} = 15 \times 200 , \text{N} = 3000 , \text{N} \]
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Adjust for efficiency: The actual output force of the hydraulic press is affected by its efficiency (\(\eta\)). The effective output force can be calculated by: \[ F_{\text{output}} = F_{\text{output, ideal}} \times \eta \] Given that the efficiency is 93% (or 0.93 when expressed as a decimal): \[ F_{\text{output}} = 3000 , \text{N} \times 0.93 = 2790 , \text{N} \]
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Relate the piston areas and the diameter: We are given that the diameter of the plunger (\(D_p\)) is 0.1 times the diameter of the ram (\(D_r\)). The area of a circular piston can be calculated with the formula: \[ A = \pi \left(\frac{D}{2}\right)^2 = \frac{\pi D^2}{4} \]
Therefore, the area of the plunger (\(A_p\)) and the ram (\(A_r\)) can be expressed as: \[ A_p = \frac{\pi (0.1 D_r)^2}{4} = \frac{\pi D_r^2}{4} \times 0.01 = 0.01 A_r \] Here, \(A_r\) is the area of the ram.
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Apply Pascal’s principle: In a hydraulic system, the pressure applied to the fluid is the same throughout. The relationship between force and area is given by: \[ F_p = P \cdot A_p \quad \text{and} \quad F_r = P \cdot A_r \] Since pressures are equal: \[ \frac{F_p}{A_p} = \frac{F_r}{A_r} \quad \Rightarrow \quad F_r = F_p \frac{A_r}{A_p} \]
Substituting \(A_p = 0.01 A_r\): \[ F_r = F_p \cdot \frac{A_r}{0.01 A_r} = 100 F_p \]
Using \(F_p = 2790 , \text{N}\): \[ F_r = 100 \times 2790 , \text{N} = 279000 , \text{N} \]
Thus, the force exerted by the ram piston is 279,000 N.