To determine the angle \(\theta\) of the inclined plane, we can use the relationship between load, effort, efficiency, and the mechanical advantage (MA).
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Mechanical Advantage (MA): \[ \text{Efficiency} = \frac{\text{MA}}{\text{Velocity Ratio (VR)}} \] Knowing the efficiency is 80%, we express it as a decimal: \[ \text{Efficiency} = 0.8 \]
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Expressing MA: The mechanical advantage can be defined as: \[ \text{MA} = \frac{\text{Load}}{\text{Effort}} = \frac{500 , \text{N}}{200 , \text{N}} = 2.5 \]
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Calculating the Velocity Ratio (VR): Rearranging the efficiency formula, we have: \[ \text{MA} = \text{Efficiency} \times \text{VR} \] Thus, \[ \text{VR} = \frac{\text{MA}}{\text{Efficiency}} = \frac{2.5}{0.8} = 3.125 \]
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Finding the Angle \(\theta\): The Velocity Ratio for an inclined plane is given by: \[ \text{VR} = \frac{1}{\sin(\theta)} \] Therefore, \[ \sin(\theta) = \frac{1}{\text{VR}} = \frac{1}{3.125} \] \[ \sin(\theta) ≈ 0.32 \]
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Calculating \(\theta\): Now, to find \(\theta\), we take the inverse sine (arcsin): \[ \theta ≈ \arcsin(0.32) \] Using a calculator, we find: \[ \theta ≈ 18.75^\circ \]
Thus, the angle \(\theta\) of the inclined plane is approximately 18.75 degrees.