To find the magnitude of the force \( F \) holding the 10.0 kg mass at rest on a frictionless incline of 30.0°, we need to analyze the forces acting on the mass.
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Identify Forces: The forces acting on the mass on the incline are:
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Gravitational force (\( W \)), which acts vertically down and can be calculated as: \[ W = mg \] Where:
- \( m = 10.0 , \text{kg} \) (mass)
- \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity)
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The component of the gravitational force acting down the incline can be calculated using the sine of the incline angle: \[ W_{\parallel} = W \sin(\theta) \]
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The component of the gravitational force acting perpendicular to the incline (normal force) can be calculated using the cosine of the incline angle: \[ W_{\perpendicular} = W \cos(\theta) \]
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Calculate the Weight: \[ W = mg = 10.0 , \text{kg} \times 9.81 , \text{m/s}^2 = 98.1 , \text{N} \]
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Calculate the Parallel Component of Weight: \[ W_{\parallel} = W \sin(30°) = 98.1 , \text{N} \times 0.5 = 49.05 , \text{N} \]
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Determine the Force \( F \): Since the mass is at rest, the force \( F \) must balance the component of the weight acting down the incline: \[ F = W_{\parallel} = 49.05 , \text{N} \]
Thus, the approximate magnitude of force \( F \) is 49.1 N (rounded to three significant figures).