Example Question - torque
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Finding Equilibrium Force and Fulcrum Reaction in a Leverage System
Given: Mass, \( m = 72 \text{ kg} \) Distance to fulcrum, \( b = 12 \text{ cm} = 0.12 \text{ m} \) Fulcrum reaction, \( R = 807.2 \text{ N} \) Gravity, \( g = 9.81 \text{ m/s}^2 \) Total length of lever, \( L = 96 \text{ cm} = 0.96 \text{ m} \) Distance \( a = L - b = 0.96 \text{ m} - 0.12 \text{ m} = 0.84 \text{ m} \) To find the equilibrium force \( F \), we use the principle of moments (torque \( \tau = Fd \)), where \( d \) is the distance from the fulcrum: <p>\( \tau_{\text{clockwise}} = \tau_{\text{counterclockwise}} \)</p> <p>\( F \cdot a = m \cdot g \cdot b \)</p> <p>\( F \cdot 0.84 \text{ m} = 72 \text{ kg} \cdot 9.81 \text{ m/s}^2 \cdot 0.12 \text{ m} \)</p> <p>\( F = \frac{72 \cdot 9.81 \cdot 0.12}{0.84} \)</p> <p>\( F = \frac{847.872}{0.84} \)</p> <p>\( F \approx 1009.61 \text{ N} \)</p> To confirm that the given \( F = 100.9 \text{ N} \) is not correct, and the correct \( F \) is approximately \( 1009.61 \text{ N} \).
Torque Balance in Lever System
<p>Given that the mass \( m = 72 \) kg, the distance from the fulcrum to the mass \( b = 12 \) cm, and the total length of the lever \( L = 96 \) cm, we need to find the force \( F \) to balance the lever and the reaction \( R \) at the fulcrum.</p> <p>To find the force \( F \), we use the principle of moments (torque balance) where the clockwise moments equal the anticlockwise moments about the pivot.</p> <p>The weight of the mass \( W \) acting at a distance \( b \) from the fulcrum is given by:</p> <p>\( W = m \cdot g \)</p> <p>\( W = 72 \cdot 9.8 \) (taking \( g = 9.8 \, \text{m/s}^2 \))</p> <p>\( W = 705.6 \) N</p> <p>The anticlockwise moment due to \( W \) is \( W \cdot b \).</p> <p>Let \( a = L - b \). Then \( a = 96 - 12 = 84 \) cm \( = 0.84 \) m.</p> <p>The clockwise moment due to \( F \) is \( F \cdot a \).</p> <p>Setting the moments equal for balance:</p> <p>\( W \cdot b = F \cdot a \)</p> <p>\( F = \frac{W \cdot b}{a} \)</p> <p>\( F = \frac{705.6 \cdot 0.12}{0.84} \)</p> <p>\( F = 100.8 \) N</p> <p>Now, to find the reaction \( R \) at the fulcrum, we use equilibrium of vertical forces:</p> <p>\( R = W + F \)</p> <p>\( R = 705.6 + 100.8 \)</p> <p>\( R = 806.4 \) N</p> <p>Since \( F \) is 100.8 N which is approximately 100.9 N, and \( R \) is 806.4 N which is approximately 806.7 N.</p>
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