To calculate the force exerted by the man on the floor of the lift when it accelerates, we can use Newton's second law of motion. The force exerted by the man on the floor of the lift will depend on both his weight and the acceleration of the lift.
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Calculating the force when the lift accelerates downward at \(0.4 , \text{m/s}^2\):
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First, calculate the weight of the man (force due to gravity): \[ F_g = m \cdot g \] where:
- \( m = 90 , \text{kg} \) (mass of the man)
- \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity)
\[ F_g = 90 , \text{kg} \times 9.81 , \text{m/s}^2 = 882.9 , \text{N} \]
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When the lift accelerates downward, the net force \( F \) exerted by the man on the floor can be calculated using: \[ F = F_g - m \cdot a \] where \( a = 0.4 , \text{m/s}^2 \).
\[ F = 882.9 , \text{N} - 90 , \text{kg} \cdot 0.4 , \text{m/s}^2 = 882.9 , \text{N} - 36 , \text{N} = 846.9 , \text{N} \]
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Calculating the force when the lift accelerates upward at \(0.5 , \text{m/s}^2\):
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Again, we start with the weight of the man, which we already calculated as \( 882.9 , \text{N} \).
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When the lift accelerates upward, we use: \[ F = F_g + m \cdot a \]
\[ F = 882.9 , \text{N} + 90 , \text{kg} \cdot 0.5 , \text{m/s}^2 = 882.9 , \text{N} + 45 , \text{N} = 927.9 , \text{N} \]
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In summary, the forces exerted by the man on the floor of the lift are:
- When accelerating downward at \(0.4 , \text{m/s}^2\): \( 846.9 , \text{N} \)
- When accelerating upward at \(0.5 , \text{m/s}^2\): \( 927.9 , \text{N} \)