To determine the mass of the elevator, we first need to consider the forces acting on it. The tension in the rope is one of these forces, and the weight of the elevator (which is the product of its mass and the acceleration due to gravity) is another.
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Identify the forces:
- Tension \( T = 6000 , \text{N} \)
- Weight \( W = mg \) (where \( m \) is the mass and \( g \) is the acceleration due to gravity, approximately \( 9.81 , \text{m/s}^2 \))
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Using Newton's second law: The net force acting on the elevator can be expressed as: \[ F_{\text{net}} = T - W = ma \] where \( a \) is the acceleration of the elevator.
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Calculate the acceleration: Since the elevator starts from rest and moves upwards with a speed of 4 m/s over 2 seconds, we can find the acceleration using the formula: \[ a = \frac{\Delta v}{\Delta t} = \frac{4 , \text{m/s} - 0 , \text{m/s}}{2 , \text{s}} = \frac{4}{2} = 2 , \text{m/s}^2 \]
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Set up the equation: Now we can substitute \( W = mg \) and \( a \) into the forces equation: \[ T - mg = ma \]
Rearranging gives us: \[ T = mg + ma \]
Factoring out \( m \): \[ T = m(g + a) \]
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Replace \( T \), \( g \), and \( a \): \[ 6000 , \text{N} = m(9.81 , \text{m/s}^2 + 2 , \text{m/s}^2) \] \[ 6000 , \text{N} = m(11.81 , \text{m/s}^2) \]
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Solve for \( m \): \[ m = \frac{6000 , \text{N}}{11.81 , \text{m/s}^2} \approx 508.5 , \text{kg} \]
Therefore, the mass of the elevator is approximately 508.5 kg.