To determine the magnitude of the tension force in the first rope connecting the airplane to the first glider, we can break the problem down into the following steps:
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Identify the masses and the acceleration:
- Mass of the airplane (\(m_a\)): 2200 kg
- Mass of the first glider (\(m_1\)): 350 kg
- Mass of the second glider (\(m_2\)): 250 kg
- Acceleration (\(a\)): 1.3 m/s²
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Calculate the total mass being accelerated: The total mass of the airplane and both gliders is: \[ m_{\text{total}} = m_a + m_1 + m_2 = 2200 , \text{kg} + 350 , \text{kg} + 250 , \text{kg} = 2800 , \text{kg} \]
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Calculate the total force exerted by the airplane: Using Newton's second law: \[ F_{\text{total}} = m_{\text{total}} \times a = 2800 , \text{kg} \times 1.3 , \text{m/s}^2 = 3640 , \text{N} \]
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Calculate the force acting on the second glider: The force acting on the second glider also needs to be considered, as it will be affected by the tension in the rope connecting the first glider and the second glider. The net force on the second glider can be calculated using: \[ F_{\text{net}} = m_2 \times a = 250 , \text{kg} \times 1.3 , \text{m/s}^2 = 325 , \text{N} \] This net force is equal to the tension (\(T\)) in the rope connecting the first glider to the second glider.
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Calculate the force acting on the first glider: Now, for the first glider, we consider the net force acting on it. The net force on the first glider is the sum of the tension \(T\) (from the rope to the second glider) and the tension \(T_1\) (the rope connecting to the airplane). The equation for the first glider will be: \[ T_1 - T = m_1 \times a \] Substituting for \(T\) from the previous step: \[ T_1 - 325 = 350 , \text{kg} \times 1.3 , \text{m/s}^2 \] So, we calculate \(350 \times 1.3\): \[ T_1 - 325 = 455 , \text{N} \] Therefore: \[ T_1 = 455 + 325 = 780 , \text{N} \]
Thus, the magnitude of the tension force in the first rope, \(T_1\), is 780 N.