A 2200-kg

airplane pulls two gliders, the first of mass 350 kg
and the second of mass 250 kg
, down the runway with an acceleration of 1.3 m/s2
(Figure 1). Neglect the mass of the two ropes and any frictional forces. Determine the magnitude of the tension force in the second rope.

1 answer

To determine the tension force in the second rope, we can use Newton's second law of motion, \(F = ma\).

Step 1: Calculate the total mass being pulled.

The total mass consists of the airplane and both gliders:

\[ m_{\text{total}} = m_{\text{airplane}} + m_{\text{glider 1}} + m_{\text{glider 2}} = 2200, \text{kg} + 350, \text{kg} + 250, \text{kg} = 2800, \text{kg} \]

Step 2: Calculate the total force acting on the system.

Using Newton's second law, we can find the total force required to accelerate this mass:

\[ F_{\text{total}} = m_{\text{total}} \cdot a = 2800, \text{kg} \cdot 1.3, \text{m/s}^2 = 3640, \text{N} \]

Step 3: Consider the forces on the second glider.

Now, we examine the forces acting on the second glider (the glider with mass \(250, \text{kg}\)). The total force acting on it can be expressed as:

\[ F_{\text{tension}} - F_{\text{glider}} = m_{\text{glider}} \cdot a \]

Where:

  • \(F_{\text{tension}}\) is the tension in the second rope (which we need to find),
  • \(F_{\text{glider}} = m_{\text{glider}} \cdot g\) only applies if we are considering weight in a vertical direction, and since we neglect friction and are in horizontal linear motion, we skip that consideration here.

Step 4: Apply Newton's second law to the second glider.

Substituting our values for the second glider \(F_{\text{tension}} - 0 = m_{\text{glider}} \cdot a\):

\[ F_{\text{tension}} = m_{\text{glider}} \cdot a \]

Where:

  • \( m_{\text{glider}} = 250, \text{kg} \)
  • \( a = 1.3, \text{m/s}^2 \)

Step 5: Calculate the tension force.

\[ F_{\text{tension}} = 250, \text{kg} \cdot 1.3, \text{m/s}^2 = 325, \text{N} \]

Conclusion

The magnitude of the tension force in the second rope is:

\[ \boxed{325, \text{N}} \]