A model airplane of mass 0.700 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 62.0-m-long control wire as shown in Figure (a). The forces exerted on the airplane are shown in Figure (b): the tension in the control wire, the gravitational force, and aerodynamic lift that acts at 𝜃 = 20.0° inward from the vertical. Compute the radius, lift, and tension in the wire, assuming it makes a constant angle of 𝜃 = 20.0° with the horizontal. Use *g*=9.8 m/s2.
radius

Let's begin by analyzing the problem and solving for the radius of the horizontal circular path.

### Known quantities:
- Mass of the model airplane: \( m = 0.700 \, \text{kg} \)
- Speed of the airplane: \( v = 35.0 \, \text{m/s} \)
- Length of the control wire: \( L = 62.0 \, \text{m} \)
- Angle between the control wire and the horizontal: \( \theta = 20.0^\circ \)
- Gravitational acceleration: \( g = 9.8 \, \text{m/s}^2 \)

### Step 1: Determine the radius of the horizontal circular path

The airplane flies in a horizontal circle, but the control wire makes an angle of \( \theta \) with the horizontal, as shown in the problem. The horizontal component of the length of the control wire is the radius of the circular path, denoted as \( r \).

The radius \( r \) can be calculated from the geometry of the situation:

\[
r = L \cdot \cos(\theta)
\]

Substitute the given values:

\[
r = 62.0 \, \text{m} \cdot \cos(20.0^\circ)
\]

Let me calculate this.The radius of the horizontal circular path is approximately \( 58.26 \, \text{m} \).

Would you like to proceed with the calculations for the lift and tension in the wire, or do you have any questions so far?

Here are 5 related questions:
1. How does the angle \( \theta \) affect the tension in the wire?
2. How would the radius change if the angle increased to 30 degrees?
3. What is the relationship between the speed and the tension in the wire?
4. How does the mass of the airplane affect the forces involved?
5. What happens to the lift force if the airplane flies faster?

**Tip**: Always ensure you understand how to decompose forces when dealing with circular motion problems.

A model airplane of mass 0.700 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 62.0-m-long control wire. The forces exerted on the airplane are the tension in the control wire, gravitational force, and aerodynamic lift acting at an inward angle of 𝜃 = 20.0° from the vertical. Compute the radius, lift, and tension in the wire, assuming the wire makes a constant angle of 𝜃 = 20.0° with the horizontal. Use g = 9.8 m/s².

Solve a physics problem involving the circular motion of a model airplane with mass 0.700 kg and speed 35.0 m/s. Learn how to compute the radius of the circle, the lift force, and the tension in the control wire using key concepts in circular motion, trigonometry, and Newton's laws. This problem is suitable for Grades 10-12.

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