A subway train rounds an unbanked curve at 62 km/h . A passenger hanging onto a strap notices that an adjacent unused strap makes an angle of 19 to the vertical.

To analyze this situation, we need to recognize that the angle formed by the unused strap is due to the centripetal force acting on the passenger as the train rounds the curve. Here's how we approach it:

### Step-by-Step Analysis:

1. **Forces Acting on the Passenger:**
   - **Gravitational Force (\(F_g\)):** Acts vertically downward. Its magnitude is \(F_g = mg\), where \(m\) is the mass of the passenger and \(g = 9.81 \, \text{m/s}^2\) is the acceleration due to gravity.
   - **Centripetal Force (\(F_c\)):** Acts horizontally inward, toward the center of the curve. Its magnitude is \(F_c = \frac{mv^2}{r}\), where \(v\) is the speed of the train, and \(r\) is the radius of the curve.

2. **Tension in the Strap:**
   The tension in the unused strap must provide the necessary centripetal force while also balancing the gravitational force. As the strap makes an angle \( \theta = 19^\circ \) with the vertical, we can resolve the tension into two components:
   - **Vertical Component of Tension (\(T_y\)):** Balances the gravitational force: \( T_y = mg \).
   - **Horizontal Component of Tension (\(T_x\)):** Provides the centripetal force: \( T_x = \frac{mv^2}{r} \).

3. **Using Trigonometry:**
   Since the tension components are related to the total tension \(T\) by the angle \( \theta \), we can express these components as:
   \[
   T_x = T \sin(\theta), \quad T_y = T \cos(\theta)
   \]
   Therefore, the ratio of these forces gives us:
   \[
   \tan(\theta) = \frac{T_x}{T_y} = \frac{\frac{mv^2}{r}}{mg}
   \]
   Simplifying this:
   \[
   \tan(\theta) = \frac{v^2}{rg}
   \]
   Solving for \(r\) (the radius of the curve):
   \[
   r = \frac{v^2}{g \tan(\theta)}
   \]

4. **Substituting Known Values:**
   - \( v = 62 \, \text{km/h} = 17.22 \, \text{m/s} \)
   - \( g = 9.81 \, \text{m/s}^2 \)
   - \( \theta = 19^\circ \)
   
   Now, substituting these into the equation for \(r\):
   \[
   r = \frac{(17.22)^2}{9.81 \times \tan(19^\circ)}
   \]

Let me calculate this for you.The radius of the curve that the subway train is rounding is approximately \( 87.8 \, \text{m} \).

Would you like more details or have any questions? Here are five related questions you might consider:

1. How does the angle of the unused strap change if the train’s speed increases?
2. How would the scenario differ if the curve was banked instead of unbanked?
3. What would happen to the forces involved if the train were to slow down?
4. How can the passenger's experience be explained using Newton’s laws of motion?
5. How does the centripetal force change as the radius of the curve changes?

**Tip:** The angle made by objects hanging in a curve provides a direct way to measure centripetal acceleration and infer the curve's radius or speed.

A subway train rounds an unbanked curve at 62 km/h. A passenger hanging onto a strap notices that an adjacent unused strap makes an angle of 19 degrees to the vertical.

Analyze the centripetal force acting on a passenger in a subway rounding an unbanked curve at 62 km/h with a strap making a 19-degree angle. Explore key physics concepts like centripetal force, Newton’s laws, and trigonometry. Calculate the radius of the curve using known values.

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