An airplane flying at a velocity of `900 km h^(-1)` loops the loop. If the maximum force pressing the pilot against the seat is five times its weight, the loop radius should be

Options

A. 1562 m
B. 1402 m
C. 1315 m
D. 1167 m

Answer

To solve the problem, we need to determine the radius of the loop that the airplane is flying through, given that the maximum force pressing the pilot against the seat is five times the pilot's weight.

### Step-by-step Solution:

1. **Convert Velocity to m/s**:
The airplane's velocity is given as \( 900 \, \text{km/h} \). We need to convert this to meters per second (m/s).
\[ \text{Velocity} = 900 \, \text{km/h} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{h}}{3600 \, \text{s}} = 250 \, \text{m/s} \]

2. **Identify Forces at the Top of the Loop**:
At the top of the loop, the forces acting on the pilot are:
- Weight (\( mg \)) acting downwards.
- Normal force (\( N \)) acting downwards (the force pressing the pilot against the seat).
- Centripetal force (\( F_c \)) required to keep the airplane in circular motion.

The centripetal force is given by:
\[ F_c = \frac{mv^2}{r} \]

3. **Set Up the Equation**:
According to the problem, the normal force is five times the weight of the pilot:
\[ N = 5mg \]
At the top of the loop, the net force acting towards the center (downwards) is the sum of the weight and the normal force:
\[ N + mg = F_c \]
Substituting the expressions we have:
\[ 5mg + mg = \frac{mv^2}{r} \]
Simplifying this gives:
\[ 6mg = \frac{mv^2}{r} \]

4. **Cancel Mass (m)**:
Since \( m \) appears in all terms, we can cancel it out:
\[ 6g = \frac{v^2}{r} \]

5. **Rearranging for Radius (r)**:
Rearranging the equation to solve for \( r \):
\[ r = \frac{v^2}{6g} \]

6. **Substituting Values**:
Now, substitute \( v = 250 \, \text{m/s} \) and \( g = 10 \, \text{m/s}^2 \):
\[ r = \frac{(250)^2}{6 \times 10} = \frac{62500}{60} = 1041.67 \, \text{m} \]

7. **Final Calculation**:
Simplifying further:
\[ r \approx 1041.67 \, \text{m} \]

### Conclusion:
The radius of the loop should be approximately \( 1041.67 \, \text{m} \).