(a) The apparent weight at the highest point is approximately \(369 \, \text{N}\).
(b) The speed for weightlessness is approximately \(44.7 \, \text{m/s}\).
(c) The apparent weight at the lowest point is approximately \(1104 \, \text{N}\).
(a) At the highest point of the loop, the apparent weight (\(N_{\text{app}}\)) can be calculated using the centripetal force equation:
\[ N_{\text{app}} = N - \frac{m \cdot v^2}{r} \]
Where:
- \( N \) is the actual weight (gravity acting on the pilot),
- \( m \) is the mass of the pilot,
- \( v \) is the speed of the pilot,
- \( r \) is the radius of the vertical circle.
First, calculate the mass (\(m\)) of the pilot:
\[ m = \frac{N}{g} \]
\[ m = \frac{726 \, \text{N}}{9.8 \, \text{m/s}^2} \]
\[ m \approx 74.08 \, \text{kg} \]
Now, calculate the apparent weight at the highest point:
\[ N_{\text{app}} = 726 - \frac{(74.08 \, \text{kg}) \cdot (200 \, \text{m/s})^2}{3060 \, \text{m}} \]
\[ N_{\text{app}} \approx 369 \, \text{N} \]
(b) At the point of weightlessness, the apparent weight (\(N_{\text{app}}\)) is zero. This occurs when the centrifugal force equals the gravitational force. The formula for apparent weight is \(N_{\text{app}} = N - \frac{m \cdot v^2}{r}\). Setting \(N_{\text{app}}\) to zero and solving for \(v\):
\[ 0 = N - \frac{m \cdot v^2}{r} \]
\[ \frac{m \cdot v^2}{r} = N \]
\[ v^2 = \frac{N \cdot r}{m} \]
\[ v = \sqrt{\frac{N \cdot r}{m}} \]
Plug in the known values:
\[ v = \sqrt{\frac{(726 \, \text{N}) \cdot (3060 \, \text{m})}{74.08 \, \text{kg}}} \]
\[ v \approx 44.7 \, \text{m/s} \]
(c) At the lowest point, the apparent weight is the sum of the actual weight and the centrifugal force:
\[ N_{\text{app}} = N + \frac{m \cdot v^2}{r} \]
Substitute the values:
\[ N_{\text{app}} = 726 + \frac{(74.08 \, \text{kg}) \cdot (200 \, \text{m/s})^2}{3060 \, \text{m}} \]
\[ N_{\text{app}} \approx 1104 \, \text{N} \]
So, the answers are:
(a) The apparent weight at the highest point is approximately \(369 \, \text{N}\).
(b) The speed for weightlessness is approximately \(44.7 \, \text{m/s}\).
(c) The apparent weight at the lowest point is approximately \(1104 \, \text{N}\).