A pilot flies his plane in a vertical loop of radius 500m at a speed where at the top he feels no force from either the seat or the seatbelt.At what speed is the plane the flying?

At the top of the loop, the pilot is experiencing weightlessness, which means that the centripetal force is equal to the gravitational force acting on the pilot.

The centripetal force required to keep the pilot moving in a circular path is given by:

F_c = m*v^2 / r

Where:
F_c = centripetal force
m = mass of the pilot
v = speed of the plane
r = radius of the loop

The gravitational force acting on the pilot is given by:

F_g = m*g

Where:
F_g = gravitational force
g = acceleration due to gravity

Since the pilot is weightless at the top of the loop, the centripetal force is equal to the gravitational force:

m*v^2 / r = m*g

Solving for v:

v^2 = r*g
v = sqrt(r*g)

Given that the radius of the loop is 500m and the acceleration due to gravity is approximately 9.81 m/s^2, we can calculate the speed of the plane:

v = sqrt(500 * 9.81)
v = sqrt(4905)
v ≈ 70 m/s

Therefore, the plane is flying at a speed of approximately 70 m/s.

At the top of the loop, the only forces acting on the pilot are gravity and the normal force from the seat. The normal force is zero, so the pilot feels no force.

$$n-mg=\frac{mv^2}{r}$$

$$n=mg-\frac{mv^2}{r}$$

$$0=mg-\frac{mv^2}{r}$$

$$v^2=\frac{rg}{$$

$$v=\sqrt{\frac{rg}{}}$$

$$v=\sqrt{(500m)(9.8m/s^2)}$$

$$v=70m/s$$