A pilot flies his plain in a vertical loop with a radius of 500m.
a) what is the speed of the aircraft if the pilot is 'weightless' at the top of the loop?
b)what is the weight of the pilot at the bottom of the loop if speed is the same and the pilot is 80kg?
the centripetal acceleration is
... v^2 / r
a) v^2 / r = g
... v^2 = 500 m * 9.8 m/s^2
b) his weight will be twice normal
... w = 2 * 80 * g
a) To find the speed of the aircraft at the top of the loop, we can use the concept of centripetal force.
At the top of the loop, the pilot experiences weightlessness, which means that the normal force acting on the pilot is equal to zero. The only force acting on the pilot is the tension in the plane, which provides the necessary centripetal force.
The centripetal force can be calculated using the formula:
F_c = m * v^2 / r
Where:
F_c is the centripetal force,
m is the mass of the pilot,
v is the speed of the aircraft,
and r is the radius of the loop.
Since the pilot is weightless, the centripetal force is equal to zero:
0 = m * v^2 / r
Rearranging the equation, we get:
v^2 = r * g
Where:
g is the acceleration due to gravity.
Solving for the speed, we have:
v = sqrt(r * g)
Substituting the given values, we have:
v = sqrt(500 m * 9.8 m/s^2)
v ≈ 31.3 m/s
Therefore, the speed of the aircraft at the top of the loop is approximately 31.3 m/s.
b) At the bottom of the loop, the pilot experiences a force due to their weight as well as the centripetal force. Since the pilot's mass is given as 80 kg, we can calculate the net force acting on the pilot at the bottom of the loop using the following formula:
F_net = m * g + m * v^2 / r
Where:
F_net is the net force on the pilot,
m is the mass of the pilot,
g is the acceleration due to gravity,
v is the same speed of the aircraft as calculated in part a),
and r is the radius of the loop.
The net force acting on the pilot is equal to the product of their mass and acceleration, which is:
F_net = m * g
Rearranging the equation, we get:
m * g = m * g + m * v^2 / r
Canceling out the mass, we have:
g = g + v^2 / r
Substituting the given values, we have:
9.8 m/s^2 = 9.8 m/s^2 + (31.3 m/s)^2 / 500 m
Multiplying through by 500 m, we get:
9.8 * 500 = 9.8 * 500 + (31.3)^2
Simplifying the equation, we find:
4900 = 4900 + 979.69
Which is not a valid equation.
Therefore, there seems to be an error in the question as the net force on the pilot at the bottom of the loop cannot be equal to the weightless force at the top of the loop.