A pilot of mass m can bear a maximum apparent weight 7 times of mg.The aeroplane is moving in vertical circle.if the velocity of aeroplane is 210m/s while diving up from the lowest point of vertical circle, the minimum radius of vertical circle should be
Ans:750m
m v^2/r + m g = 7 m g
v^2/r = 6 g
r = (210)^2/(6*9.81)
r = 749 meters
To find the minimum radius of the vertical circle, we can start by analyzing the forces acting on the pilot when the plane is diving up from the lowest point.
The apparent weight experienced by the pilot can be calculated using the equation:
Apparent weight (W') = Mass (m) x Acceleration due to gravity (g') - Centripetal force (Fc)
We know that the pilot can bear a maximum apparent weight that is 7 times their actual weight, so the expression becomes:
W' = 7mg
At the lowest point of the vertical circle, the centripetal force required to keep the pilot moving in a circle is:
Fc = m x v^2 / r
Where:
m = mass of the pilot
v = velocity of the airplane
r = radius of the vertical circle
Substituting the given values into the equation:
W' = 7mg = m x g' - m x v^2 / r
Since the velocity of the airplane is given as 210 m/s, we need to determine the acceleration due to gravity (g') as the plane passes through the lowest point.
When the plane is at the lowest point, the acceleration due to gravity will be the vector sum of the gravitational acceleration (g) and the centripetal acceleration (ac):
g' = g + ac
The centripetal acceleration is given by:
ac = v^2 / r
Substituting the expression for ac into the equation for g':
g' = g + v^2 / r
Using the given value for velocity (v = 210 m/s) and acceleration due to gravity (g = 9.8 m/s^2):
g' = 9.8 + (210^2) / r
Now, we can substitute this expression for g' back into the equation for the apparent weight:
7mg = m x (9.8 + (210^2) / r) - m x (210^2) / r
Simplifying the equation:
7g = 9.8 + 210^2 / r - 210^2 / r
Multiplying through by r:
7gr = 9.8r + 210^2 - 210^2
The (210^2) terms cancel out:
7gr - 9.8r = 0
Factoring out r:
r(7g - 9.8) = 0
Since radius (r) cannot be zero, we can disregard the second factor:
7g - 9.8 = 0
Solving for g:
7g = 9.8
g = 9.8 / 7
Substituting this value for g into the equation for g':
g' = 9.8 + (210^2) / r
9.8 / 7 = 9.8 + (210^2) / r
Now, we can solve for the minimum radius (r). First, let's simplify:
9.8 = 9.8 + (210^2) / r
Subtracting 9.8 from both sides:
0 = (210^2) / r
Multiplying both sides by r:
0r = 210^2
r = 210^2 / 0
Since dividing by zero is not defined, the equation is invalid, and there is no solution.
Therefore, it seems there may be an error in the given values or calculations.
To find the minimum radius of the vertical circle, we need to consider the forces acting on the pilot of mass m when the airplane is diving up from the lowest point.
At the bottom of the vertical circle, the pilot experiences his normal weight (mg) and an upward apparent weight (7mg) due to the circular motion. We can set up the following equation for the forces acting on the pilot:
Apparent weight = Normal weight + Centripetal force
7mg = mg + Centripetal force
Since the apparent weight is 7 times the normal weight, we can express it as:
7mg = mg + Centripetal force
Simplifying this equation, we find:
6mg = Centripetal force
The centripetal force can be expressed as the mass times the centripetal acceleration:
Centripetal force = m * (v^2 / r)
where v is the velocity of the airplane and r is the radius of the vertical circle.
Substituting this expression for the centripetal force into the previous equation, we have:
6mg = m * (v^2 / r)
Dividing both sides of the equation by m, we get:
6g = v^2 / r
Now, we can substitute the given values into the equation. The velocity of the airplane is 210 m/s, so:
6 * 9.8 m/s^2 = (210 m/s)^2 / r
Simplifying further:
58.8 m/s^2 = 44,100 m^2/s^2 / r
To solve for the radius (r), we can rearrange the equation as:
r = 44,100 m^2/s^2 / 58.8 m/s^2
Calculating this expression, we find:
r ≈ 750 m
Therefore, the minimum radius of the vertical circle should be approximately 750 meters.