he a body is moving in a circular track of radius 100m with a velocity 120m/s.determine the banking angle of the track (g=9.8m/s)
To determine the banking angle of the track, we need to consider the balance of forces acting on the body as it moves in a circular path. The main forces involved here are the gravitational force (weight) and the centripetal force.
The centripetal force is responsible for keeping the body in circular motion and is provided by the friction between the body and the track. To achieve this, the track is usually inclined or banked at a specific angle.
Let's break down the forces acting on the body:
1. Weight (mg): This is the gravitational force acting on the body due to its mass (m) and the acceleration due to gravity (g). The weight acts vertically downward.
2. Centripetal force (Fc): This force is directed towards the center of the circular path and is responsible for keeping the body in circular motion. It is given by the equation Fc = (mv^2) / r, where m is the mass of the body, v is the velocity, and r is the radius of the circular track.
To determine the banking angle, we need to find the horizontal and vertical components of the weight and equate them to the horizontal and vertical components of the centripetal force.
1. Horizontal components:
The horizontal component of weight is mg sinθ, where θ is the banking angle.
The horizontal component of the centripetal force is Fc cosθ.
2. Vertical components:
The vertical component of weight is mg cosθ.
The vertical component of the centripetal force is Fc sinθ.
Since the body is not slipping, the vertical component of the centripetal force should balance the weight of the body. Therefore:
mg cosθ = Fc sinθ
We can now solve for the banking angle (θ) using the given information:
Radius (r) = 100 m
Velocity (v) = 120 m/s
Acceleration due to gravity (g) = 9.8 m/s^2
First, let's find the centripetal force (Fc):
Fc = (mv^2) / r
Fc = (m(120 m/s)^2) / 100 m
Fc = 1440m N
Substituting this value in the equation:
mg cosθ = Fc sinθ
(mg cosθ) / (Fc sinθ) = 1
Now we can solve for θ:
(mg / Fc) cotθ = 1
cotθ = (Fc / mg)
θ = arccot (Fc / mg)
Substituting the known values:
θ = arccot (1440m / (m * 9.8m/s^2))
Finally, we obtain the banking angle (θ) by evaluating the arccot function. Since we don't have a specific value for the mass of the body, we cannot provide an exact answer. You would need to substitute the mass value to get the actual banking angle.