If a body is moving in a circular track of radius 100m with a velocity of 120m/s determine the banking angle of the track (g=9•8m/s)
We can start by finding the acceleration of the body using the centripetal acceleration formula:
a = v^2 / r
Where:
a = acceleration
v = velocity = 120 m/s
r = radius = 100 m
Substituting the given values:
a = (120^2) / 100
a = 14400 / 100
a = 144 m/s^2
Now, we can determine the banking angle (θ) using the equation:
tan(θ) = a / g
Where:
θ = banking angle
a = acceleration = 144 m/s^2
g = acceleration due to gravity = 9.8 m/s^2
Substituting the given values:
tan(θ) = 144 / 9.8
Taking the inverse tangent on both sides:
θ = tan^(-1)(144 / 9.8)
θ ≈ 83.6 degrees
Therefore, the banking angle of the track is approximately 83.6 degrees.
To determine the banking angle of the track, we can use the concept of centripetal force.
The formula for centripetal force is given by:
Fc = (m * v^2) / r
where,
Fc is the centripetal force,
m is the mass of the body,
v is the velocity of the body, and
r is the radius of the circular track.
In this case, the centripetal force is provided by the normal force (N).
At the banking angle, the normal force can be resolved into two components:
1. Ncosθ, which acts perpendicular to the surface of the track, providing the required centripetal force.
2. Nsinθ, which acts vertically downwards, balancing the force of gravity (mg).
The formula to find the banking angle is given by:
tanθ = (v^2) / (g * r)
where,
θ is the banking angle,
v is the velocity of the body,
g is the acceleration due to gravity, and
r is the radius of the circular track.
Let's substitute the given values into the formula:
tanθ = (120^2) / (9.8 * 100)
Simplifying,
tanθ = 1.4693877551020408
To find the value of θ, we take the inverse tangent (tan^-1) of both sides:
θ = tan^-1(1.4693877551020408)
Using a calculator, we find:
θ ≈ 57.6 degrees
Therefore, the banking angle of the track is approximately 57.6 degrees.