(I) Derive the expression for the angle of banking.

(II) Derive the relation between v and ω.

(i) Derive expression for angle of banking:

When an object, like an airplane, is turning in a horizontal circle, it experiences centripetal force (Fc) directed towards the center of the circle, which helps it maintain the circular path. To counteract this force and keep the object in balance, an additional force called the "banking force" (Fb) is applied perpendicular to the wings. This banking force is responsible for the angle of banking (θ) of the object.

Consider an object moving in a horizontal circle of radius (r) with a constant speed (v). Let's assume the mass of the object is (m), and the angle of banking is (θ). The force components acting on the object are:

Weight force (mg), acting vertically downward.
Normal force (N), perpendicular to the wings (acting upward).
Banking force (Fb), acting perpendicular to the wings (directed inward or outward depending on the banking direction).
To maintain horizontal circular motion, the centripetal force (Fc) can be calculated as:

Fc = m * v^2 / r

In the vertical direction, the sum of the vertical components of forces should be zero:

N - mg * cos(θ) = 0 ----(1)

In the horizontal direction, the sum of the horizontal components of forces should provide the necessary centripetal force:

Fb - mg * sin(θ) = m * v^2 / r ----(2)

Now, we need to find the relation between Fb and θ. To do that, let's rearrange equation (1) to solve for N:

N = mg * cos(θ)

Now, substitute the expression for N into equation (2):

Fb - mg * sin(θ) = m * v^2 / r

Fb = m * v^2 / r + mg * sin(θ)

Finally, solve for θ:

θ = arctan((m * v^2) / (r * Fb - m * g))

This equation gives you the angle of banking (θ) required to maintain the circular motion with a given speed (v), radius (r), mass (m), and banking force (Fb).

(ii) Derive the relation between velocity (v) and angular velocity (ω):

To relate linear velocity (v) and angular velocity (ω), consider a point object moving in a circle of radius (r) with an angular velocity ω (measured in radians per second). The linear velocity (v) of the object can be obtained by multiplying the angular velocity (ω) by the radius (r) of the circle.

The linear velocity (v) of an object moving in a circle is given by:

v = r * ω

Where:
v = Linear velocity (in meters per second, m/s)
r = Radius of the circle (in meters, m)
ω = Angular velocity (in radians per second, rad/s)

This relationship shows that the linear velocity of an object moving in a circle is directly proportional to the product of the angular velocity and the radius of the circle. As the angular velocity increases, the linear velocity increases proportionally. Similarly, a larger radius will result in a higher linear velocity for the same angular velocity.