If the ball were stationary, the tension would be the force caused by gravity, m g, where m is the mass in kg and g is the acceleration of gravity 9.8 m/s^2. (approx. 10 m/s^2 or 10 N/kg)
If there were no gravity, the force would be the "centrifugal" force (mass times acceleration), m (v^2)/r, where v is the velocity and r the length of the rope. The change in direction requires a force.
In this case, both forces are acting in the same direction and the total force is the sum of the two
F = (10 kg) (9.8 N/kg) + (10 kg) (6 m/s)^2 / (4.5 m) = 10 (9.8 + 8) = 178 N
The tension is the sum of the weight of the ball and the force needed to change the ball's path.
If the train curves to the left, the ball will have its momentum initially in the original direction and thus will be displaced to the right in the car.
The vertical force vector has a magnitude of m g.
The horizontal force vector has a magnitude of m (v^2)/r.
The resultant force is the vector sum of the two, with an angle of a,
where the tangent is the ratio of the horizontal to the vertical forces:
tan a = (v^2) / g r
since a = 15o, tan a = 0.27
Using SI units (mks):
0.27 r = (v^2)/g = (60000 m / 3600 s)^2 / (9.8 m/s^2) = 28.3 m
r = (28.3 m) / 0.27 = 105 m
Using g = 10 N/kg rather than 9.8 would be simpler and nearly as correct.
