For a train moving around a curve, the required banking angle θ is given by:
\[\tan \theta = \frac{v^2}{Rg}\]
where \(v = 12 \, \text{m/s}\), \(R = 400 \, \text{m}\), and \(g = 10 \, \text{m/s}^2\).
Substitute the values:
\[\tan \theta = \frac{12^2}{10 \times 400} = \frac{144}{4000} = \frac{h}{1.5}\]
where \(h\) is the height by which the outer rail should be raised over the inner rail, and the distance between the rails is 1.5 m.
Solving for \(h\):
\[h = \frac{144 \times 1.5}{4000} = 5.4 \, \text{cm}\]
Thus, the required height is 5.4 cm.
A uniform circular disc of radius \( R \) and mass \( M \) is rotating about an axis perpendicular to its plane and passing through its center. A small circular part of radius \( R/2 \) is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.