Irodov Problem 1.290

Let the coefficient of linear thermal expansion be . When the temperature increases by the length of the rod increases by a fraction . So if the length of the rod is l then the increase in length of the rod will be . In order for the length of the rod to be constant enough stress should be applied to compress the rod by . The strain will thus be . The stress (pressure) required will be .

Irodov Problem 1.289

The angular momentum vector of the shaft is directed radially inwards (or outwards depending on the direction of motion of the train) and is equal to Iw.

As the train turns in the circle, the shaft turns too and so does the angular momentum vector L. The change in direction of the angular momentum vector must be induced by a moment (torque dL/dt) that is acts tangentially to the circle. This, torque is in turn generated by the vertical forces F acting on the rails as shown in the figure.

Irodov Problem 1.288

The gyroscope rotates at a rate and thus has an angular momentum of Iw oriented along the direction of the ship's length. As the ship turns in the circular path, so does the angular momentum vector. This change in angular momentum is caused by the Gyroscopic moment (rate of change of angular momentum).The ship rotates in the circular path at a rate (v/R). Thus, the angular momentum vector is given by,