Irodov Problem 1.112

In this problem because the particle is moving linearly on a uniformly rotating disc it will experience the Coriolis psuedo force in the frame fixed to the particle's motion in addition to the centrifugal force. Centrifugal force is responsible for rotating the body around its center of rotation. It does not alter its magnitude. However, as the body moves radially outwards (or inwards) the magnitude of its tangential velocity must also change as it is given by angular speed times the radius. The Coriolis force is responsible to bring about this change in magnitude of tangential velocity.

Rather than using the expression for the force I will derive it from first principles so that the origin of the force is clear.

The position vector p of the particle (with the center of the disc as the origin) with respect to a stationary observer can be written as,



Here, the particle is assumed to have started at the center and move radially outwards so it will be at a radius v't from the center. The nature or the value of the force will not change even if it started somewhere else rather than the center since it will simply be (v't - k) instead of v't and the constant will disappear when we differentiate.
















From the point of view of the observer on the moving particle both these accelerations will seem like forces acting in the opposite direction to the accelerations in (2).

The force exerted by the body on the particle on the disc is same as the reaction the disc exerts on the particle and this must provide for both the net acceleration of the particle. Further there is also the force of gravity that is acting on the body acting downwards with which the particle pushes the disc downwards. All three forces act in mutually perpendicular directions. Hence the net force exerted by the particle on the disc is given by,











So while the centrifugal force is responsible for changing the direction of motion of the particle, the Coriolis force is responsible for changing the tangential velocity of the particle as it radius of rotation changes.

Irodov Problem 1.111

The earth is a rotating sphere (well almost). Suppose that its radius is R
and it is rotating at an angular speed of and so are all things on it - the person, the gun, the bullet in it and the target.


The bullets's motion
The person stands at a longitude of and hence his distance from the Earth's axis of rotation is . Because of the Earth's rotation about its axis the person and hence the gun and the bullet have a tangential velocity . After the bullet is shot it will continue to have this tangential velocity. Further, it will also be acted upon by gravity that will pull it radially inwards towards the ground. This is indicated in the figure.

The target's motion
The target that is s units away from the person further along the north. Suppose that the latitude here is . This means that,




The target is thus at a distance from the axis of rotation. This means that from the point of view of the shooter, the target is below his horizontal level as shown in the figure. Also the tangential velocity of the target as it rotates with the earth is which is slightly less than the tangential velocity of the bullet. This is also depicted in the figure.

The time taken by the bullet to reach the target is s/v, and during this time it will fall a distance of below the horizontal. The target is moving a bit slower tangentially then the bullet by a value and so the bullet will move farther than the target tangentially.

Vertical Offset
Since 1km is much much smaller than the earth's radius will be very very small. The vertical offset will thus be,





Horizontal offset
The horizontal offset is due to the difference in tangential velocities of the target and bullet given by,







Also in the problem this means that will be negligibly small and we can neglect it. Given this the only displacement becomes