Irodov Problem 1.120

There are two forces acting on the body as it moves along the circle, i) one tangential force that is tangentially accelerating the body as it moves given by and ii) the centripetal force that making the body turn given by .

So we have,

Irodov Problem 1.119

The entire work done on the mass is converted into kinetic energy and so not surprisingly the work done is equal to its kinetic energy as derived beside. Now,

Irodov Problem 1.118

From the definition of work dW = F.dr. Here both F is the force vector and dr is the displacement vector. Since F was constant throughout the motion W = F.(r2-r1).

Irodov Problem 1.117

The entire situation is depicted in the figure beside from various view points. As the body falls downwards (z-direction) with a velocity vz , it experiences a Coriolis force (in the reference frame of Earth) of magnitude . This force is responsible for the apparent deflection of the body along the latitude along the direction of Earth's rotation (lets call this x-direction).

In other words,




Since the body is under free fall,














The body falls a height of h units, and so

Irodov Problem 1.116

a) the picture for this part is shown beside. The body moves along the Meridian i.e. straight towards the north pole. There are two inertial forces acting on the body i) centrifugal force and ii) the Coriolis force. The centrifugal force acts outwards from the surface of the earth and thus does not put any pressure on the rails. The Coriolis force on the other hand acts along the surface of the earth and thus can cause pressure on the rails. The Coriolis force is given by . Here is the angular velocity of the earth (you can determine the direction of rotation based on "sun rises in the east"). The component of velocity of the train in the plane perpendicular to the earth's angular velocity is as shown in the figure. The Coriolis force thus acts to the right rail as shown in the figure and has a magnitude of .

b) If the total inertial force has to be zero, then the Coriolis force must act in the opposite direction of the centrifugal force and completely cancel it out. Now centrifugal force acts outwards from the Earth's surface as shown in the second figure and so the Coriolis force Fco must act inwards as shown in the figure. For the Coriolis force to act in that direction, the velocity vector must be oriented perfectly along the latitude and in the opposite direction of the Earth's rotation as shown in the figure. Its magnitude will then be . So in order for the two to exactly cancel out we have,

Irodov Problem 1.115

First let us start looking the problem from a stationary frame (a non-rotating frame). There are two forces i) acting on the mass, i) the normal reaction from the surface N and the force of gravity mg as shown in the figure. Let us consider the forces acting in a direction normal to the surface of the sphere. There are two forces acting in this direction, N and . The body is subject to centrifugal acceleration of as it moves (slides) along the curvature of the sphere in a circular path. Thus we have,




As the body slides down it looses potential energy and gains speed (kinetic energy). When it is at an angle , the mass has fallen a height of and looses a potential energy of . This will get converted into kinetic energy of the body, hence we have,





At the break-off point, the normal reaction N offered by the surface becomes 0. From (1) and (2) and setting N=0 in (1) we have,




Now coming to the problem. The centrifugal force seen by the observer in the rotating frame is given by (see here). Here is the angular velocity vector of the rotating frame as seen from a stationary reference and r is the radius vector of the particle as shown in the figure in the rotating reference . For this problem,




Thus, the centrifugal force experienced by the mass in the rotating frame is .

The Coriolis force is defined as (see here.) Here is the angular velocity of the rotating frame of reference with respect to a stationary reference and is the velocity of the mass relative to the rotating frame.
Relative to the rotating frame the mass's velocity is a sum of two motions, i) the tangential velocity of the mass on the surface of the sphere as it slides down v and ii) the apparent rotation of the mass along the horizontal plane with angular velocity in the opposite direction to the rotation of the sphere (basically to someone on the sphere the mass rotates with an angular velocity in the opposite direction). This is depicted in the picture.

If we write the velocity vector as,


as shown in the figure beside and the angular velocity as k then the Coriolis force can be determined as,