Irodov Problem 1.50

From the problem definition we have,

Irodov Problem 1.49

From the problem definition,















The problem is essentially almost an exercise in Calculas rather than a physics problem.

Irodov Problem 1.48

From the problem definition we have,

















When the rotating body stops, its angular velocity will be 0, thus from (1) we can determine how long it will take for the body to stop as,


The average angular velocity is the total angle turned by total time and is given by,




The mean angular velocity averaged over the entire time can now be determined as,

Irodov Problem 1.47

Let the radius of the circle of rotation be r. The tangential acceleration is given by,



The angular velocity of the particle at any time t is given by,



The centripetal acceleration (normal to direction of motion) is given by,



The angle between the net acceleration vector and the velocity vector as indicated in the figure and given by,

Irodov Problem 1.46

(a) The average angular velocity is simply the angle traversed by the time it took to traverse this angle and thus is given by,



The particle stops when its angular velocity is 0, thus it stops at time given by,









Thus, the mean angular velocity will be,



The mean angular acceleration is the amount of angular velocity gained by the time it took to gain it. This is given by,



Thus, when the particle stops, its mean angular acceleration will be,




(b) The angular acceleration at any given time is the derivative of its angular velocity given by,




Thus, when the particle stops it will be,

Irodov Problem 1.45

Suppose that a is the linear acceleration of the particle and angular acceleration. Suppose that it took t time for the particle to escape out of the barrel. Thus, from the information provided in the problem,








Now suppose that the final angular velocity of the shell while its exiting out of the barrel is , then,

Irodov Problem 1.44

If the radius of the wheel is given by r then we have,






The tangential acceleration of the particle is given by,




The normal (radial) acceleration is simply the centripetal acceleration and is given by,



The total acceleration is thus given by,

Irodov Problem 1.43

For solving this problem we will rely heavily on the figure provided. The particle moves along the circle shown in the figure and its angular velocity relative to point O is constant. Let be the instantaneous tangential velocity of the particle. The component of tangential velocity in a direction perpendicular to the vector OA (along AB) is given by, . Thus, from the basic definition of angular velocity of a particle relative to a point, the angular velocity of the particle relative to O is given by, .

Form basic geometry in the figure we have,




In other words,






Now the particle moves along the circle thus, there is no component of the velocity normal to its direction of motion, ie Vn = 0. In other words, the velocity of the particle is same as the tangential component. Since there is no tangential acceleration for the particle (since it moves at a constant tangential velocity), the only acceleration experienced by the body is the centripetal acceleration given by,