Distance over which friction acts to make object roll

[Abhishek11235]

In Morin's textbook,he ask a problem of the work done by friction in making body roll if the body initially starts with pure translation (i.e sliding motion). However I didn't get require answer(via Work Energy theorem) and I gave up. I checked solution and I became confused. He tells that friction doesn't work over the distance d

(The distance from start position to the position at which body begins to roll) but at a distance given in screenshot. Now the point at which friction acts will travel the same distance as other points on body travel. Then,why friction acts shorter distance?

 

[haruspex]

why friction acts shorter distance?

Friction arises from the relative motion of surfaces in contact. Since the ball has started to roll, the relative motion is less than if it had not rolled. The difference is the angle of rotation (in radians) multiplied by the radius.

 

[Abhishek11235]

Friction arises from the relative motion of surfaces in contact. Since the ball has started to roll, the relative motion is less than if it had not rolled. The difference is the angle of rotation (in radians) multiplied by the radius.

Consider ball initially in pure translation. Friction acts over this ball. Now we can find the distance when it begins to roll. Call it D. At this friction become static. Now consider the point in contact with ground. This will have 0 velocity once it starts to roll after a certain distance. Now this distance shouldn't be equal to distance D? It comes at 0 velocity after it has covered same distance as COM? I think I am not able to visualize it. Can you elaborate it?

 

[haruspex]

Consider ball initially in pure translation. Friction acts over this ball. Now we can find the distance when it begins to roll.

Pure translation lasts for no time. It will immediately start to turn, but not fast enough to be rolling. It will lose linear velocity but gain angular velocity until the two match and it transits to rolling.