Chapter 10, Problem 18A

The three cups shown below are rolled on a level surface. Rank the cups by the amount they depart from a straight-line path (most curved to least curved).

To determine

To rank: The cups on the basis of the amount they depart from a straight-line path.

Answer to Problem 18A

The rank of the cups by the amount they depart from a straight-line path is $B>C>A$ .

Explanation of Solution

Given:

The given figure is shown below.

  

Formula used:

Linear speed varies with the distance from the axis of rotation. The expression for the linear or tangential speed is,

  $v=r\omega$

Here, $r$ is the radial distance and $\omega$ is the angular velocity.

Tangential speed depends on rotational speed and the distance from the axis of rotation.

Calculation:

The linear speed is directly proportional to the radius of its circular path.

For case A:

Both the ends of cup A have the same radius. So the ends of cup A are moving with the equal speed.

Hence, the cup A is moving along the flat surface.

For case B:

The radius of the top end of cup B is greater than the radius of the bottom end of cup B. So the speed at the top end of cup B is more when compared to the bottom end of cup B.

Hence, the cup B is moving along the flat surface and it makes a most circular path.

For case C:

The radius of the top end of cup C is greater than the radius of the bottom end of cup B and it is nearer to the top radius. So the speed at the top end of cup B is nearer when compared to the bottom end of cup C.

Hence, the cup C is moving along the flat surface and it makes a least circular path compared to cup B.

By comparing the circular path is $B>C>A$ .

Conclusion:

Thus, the rank of the cups by the amount they depart from a straight-line path is $B>C>A$ .