Disk rolls around another disk /

The radius of the disk A is 4 cm and that of the disk B is 3 cm.

As shown in a figure, the arrow is written on the disk B.

The disk A is not moved.

The disk B rolls clockwise without sliding along with the circumference of the disk A until the arrow and the disk B come back to the original position same as in the figure.

Find the number of times that the disk B rotates around the center B.

Answer

Seven times

Solution

The formula in order to find the numbers of rotation of the circle is

(the distance the center of the circle moved) / (the length of the circumference).

For example, as shown in a figure, a circle is rolled on a straight line.

If the center of the circle of B moves 6 x 3.14 cm, a circle will rotate one time.

This is the same when rolling on a straight line or when rolling on a curve.

As shown in a figure, B rolls around A one time.

Since the center of a circle moves the distance of a dotted line which is (radius of A + radius of B) 2 x 3.14, it is (4+3) x 2 x 3.14 = 14 x 3.14 cm.

Since the circumference of B is 6 x 3.14, by the time B rolls and come back to the original position, it will rotate 14 x 3.14 / (6 x 3.14) = 7/3 times.

If B rolls integer times and comes to the position of just origin, the arrow will also come to the original position.

According to 7/3 x 3 = 7, when B rolls 3 round, the arrow also comes to the original position.

The number of times that B is rotating is seven times.