A uniform disc of radius R is put over another uniform disc of radius 2R of the same thickness and density. The peripheries of the two discs touch each other. The position of their centre of mass is

To find the position of the center of mass of the system consisting of two uniform discs, one with a radius R and the other with a radius 2R, placed such that their peripheries touch each other, you can use the principle of center of mass.

First, calculate the individual center of mass of each disc. The center of mass of a uniform disc is at its geometric center.

For the disc with radius R, the center of mass (CM1) is at its center:
CM1 = (0, 0)

For the disc with radius 2R, the center of mass (CM2) is at its center:
CM2 = (0, 0)

Now, since the two discs touch each other at their peripheries, their combined center of mass (CM_total) can be found by taking the weighted average of their individual center of masses, where the weights are proportional to the masses of the discs.

The mass of a uniform disc is directly proportional to the area. Since they have the same thickness and density, the ratio of the areas is equal to the ratio of their masses. Therefore, the larger disc (2R) has 4 times the area (and mass) of the smaller disc (R).

So, you can calculate the combined center of mass as follows:

CM_total = (m1 * CM1 + m2 * CM2) / (m1 + m2)

Where:

m1 is the mass of the disc with radius R.
m2 is the mass of the disc with radius 2R.
Since m2 is 4 times larger than m1 (due to the area ratio), you have:

m1 = m
m2 = 4m

Now, plug in these values and calculate CM_total:

CM_total = (m * CM1 + 4m * CM2) / (m + 4m)

CM_total = (m * (0, 0) + 4m * (0, 0)) / (5m)

CM_total = (0, 0)

So, the center of mass of the combined system of two discs is located at the origin (0, 0).