A U.S quarter is rolling on the floor without slipping in such a way that it describes a circular path of radius R=4 cm. The plane of the coin is tilted at an angle of θ=45∘ with respect to the horizontal plane. Find the coin's period T in seconds, that is, the time it takes for the coin to go around the circle of radius R. The radius of a U.S quarter is r=1.2 cm.

To find the coin's period T, we can use the equation for the period of a rotating object. The period is the time it takes for the coin to complete one revolution around the circle.

The period T is given by the equation:

T = 2π√(I / mgh)

Where:
- T is the period
- π is a mathematical constant approximately equal to 3.14159
- I is the moment of inertia of the coin
- m is the mass of the coin
- g is the acceleration due to gravity
- h is the height of the center of mass of the coin above the pivot point

First, we need to calculate the moment of inertia I of the coin. The moment of inertia depends on the shape of the object and how the mass is distributed around the axis of rotation.

Assuming the U.S quarter is a flat disk, the moment of inertia can be calculated using the formula for a solid disk:

I = (1/2)mr^2

Where:
- m is the mass of the coin
- r is the radius of the coin

Given that the radius of the U.S quarter is r = 1.2 cm, we can convert it to meters by dividing it by 100:

r = 1.2 cm = 0.012 m

Next, we need to calculate the mass m of the coin. The mass of a U.S quarter is approximately 5.67 grams.

m = 5.67 grams

Since the acceleration due to gravity g is approximately 9.8 m/s^2 and the center of mass of the coin is at the same height as the pivot point, h = 0.

Substituting the values into the equation for the period T:

T = 2π√((1/2)m(0.012 m)^2 / (5.67 grams)(9.8 m/s^2)(0))

Simplifying the equation:

T = 2π√((1/2)(5.67 grams)(0.012 m)^2 / (5.67 grams)(9.8 m/s^2)(0))

Since the height h is zero, the denominator becomes zero, which means the period is undefined. Therefore, we cannot calculate the period of the rolling coin under these conditions.