A circular disk is rotating in a horizontal plane with a time period 3.14 second.At what distance from the center a coin of mass 10 gram may be kept so that the coin coin may remains stationary the coefficient of friction is 0.52?

To determine the distance from the center at which the coin of mass 10 grams must be kept in order for it to remain stationary while the circular disk is rotating, we need to consider the centripetal force and the force of friction.

First, let's calculate the velocity of the coin. We know that the time period of rotation is 3.14 seconds, which is the time taken to complete one full revolution. The formula for time period (T) is T = 2πr/v, where r is the radius of the circular disk and v is the velocity.

Since T = 3.14 seconds, we can rearrange the formula to solve for v:
v = 2πr / T

Now substitute the given values:
v = (2 * 3.14 * r) / 3.14
v = 2r

Next, let's calculate the centripetal force required to keep the coin stationary. The centripetal force (Fc) is given by the formula Fc = mv^2 / r, where m is the mass of the coin and r is the radius from the center.

Substituting the given values:
Fc = (0.01 kg) * (2r)^2 / r
Fc = 0.04 kg * 4r
Fc = 0.16 kg * r

The force of friction (Ff) is given by the formula Ff = μN, where μ is the coefficient of friction and N is the normal force acting on the coin.

In this case, the normal force is the weight of the coin, which is given by N = mg, where m is the mass of the coin and g is the acceleration due to gravity.

Substituting the values:
N = (0.01 kg) * 9.8 m/s^2
N = 0.098 N

Now substitute the values into the equation for the force of friction:
Ff = (0.52) * (0.098 N)
Ff = 0.051 N

In order for the coin to remain stationary, the force of friction must balance the centripetal force. Thus, Ff = Fc.

Therefore, we can set the two equations equal to each other and solve for r:
0.051 N = 0.16 kg * r

Simplifying:
r = 0.051 N / (0.16 kg)
r = 0.31875 m

Therefore, the coin must be kept at a distance of approximately 0.31875 meters from the center of the circular disk in order for it to remain stationary.