A coin is placed 14.0 cm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 31.0 rpm (revolutions per minute) is reached, at which point the coin slides off.

Part A
What is the coefficient of static friction between the coin and the turntable?

friction force = centripetal force

μ m g = m v^2 / r

μ = v^2 / (g r)

What is a wheels velocity if it rotates through 720 degrees every two seconds

To determine the coefficient of static friction between the coin and the turntable, we can use the concept of centripetal force.

The centripetal force required to keep the coin fixed on the turntable is provided by the static friction between the coin and the turntable surface.

The centripetal force is given by the equation:

Fc = m * v^2 / r

where Fc is the centripetal force, m is the mass of the coin, v is the linear speed, and r is the distance of the coin from the axis of rotation.

In this case, we are given that the coin remains fixed on the turntable until a rate of 31.0 rpm is reached, which is equivalent to:

v = 2π * r * (31.0 rev/min) / 60

We are also given that the distance from the axis of rotation to the coin is 14.0 cm, which is equivalent to 0.14 m.

Plugging in these values, we can determine the centripetal force required to keep the coin fixed:

Fc = m * (2π * (0.14 m) * (31.0 rev/min) / 60)^2 / (0.14 m)

The coefficient of static friction (μs) can be found using the following equation:

μs = Ff / Fn

where Ff is the force of friction and Fn is the normal force.

Since the coin is stationary, the normal force and the gravitational force balance each other:

m * g = Fn

where m is the mass of the coin and g is the acceleration due to gravity.

We can now substitute the value of Fn into the equation for μs:

μs = Ff / (m * g)

To determine the coefficient of static friction, we need to find the maximum value of the force of friction before the coin starts sliding.

The maximum value of the force of friction is given by:

Ff = μs * Fn

Substituting the value of Fn and rearranging the equation, we get:

μs = Fc / (m * g)

Now we can substitute the value of Fc (centripetal force) and solve for the coefficient of static friction (μs).

To determine the coefficient of static friction between the coin and the turntable, we can use the concept of centripetal force.

When the coin starts sliding, the centripetal force acting on the coin is provided by the friction force between the coin and the turntable. At this point, the friction force equals the maximum static friction force, which can be defined as:

F_friction = μ_s * N

Where:
F_friction is the friction force
μ_s is the coefficient of static friction
N is the normal force

To find the normal force, we need to consider the forces acting on the coin. The normal force is the force exerted by the turntable perpendicular to its surface, and it balances the weight of the coin. Since the coin remains fixed on the turntable until a certain speed, the normal force is equal to the weight of the coin:

N = m * g

Where:
m is the mass of the coin
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Next, we need to find the centripetal force acting on the coin when it slides off. The centripetal force is given by:

F_centripetal = m * ω^2 * r

Where:
m is the mass of the coin
ω is the angular velocity (converted from 31.0 rpm to radians/second)
r is the radius (14.0 cm converted to meters)

The centripetal force is equal to the friction force at the point of sliding off:

F_centripetal = F_friction

Now we can solve for the coefficient of static friction:

μ_s * N = m * ω^2 * r

Substituting N = m * g:

μ_s * m * g = m * ω^2 * r

Simplifying and canceling out the mass:

μ_s * g = ω^2 * r

Finally, we can calculate the coefficient of static friction:

μ_s = (ω^2 * r) / g

Plug in the values for ω (converted to radians/second), r, and g, then solve for μ_s.