A coin placed on a disc rotates with a speed of 100/3 rev/min provided that the coin is not more than 10cm from the axis. Calculate the coefficient of static friction between the coin and disc answer this question plz

To determine the coefficient of static friction between the coin and the disc, we need to analyze the forces acting on the coin as it rotates. The key here is to understand how the centripetal force required to keep the coin moving in a circle relates to the frictional force between the coin and the disc.

Understanding the Forces at Play

When the disc rotates, the coin experiences a centripetal force that keeps it moving in a circular path. This force is provided by the friction between the coin and the surface of the disc. If the frictional force is not sufficient to provide the necessary centripetal force, the coin will slip off the disc.

Calculating Centripetal Force

The formula for centripetal force (\(F_c\)) is given by:

• F_c = m * a_c

Where:

• m = mass of the coin (in kg)
• a_c = centripetal acceleration (in m/s²)

The centripetal acceleration can be calculated using the formula:

• a_c = r * ω²

Here, \(r\) is the radius (distance from the axis of rotation), and \(ω\) is the angular velocity in radians per second.

Finding Angular Velocity

First, we need to convert the rotational speed from revolutions per minute (rev/min) to radians per second. The conversion is as follows:

• ω = (100/3 rev/min) * (2π rad/rev) * (1 min/60 s)

Calculating this gives:

• ω = (100/3) * (2π/60) = (100 * 2π) / (3 * 60) = (200π/180) = (10π/9) rad/s

Calculating Centripetal Acceleration

Now, substituting \(r = 0.1\) m (10 cm) and \(ω = (10π/9)\) rad/s into the centripetal acceleration formula:

• a_c = 0.1 * (10π/9)²

This simplifies to:

• a_c = 0.1 * (100π²/81) = (10π²/81) m/s²

Relating Forces to Friction

The frictional force (\(F_f\)) that keeps the coin from slipping is given by:

• F_f = μ * N

Where:

• μ = coefficient of static friction
• N = normal force (which equals the weight of the coin, \(mg\))

Setting the centripetal force equal to the frictional force gives us:

• m * a_c = μ * mg

We can cancel \(m\) from both sides (assuming \(m \neq 0\)):

• a_c = μ * g

Solving for the Coefficient of Static Friction

Now we can solve for \(μ\):

• μ = a_c / g

Substituting \(a_c = (10π²/81)\) m/s² and \(g ≈ 9.81\) m/s²:

• μ = (10π²/81) / 9.81

Calculating this gives:

• μ ≈ (10 * 9.87) / 81 ≈ 1.22 / 81 ≈ 0.015

Final Result

The coefficient of static friction between the coin and the disc is approximately 0.015. This value indicates that the frictional force is quite low, which is typical for a smooth surface like a disc. If you have any further questions or need clarification on any part of this process, feel free to ask!