A coin is placed 20 cm from the axis of a rotating turntable of variable speed. This speed is gradually increased and the coin remains fixed on the horizontal turntable until a rate of 36 rpm is reached. At this point, the coin slides off. What is the coefficient of static friction between the coin and the turntable?
To find the coefficient of static friction between the coin and the turntable, we can use the concept of centripetal acceleration.
The centripetal acceleration is given by the equation:
a = ω^2 * r
where "a" is the centripetal acceleration, ω is the angular velocity in radians per second, and "r" is the radius of the circular path.
In this case, we need to convert the given angular velocity from revolutions per minute (rpm) to radians per second. The conversion factor is:
1 rpm = 2π/60 radians/second
So, the angular velocity is:
ω = 36 rpm * (2π/60 radians/second) = 12π/5 radians/second
Now, we can substitute the values into the centripetal acceleration equation:
a = (12π/5)^2 * 0.20 meters = (144π^2/25) meters/second^2
Next, we need to consider the forces acting on the coin. There are two forces at play: the force of static friction and the weight of the coin.
The force of static friction provides the necessary centripetal force to keep the coin on a circular path. The centripetal force is given by:
F = m * a
where "F" is the centripetal force, "m" is the mass of the coin, and "a" is the centripetal acceleration.
The weight of the coin is given by:
W = m * g
where "W" is the weight, "m" is the mass of the coin, and "g" is the acceleration due to gravity.
Since the coin remains fixed on the turntable until it slides off, the force of static friction is equal to the weight of the coin. Therefore, we can set F equal to W:
m * a = m * g
The mass of the coin cancels out, giving us:
a = g
Substituting the value for "a" that we found earlier:
(144π^2/25) = g
Solving for "g", we find:
g ≈ 57.2 meters/second^2
Now, we can find the coefficient of static friction using the equation:
μ = F_s / N
where "μ" is the coefficient of static friction, "F_s" is the force of static friction, and "N" is the normal force.
In this case, the normal force is equal to the weight of the coin:
N = m * g
Substituting the value of "g" that we found earlier:
N = m * 57.2
The force of static friction is given by:
F_s = μ * N
Substituting the values, we have:
(144π^2/25) = μ * (m * 57.2)
The mass of the coin cancels out again, giving us:
(144π^2/25) = μ * 57.2
Solving for μ, we find:
μ ≈ (144π^2/25) / 57.2
So, the coefficient of static friction between the coin and the turntable is approximately (144π^2/25) / 57.2.