A coin is placed 11.0 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 rpm is reached, at which point the coin slides off. What is the coefficient of static friction between the coin and the turntable?
wouldnt w be 2pir/period in s
To find the coefficient of static friction between the coin and the turntable, we can use the following steps:
Step 1: Convert the speed of the turntable to radians per second (rad/s).
To convert from rpm (revolutions per minute) to rad/s, we multiply by 2π/60.
Speed in rad/s = 31 rpm × (2π/60) = 31 × 2π/60 = 31π/30 rad/s
Step 2: Find the acceleration of the coin at the point of sliding off.
The acceleration of the coin is given by the centripetal acceleration formula: a = ω^2r, where ω is the angular velocity and r is the distance from the axis.
ω = 31π/30 rad/s (as found in step 1)
r = 11.0 cm = 0.11 m (converted to meters)
a = (31π/30)^2 × 0.11 ≈ 3.245 m/s^2
Step 3: Calculate the net force acting on the coin.
The net force acting on the coin is the product of its mass (m) and acceleration (a): Fnet = ma.
Step 4: Determine the gravitational force acting on the coin.
The gravitational force is given by Fg = mg, where m is the mass of the coin and g is the acceleration due to gravity (approximated as 9.8 m/s^2).
Step 5: Calculate the normal force exerted on the coin.
The normal force (N) is equal to the gravitational force (Fg) since the coin is not accelerating vertically.
Step 6: Determine the maximum static friction force.
The maximum static friction force (Fs) is given by Fs = μsN, where μs is the coefficient of static friction.
Step 7: Equate the maximum static friction force to the net force to find the coefficient of static friction.
Since the coin slides off when the net force exceeds the maximum static friction force, we have Fs = Fnet.
μsN = ma
Step 8: Solve for the coefficient of static friction.
Divide both sides of the equation by N to solve for μs:
μs = ma / N = ma / (mg)
By plugging in the values and calculating, we can find the coefficient of static friction between the coin and the turntable.
To determine the coefficient of static friction between the coin and the turntable, we can analyze the forces acting on the coin at the point it slides off.
First, let's calculate the acceleration of the coin at the point of sliding. We know that the rotational speed of the turntable is 31 rpm (revolutions per minute), but we need to convert it to radians per second (rad/s) to analyze the forces. Since one revolution is equal to 2π radians, we have:
Angular speed = (31 rev/min) * (2π rad/rev) * (1 min/60 s) ≈ 3.26 rad/s
Next, we can find the linear tangential speed of the coin at the point of sliding. The tangential speed is equal to the radius of rotation multiplied by the angular speed:
Linear tangential speed = (11.0 cm) * (3.26 rad/s) ≈ 35.86 cm/s
Now, let's consider the forces acting on the coin:
1. The gravitational force acts vertically downward with a magnitude of mg, where m is the mass of the coin and g is the acceleration due to gravity.
2. The normal force acts vertically upward and is equal in magnitude but opposite in direction to the gravitational force. This force prevents the coin from sinking into the turntable.
3. The static friction force acts horizontally between the coin and the turntable, opposing the relative motion between them.
If the coin is just about to slide, the static friction force reaches its maximum value, which can be expressed as:
Maximum static friction force (Ff_max) = coefficient of static friction (μs) * normal force
At the point of sliding, the tangential (centripetal) acceleration of the coin equals the square of the linear tangential speed divided by the radius of rotation. Thus, we have:
Centripetal acceleration = (Linear tangential speed)^2 / radius of rotation
To prevent the coin from sliding, the static friction force must provide the necessary centripetal force:
Centripetal force = mass of the coin * centripetal acceleration
Equating the maximum static friction force to the centripetal force, we can solve for the coefficient of static friction:
Ff_max = Centripetal force
μs * normal force = mass of the coin * [(Linear tangential speed)^2 / radius of rotation]
μs = [mass of the coin * (Linear tangential speed)^2 / (radius of rotation * normal force)]
Since we don't know the mass of the coin, we can't determine the exact value of the coefficient of static friction. However, we can still calculate the expression that represents it.
Note: To solve for the coefficient of static friction, we need additional information such as the mass of the coin or the normal force applied by the turntable.
forcefriction=mg*mu
centripetalforce= mw^2r
w= 31*2PI/60 in rad/sec
set them equal, solve for mu.