A 5.0 gram coin is placed 5.0 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 36 RPM is reached and the coin slides off.
A.) What is the coefficient of static friction between the coin and the turntable?
B.)What is the acceleration at that moment?
C.) What is the centripetal force then?
At 36 rpm, the angular velocity is
w = 36*(2 pi)/60 = 6pi/5 = 3.770 radians/sec.
A) mus*M g = M r w^2
mus = rw^2/g = (0.05)(3.77)^2/9.81 = 0.072
B)Before it starts slipping, the acceleration is
rw^2 = (0.05)(3.77)^2 = 0.71 m/s^2
After it starts slipping, it depends upon the kinetic coefficient of friction. More information is needed to get that.
c)centripetal force = M r w^2
To answer these questions, we need to use the concepts of friction, centripetal force, and acceleration.
A.) To find the coefficient of static friction between the coin and the turntable, we need to equate the centripetal force required to keep the coin in circular motion with the maximum static friction force that can be exerted on the coin. When the coin slides off, it means that the static friction reaches its maximum value.
The centripetal force required to keep the coin in circular motion is given by:
F_c = m * v^2 / r
Where:
- F_c is the centripetal force
- m is the mass of the coin (5.0 grams = 0.005 kg)
- v is the velocity of the coin (in meters/second)
- r is the radius of rotation (5.0 cm = 0.05 meters)
At the moment the coin slides off, the velocity is calculated using the given speed in RPM (rotations per minute). To convert RPM to meters/second, we need to multiply by 2π/60:
v = (36 RPM) * (2π/60) * (0.05 meters)
Now we can substitute the values into the equation for centripetal force:
F_c = (0.005 kg) * [(36 RPM) * (2π/60) * (0.05 meters)]^2 / (0.05 meters)
B.) To find the acceleration at that moment, we can use the formula:
a = v^2 / r
Where:
- a is the acceleration
- v is the velocity of the coin (in meters/second)
- r is the radius of rotation (0.05 meters)
We already calculated v in the previous step, so we can substitute it into the formula:
a = [(36 RPM) * (2π/60) * (0.05 meters)]^2 / (0.05 meters)
C.) The centripetal force at that moment can be calculated using the same formula as in part A:
F_c = (0.005 kg) * [(36 RPM) * (2π/60) * (0.05 meters)]^2 / (0.05 meters)
Now we have the equations to calculate the coefficient of static friction, acceleration, and centripetal force. Substitute the given values and solve the equations to find the answers.