A penny is placed at the outer edge of a disk (radius = 0.159 m) that rotates about an axis perpendicular to the plane of the disk at its center. The period of the rotation is 1.50 s. Find the minimum coefficient of friction necessary to allow the penny to rotate along with the disk
Static Friction Coefficient = Cfs
For of Static Friction = Fs
Normal Force = Fn
Centripetal Force = Fc
mass = m
radius = r
speed = v
period of time = T
Fs = Cfs * Fn so Cfs = Fs/Fn
Fc = Fs and since Fs = mv^2/r
Cfs = Fs/Fn = mv^2/r*Fn
v = 2πr/T and Fn = mg
so....
mv^2 (2πr/T)^2 4(π^2)r
Cfs = ------ = ----------- = --------
rFn r(g) g(T^2)
4(π^2)(0.159 m)
Cfs = ------------------- = 0.28467
(9.8 m/s^2)(1.50 s)^2
*** Remember there are no units on coefficients***
well that doesn't look readable. the answer should still be right but I'll rewrite how I did the math
Cfs = mv^2/rFn = ((2πr/T)^2)/r(g) = (4(π^2)r)/g(T^2)
Cfs = 4(π^2)(0.159 m) / (9.8 m/s^2)(1.50 s)^2 =
0.28467
To find the minimum coefficient of friction necessary to allow the penny to rotate along with the disk, we need to consider the forces acting on the penny and use the concept of circular motion.
1. Start by identifying the forces acting on the penny:
- The weight of the penny (mg), where m is the mass of the penny and g is the acceleration due to gravity.
- The normal force (N) exerted by the disk on the penny, which is perpendicular to the disk's surface.
- The frictional force (f) opposing the motion of the penny.
2. In order for the penny to rotate along with the disk, the frictional force must provide the necessary centripetal force to keep the penny moving in a circle. Thus, the frictional force must be equal to the centripetal force.
3. The centripetal force (Fc) is given by the formula: Fc = m * ac, where m is the mass of the penny and ac is the centripetal acceleration.
4. The centripetal acceleration (ac) can be determined using the formula: ac = (2 * π * r) / T, where r is the radius of the disk and T is the period of rotation.
5. Now we can set up the condition for the minimum coefficient of friction. The frictional force is given by f = μN, where μ is the coefficient of friction. The normal force (N) can be calculated as N = mg.
6. Replace the value of ac in the centripetal force formula and set it equal to the frictional force: ac = f / m. Substitute ac with (2 * π * r) / T, and f with μN.
7. Setting the two expressions equal gives: f / m = (2πr) / T. Now substitute f with μN and N with mg.
8. We have: (μmg) / m = (2πr) / T. Simplify the equation by canceling out the mass term.
9. The minimum coefficient of friction (μmin) necessary can be found by solving for μ: μmin = (2πr) / (gT).
10. Substitute the given values for the radius (r) and the period of rotation (T) into the equation, and use the standard value for the acceleration due to gravity (9.8 m/s^2) to calculate μmin.
By following these steps, you can find the minimum coefficient of friction necessary to allow the penny to rotate along with the disk.
Hey, Do your homework. Look at your notes and read the book. We are watching you.
Dr. C will not be happy.