Here were two tricky questions. Please provide steps. Answers are not necessary (since these are practice questions for an exam).
1.) The centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance to cause the more less dense blood serum and collect at the bottom. The question is suppose the centripetal Acceleration is 6.5*10^3 times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 5.00 cm from axis of rotation?
2.)A penny is placed at the outer edge of a disk with radius = 0.150m. that rotates about an axis perpendicular to the plane of the disk at its center. The period is 1.8 seconds. Find the minimum coeficient of friction necessary to allow the penny to rotate along with the disk.
I have a general idea of how to do the first one.. so I'll provide the necessary equations. a = v^2/r, and 2(pie)r/v. I will be bumping this thread accordingly.
1) Solve this equation for angular velocity w (radians per second), then convert w to rpm
R*w^2 = 6500*g
R = 0.05 m.
g = 9.8 m/s^2
w = sqrt[6500(g/R)]
2. Let U be the coefficient of syatice friction
M g U = M Rw^2
U = R w^2/g
If the period is P = 1.8 s,
w = 2 pi/P = 3.49 rad/s
a. a= 6500*9.8m/s
a= w^2/r solve for w
w is in radians/second, so you have to change that to rev/min (1/(2PI*60)
b. w=2PI/1.8
centripetal force wants requires the following force to keep it in place
force needed=mw^2/r
but friction to provide this is mu*mg
mu*mg=m*w^2/r
solve for mu
Sure! Let's break down the steps for each question:
1) To find the number of revolutions per minute (rpm) the sample is making, we can use the following steps:
- Firstly, let's identify the known values:
- centripetal acceleration (ac) = 6.5 * 10^3 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- radius (r) = 5.00 cm = 0.05 m (converted to meters).
- Now, let's use the formula for centripetal acceleration: ac = v^2 / r, where v is the linear velocity.
- We can rearrange the equation to solve for v: v = sqrt(ac * r).
- Substitute the known values: v = sqrt((6.5 * 10^3 * g) * 0.05).
- Calculate v using the appropriate value for g.
- To convert the linear velocity to revolutions per minute (rpm), we use the formula: v = 2 * pi * r / T, where T is the period of one revolution.
- Rearrange the equation to solve for T: T = (2 * pi * r) / v.
- Substitute the known values and the calculated value for v.
- Finally, calculate T and convert it to rpm: rpm = 1 / T * 60.
2) To find the minimum coefficient of friction necessary for the penny to rotate along with the disk, follow these steps:
- Firstly, identify the known values:
- radius (r) = 0.150 m.
- period (T) = 1.8 seconds.
- Use the formula for period: T = 2 * pi * sqrt(r / g), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Rearrange the equation to solve for g: g = (4 * pi^2 * r) / T^2.
- Substitute the known values for r and T.
- Calculate g using the appropriate units.
- To find the minimum coefficient of friction (μ) required, we need to equate it to the ratio of normal force (N) to gravitational force (mg) exerted on the penny: μ = (N / mg).
- In this case, N is equal to mg (the penny is rotating along with the disk), so the coefficient of friction becomes μ = (mg / mg) = 1.
- Therefore, the minimum coefficient of friction required is 1.
Hope this helps! Let me know if you have any further questions.