A centrifuge is revolving at the rate 10,000rpm. The radius of revolution for a 0.100-g particle being separated is 8.00 cm.

a) Identify the given quantities with their symbols and units.
b) What is the angular speed in radians per second?
c) What is the linear speed in meters per second?
d) Find the centripetal force acting on the centrifuge.

a) Given quantities:

- Angular speed (ω) = 10,000 rpm (revolutions per minute)
- Radius of revolution (r) = 8.00 cm
- Mass of particle (m) = 0.100 g

b) To convert the angular speed from rpm to radians per second:
- The conversion factor is 1 revolution = 2π radians
- 60 seconds in 1 minute

We can calculate as follows:
ω (in radians per second) = (10,000 rpm) × (2π radians/1 revolution) × (1 minute/60 seconds)
= (10,000 × 2π) radians/60 seconds
≈ 1,047.2 radians/second

So, the angular speed is approximately 1,047.2 radians per second.

c) To find the linear speed, we can use the formula:
v = rω

Since the radius is given in cm, we need to convert it to meters:
r (in meters) = 8.00 cm × (1 m/100 cm)
= 0.08 m

The linear speed is given by:
v = (0.08 m) × (1,047.2 radians/second)
= 83.8 m/s

Therefore, the linear speed is approximately 83.8 meters per second.

d) The formula for centripetal force is:
F = mω²r

- Given mass of the particle (m) = 0.100 g = 0.100 × 10⁻³ kg (since 1 g = 10⁻³ kg)
- Angular speed (ω) = 1,047.2 radians/second (from part b)
- Radius (r) = 0.08 m (from part c)

Plugging in the values, we get:
F = (0.100 × 10⁻³ kg) × (1,047.2 radians/second)² × (0.08 m)

Simplifying the equation, we have:
F ≈ 0.08 N

Therefore, the centripetal force acting on the centrifuge is approximately 0.08 Newtons.