8. In an ultracentrifuge a solution is rotated with an angular speed of 3000 rev/sec at a radius of 10 cm. How large is the centripetal acceleration of each particle in the solution? Compare the centripetal force needed to hold a particle of mass 10-5 gm with the weight of the particle.
To find the centripetal acceleration of each particle in the solution, we can use the formula for centripetal acceleration:
ac = ω^2 * r
where ac is the centripetal acceleration, ω is the angular speed, and r is the radius.
Given that the angular speed is 3000 rev/sec and the radius is 10 cm (or 0.1 m), we can plug these values into the formula:
ac = (3000 rev/sec)^2 * 0.1 m
To convert rev/sec to radians/sec, we need to multiply by 2π since there are 2π radians in one revolution:
ac = (3000 * 2π rad/sec)^2 * 0.1 m
Simplifying:
ac = (6000π rad/sec)^2 * 0.1 m
ac = (36000000π^2) m/s^2
The centripetal acceleration of each particle in the solution is approximately 113097336000 m/s^2.
To compare the centripetal force needed to hold a particle of mass 10^-5 gm with the weight of the particle, we need to consider the formula for centripetal force:
Fc = m * ac
where Fc is the centripetal force, m is the mass of the particle, and ac is the centripetal acceleration.
Given that the mass of the particle is 10^-5 gm (or 10^-8 kg), we can plug this value into the formula:
Fc = (10^-8 kg) * 113097336000 m/s^2
Fc = 1130973.36 N
The centripetal force needed to hold the particle is approximately 1130973.36 Newtons.
Now, let's compare this with the weight of the particle. The weight of an object can be calculated using the formula:
Fw = m * g
where Fw is the weight, m is the mass of the particle, and g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Given that the mass of the particle is 10^-5 gm (or 10^-8 kg), we can plug this value into the formula:
Fw = (10^-8 kg) * 9.8 m/s^2
Fw = 9.8 * 10^-8 Newtons
The weight of the particle is approximately 9.8 * 10^-8 Newtons.
Comparing the centripetal force needed (1130973.36 Newtons) with the weight of the particle (9.8 * 10^-8 Newtons), we can see that the centripetal force needed to hold the particle is much larger than the weight of the particle.