Q: A centrifuge at NASA's Ames Research Center spins a mechanical arm of length $$r = 8\ \mathrm{m}$$ such that mass $$m_1$$ experiences a centripetal acceleration of $$12g$$, where $$g = 9.81\ \mathrm{m/s^2$}. $$What is the speed $$v_1 of $$mass $$m_1$$?

$$v_1$$ = \sqrt{12 \times 9.81 \times 8}$$

Q: If mass $$m_2 is $$located $$2\ \mathrm{m}$$ farther from the center than $$m_1 (so$$ $$r_2 = 10$$\ \mathrm{m}$$), and both are attached to the same rotating arm, what is the speed v_2$ of mass m_2$?

$$v_2$$ = $$v_1$$ \times \sqrt{\dfrac{$$r_2$$}{$$r_1$$}}$$

Q: Given the arm rotates at a constant angular velocity $$\omega$$, what is the relationship between the linear speeds $$v_1$$ and $$v_2 of $$masses $$m_1$$ and $$m_2$$?

$$v_2$$ = $$v_1$$ \times \dfrac{$$r_2$$}{$$r_1$$}$$

Q: If mass $$m_2 is$$ $$10\ \mathrm{m}$$ from the center and completes $$N$$ revolutions per minute, what is the formula for $$N in $$terms of $$v_2$$?

$$N = \dfrac{v_2$}{2\pi r_2$} \times 60$$

Question 1

A centrifuge at NASA's Ames Research Center spins a mechanical arm of length $$r = 8\ \mathrm{m}$$ such that mass $$m_1$$ experiences a centripetal acceleration of $$12g$$, where $$g = 9.81\ \mathrm{m/s^2$}. $$What is the speed $$v_1 of $$mass $$m_1$$?

Accepted Answer

$$v_1$$ = \sqrt{12 	imes 9.81 	imes 8}$$

Question 2

If mass $$m_2 is $$located $$2\ \mathrm{m}$$ farther from the center than $$m_1 (so$$ $$r_2 = 10$$\ \mathrm{m}$$), and both are attached to the same rotating arm, what is the speed v_2$ of mass m_2$?

Accepted Answer

$$v_2$$ = $$v_1$$ 	imes \sqrt{\dfrac{$$r_2$$}{$$r_1$$}}$$

Question 3

Given the arm rotates at a constant angular velocity $$\omega$$, what is the relationship between the linear speeds $$v_1$$ and $$v_2 of $$masses $$m_1$$ and $$m_2$$?

Accepted Answer

$$v_2$$ = $$v_1$$ 	imes \dfrac{$$r_2$$}{$$r_1$$}$$

Question 4

If mass $$m_2 is$$ $$10\ \mathrm{m}$$ from the center and completes $$N$$ revolutions per minute, what is the formula for $$N in $$terms of $$v_2$$?

Accepted Answer

$$N = \dfrac{v_2$}{2\pi r_2$} 	imes 60$$

Question 5

If the centripetal acceleration at $$m_1 is$$ $$a_c = 12g$$, what is the angular velocity $$\omega of $$the arm?

Accepted Answer

$$\omega = \sqrt{\dfrac{12g}{r_1$}}$$

Question 6

If the arm is $$8\ \mathrm{m}$$ long and rotates such that $$m_1$$ experiences $$12g$$ acceleration, what is the period $$T of $$one revolution?

Accepted Answer

$$T = \dfrac{2\pi}{\omega}$$, where $$\omega = \sqrt{\dfrac{12g}{8}}$$

Question 7

If the rod connecting $$m_1$$ and $$m_2 is $$rigid and massless, what force does the rod exert on $$m_2 if$$ $$m_2 is $$farther from the center?

Accepted Answer

A tension force directed toward the center, equal to $$m_2$$ \$$omega^2$$ $$r_2$$

Question 8

If the arm suddenly stops, what path will $$m_2$$ follow immediately after release?

Accepted Answer

A straight line tangent to the circle at the point of release

Question 9

If $$m_1$$ has a mass of $$50\ \mathrm{kg}$$ and $$m_2$$ has a mass of $$100\ \mathrm{kg}$$, does the mass affect the speed required for $$12g$$ acceleration?

Accepted Answer

No, the required speed depends only on radius and acceleration

Question 10

If the rod between $$m_1$$ and $$m_2$$ breaks, what happens to $$m_2$$?

Accepted Answer

It moves in a straight line tangent to its circular path at the moment of breakage

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