Two small beads of mass m are free to slide on a frictionless rod of mass M and

Question

Two small beads of mass m are free to slide on a frictionless rod of mass M and length L, as shown in Fig. 13-34. Initially the beads are held together at the rod center, and the rod is set spinning freely with initial angular speed ?0 about a vertical axis coming out of the page in Fig. 13-34. The beads are released, and they slide to the ends of the rod and then off.

Find the expressions for the angular speed of the rod at the following times. (Use the following as necessary: m, M, L, and ?0.)

(a) when the beads are halfway to the ends of the rod

(b) when they're at the ends

(c) after the beads are gone

Hint: Two of the answers are the same.

Two small beads of mass m are free to slide on a frictionless rod of mass M and length L, as shown in Fig. 13-34. Initially the beads are held together at the rod center, and the rod is set spinning freely with initial angular speed ?0 about a vertical axis coming out of the page in Fig. 13-34. The beads are released, and they slide to the ends of the rod and then off. Find the expressions for the angular speed of the rod at the following times. (Use the following as necessary: m, M, L, and ?0.) (a) when the beads are halfway to the ends of the rod (b) when they're at the ends (c) after the beads are gone Hint: Two of the answers are the same.

Explanation / Answer

a)
Io = M*L^2/12

I' = (M*L^2/12)+ 2*m*(L/4)^2

= M*L^2/12 + m*L^2/8

here angular momentum is conserved.

Lo = L'

Io*wo = I'*w'

w' = Io*wo/I'

= (M*L^2/12)*wo/(M*L^2/3 + m*L^2/8)

b)

Io = M*L^2/12

I' = (M*L^2/12)+ 2*m*(L/2)^2

= M*L^2/12 + m*L^2/2

here angular momentum is conserved.

Lo = L'

Io*wo = I'*w'

w' = Io*wo/I'

= (M*L^2/12)*wo/(M*L^2/3 + m*L^2/2)

c)

Io = M*L^2/12

I' = M*L^2/12

here angular momentum is conserved.

Lo = L'

Io*wo = I'*w'

here I' = Io

so, w' = wo

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