Question

A barbell spins around a pivot at its center at A. The barbell consists of two small balls, each with mass 450 grams (0.45 kg), at the ends of a very low mass rod of length d = 50 cm (0.5 m; the radius of rotation is 0.25 m). The barbell spins clockwise with angular speed 120 radians/s.



We can calculate the angular momentum and kinetic energy of this object in two different ways, by treating the object as two separate balls, or as one barbell.

I: Treat the object as two separate balls

(a) What is the speed of ball 1?
|| = m/s

(b) Calculate the translational angular momentum _{trans, 1, A} of just one of the balls (ball 1).
|_{trans, 1, A}| = kg · m²/s

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(c) Calculate the translational angular momentum _{trans, 2, A} of the other ball (ball 2).
|_{trans, 2, A}| = kg · m²/s

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(d) By adding the translational angular momentum of ball 1 and the translational angular momentum of ball 2, calculate the total angular momentum of the barbell, _{tot, A}.
|_{tot, A}| = kg · m²/s

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(e) Calculate the translational kinetic energy of ball 1.
K_trans,1 =

1

2

m||²

= J

(f) Calculate the translational kinetic energy of ball 2.
K_trans,2 =

1

2

m||²

= J

(g) By adding the translational kinetic energy of ball 1 and the translational kinetic energy of ball 2, calculate the total kinetic energy of the barbell.
K_total = J

II: Treat the object as one barbell

(h) Calculate the moment of inertia I of the barbell.
I = kg · m²
(i) What is the direction of the angular velocity vector ?

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(j) Use the moment of inertia I and the angular speed || = 120 rad/s to calculate the rotational angular momentum of the barbell:
|_rot| = I || = kg · m²/s

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(k) How does this value, |_rot|, compare to the angular momentum |_{tot, A}| calculated earlier by adding the translational angular momenta of the two balls?

|_rot| > |_{tot, A}||_rot| = |_{tot, A}|    |_rot| < |_{tot, A}|

(l) Use the moment of inertia I and the angular speed || = 120 rad/s to calculate the rotational kinetic energy of the barbell:
K_rot =

1

2

Iω²

= J
(m) How does this value, K_rot, compare to the kinetic energy K_total calculated earlier by adding the translational kinetic energies of the two balls?

K_rot > K_totalK_rot < K_total    K_rot = K_total

Answer 1