Puck A, of inertia mm, is attached to one end of a string of length ℓℓ , and the other end of the string is attached to a pivot so that the puck is free to revolve on a smooth horizontal surface. Puck B, of inertia 20m, is attached to one end of a string of length ℓ/4, and the other end of the string is attached to a second pivot so that BB is also free to revolve. In each case, the puck is held as far as possible from the pivot so that the string is taut and then given an initial velocity v perpendicular to the string.

How does the magnitude of the angular momentum of puck A about its pivot compare with that of puck B about its pivot?

Where the bold letters indicate vectors, r is the position vector and p the moment. In the body that is in a rotation can be written as a function of the moment of inertia and the angular velocity.

L = I w

Linear and angular variables are related.

v = w L

We substitute

L = I v L

The moment of inertia of a disk with respect to its center is:

I = ½ m r²

Let's write the angular momentum for each disk.

Disc A

[tex]L_A[/tex] = ½ [tex]m_A r^2 v_A L_A[/tex]

Disc B

[tex]L_B[/tex] = ½ [tex]m_B r^2 v_B L_B[/tex]

They indicate that the mass of disk A is m and that of disk b is 20m, the length of the string is l for disk A and l / 4 for disk B.

We substitute

[tex]L_A[/tex] = ½ m r² v l

[tex]L_B[/tex] = ½ (20m) r² v ( [tex]\frac{l}{4}[/tex] )

The relationship between the angular moments is

[tex]\frac{L_B}{L_A} = \frac{20m \frac{l}{4} }{m l}\\ \frac{L_B}{L_A} = 5[/tex]

Consequently using the definitions of angular momentum we can find the result for the relationship between the two angular moments is:

[tex]\frac{L_B}{L_A} = 5[/tex]

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