# A solid disk, spinning counter-clockwise, has a mass of #13 kg# and a radius of #4/7 m#. If a point on the edge of the disk is moving at #8/5 m/s# in the direction perpendicular to the disk's radius, what is the disk's angular momentum and velocity?

The angular momentum is

The angular velocity is

The velocity at an angle is

where,

So,

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Angular momentum: ( L = I \cdot \omega ), where ( I ) is the moment of inertia and ( \omega ) is the angular velocity.

( I ) for a solid disk: ( I = \frac{1}{2} m r^2 )

( m = 13 ) kg, ( r = \frac{4}{7} ) m

( I = \frac{1}{2} \cdot 13 \cdot \left(\frac{4}{7}\right)^2 )

( I = \frac{1}{2} \cdot 13 \cdot \frac{16}{49} )

( I = \frac{104}{49} ) kg m²

Angular momentum ( L ): ( L = \frac{104}{49} \cdot \omega )

Given linear velocity ( v ): ( v = r \cdot \omega )

Solve for ( \omega ): ( \omega = \frac{v}{r} )

Substitute ( v = \frac{8}{5} ) m/s and ( r = \frac{4}{7} ) m to find ( \omega )

Angular velocity ( \omega ): ( \omega = \frac{\frac{8}{5}}{\frac{4}{7}} )

Angular velocity ( \omega ): ( \omega = \frac{56}{20} ) rad/s

Now, find angular momentum ( L ):

( L = \frac{104}{49} \cdot \frac{56}{20} )

( L = \frac{2912}{245} ) kg m²/s

Velocity ( v ): ( v = r \cdot \omega )

( v = \frac{4}{7} \cdot \frac{56}{20} )

( v = \frac{224}{140} ) m/s

Velocity ( v ): ( v = \frac{16}{5} ) m/s

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A cylinder has inner and outer radii of #8 cm# and #12 cm#, respectively, and a mass of #8 kg#. If the cylinder's frequency of rotation about its center changes from #7 Hz# to #4 Hz#, by how much does its angular momentum change?
- A solid disk with a radius of #9 m# and mass of #12 kg# is rotating on a frictionless surface. If #480 W# of power is used to increase the disk's rate of rotation, what torque is applied when the disk is rotating at #15 Hz#?
- True or false? If no external resultant torque acts on a body its angular momentum remains constant.
- A cylinder has inner and outer radii of #8 cm# and #15 cm#, respectively, and a mass of #1 kg#. If the cylinder's frequency of counterclockwise rotation about its center changes from #15 Hz# to #9 Hz#, by how much does its angular momentum change?
- A solid disk with a radius of #2 m# and mass of #2 kg# is rotating on a frictionless surface. If #640 W# of power is used to increase the disk's rate of rotation, what torque is applied when the disk is rotating at #16 Hz#?

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