A disk has a mass 3.5 kg and radius 15 cm, rotating with angular speed 15 rev/s when a second disk of 5.0 kg is dropped onto it. If second disk has diameter 18 cm and mass 5.0 kg, what is the common final angular speed of the system?

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A disk has a mass 3.5 kg and radius 15 cm, rotating with angular speed 15 rev/s when a second disk of 5.0 kg is dropped onto it. If second disk has diameter 18 cm and mass 5.0 kg, what is the common final angular speed of the system?

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Answer 1 · 2017-02-26T08:29:54

$9.9 rev per sec$ $9.9" rev per sec"$ , rounded to one decimal place.

Explanation:

Law of Conservation of angular momentum states:
The total angular momentum of a system about an axis remains constant, when the net external torque acting on the system about the given axis is zero.

This is applicable in the given question as a stationary second disk is dropped on a rotating disk. It is assumed that second disk is also mounted on the same shaft.

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Angular momentum $\to L = \to I \times \to ω$ $vecL=vecI xxvec omega$
where angular velocity of disk rotating with angular speed $f$ $f$ is $ω = 2 π f$ $omega=2pif$
Moment of inertia of disk $I = \frac{1}{2} m_{disk} R^{2}$ $I=1/2m_"disk"R^2$

Initial Angular momentum $= {\to L}_{1} + {\to L}_{2}$ $=vecL_1+vecL_2$
$= ∣ ∣ ∣ {\to L}_{1} ∣ ∣ ∣ + 0$ $=|vecL_1|+0$
$= (\frac{1}{2} m_{1} R_{1}^{2}) (2 π f)$ $=(1/2m_1R_1^2)(2pif)$
$= (3.5 \times {(0.15)}^{2}) (π \times 15)$ $=(3.5xx(0.15)^2)(pixx15)$
$= 1.18125 π$ $=1.18125pi$

If $f_{c}$ $f_c$ is the final common angular velocity of the system,
Final Angular momentum $= ({\to I}_{1} + {\to I}_{2}) \times {\to ω}_{c}$ $=(vec I_1+vec I_2)xxvecomega_c$

Inserting given values and equating scalar part with (1) we get, (remember to change the given diameter of second disk to its radius)
$(\frac{1}{2} m_{1} R_{1}^{2} + \frac{1}{2} m_{2} R_{2}^{2}) 2 π f_{c} = 1.18125 π$ $(1/2m_1R_1^2+1/2m_2R_2^2)2pif_c=1.18125pi$
$\Rightarrow (3.5 \times {(0.15)}^{2} + 5.0 \times {(\frac{0.18}{2})}^{2}) π f_{c} = 1.18125 π$ $=>(3.5xx(0.15)^2+5.0xx(0.18/2)^2)pif_c=1.18125pi$
$\Rightarrow f_{c} = \frac{1.18125}{3.5 \times {(0.15)}^{2} + 5.0 \times {(\frac{0.18}{2})}^{2}}$ $=>f_c=1.18125/(3.5xx(0.15)^2+5.0xx(0.18/2)^2)$
$f_{c} = 9.9 rev per sec$ $f_c=9.9" rev per sec"$ , rounded to one decimal place.

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A disk has a mass 3.5 kg and radius 15 cm, rotating with angular speed 15 rev/s when a second disk of 5.0 kg is dropped onto it. If second disk has diameter 18 cm and mass 5.0 kg, what is the common final angular speed of the system?

1 Answer

Explanation:

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