Question

A horizontal circular platform rotates counterclockwise about its axis at the rate of 0.821 rad/s. You, with a mass of 72.1 kg, walk clockwise around the platform along its edge at the speed of 1.19 m/s with respect to the platform. Your 20.9 kg poodle also walks clockwise around the platform, but along a circle at half the platform's radius and at half your linear speed with respect to the platform. Your 18.1 kg mutt, on the other hand, sits still on the platform at a position that is 3/4 of the platform's radius from the center. Model the platform as a uniform disk with mass 92.7 kg and radius 1.89 m. Calculate the total angular momentum of the system. kg m2/s 248.26 total angular momentum:

Answer 1

Moment of Inertia of Platform 1 I = Į MR? = 4x (92.7) (1.8934 3 165.57 Angular momentum L='IW =165-57% 6.821 - 135.93 moment of intertia of , ad person - 72.1% (1,893? = 257.55 Angular momentui , L= I w := 257.55 (0.821 - 1.19 ) = 49.2879 . I = MR? 1.89

Momentum of Inertia of mult ,I=mp? J = 18.1 (32X1-21) = 33.355 Angular momentum, L=1W = 33,355 (0.821) = 27.387 moment of Inertia of poodle 1 I = mR?. 520.9 (481)? = 17.118 Angular momentun L = I w = 17J18(0.82) - Luis = 3.28 Total angular momento L = 135.93 +49.287 +27.38 +3.28 = 215.88 kg.ms (counter Clockwie)

Answer 2

Find the angular momentum ?$L$ of each component of this system by multiplying its moment of inertia ?$I$ by its angular velocity ?.$\omega .$

?=$$L=I\omega$$

Since all the components revolve about the same axis, the total angular momentum ?tot$L_{\text{tot}}$ of the system is the algebraic sum of the angular momenta of its components with respect to that axis.

$$L_{\text{tot}}=L_{\text{plat}}+L_{\text{you}}+L_{\text{pood}}+L_{\text{mutt}}$$

Now, deal with each component's angular moment in turn. Start with that of the platform. Its moment of inertia ?plat$I_{\text{plat}}$ is

$$I_{\text{plat}}=\frac{1}{2}{M}_{\text{plat}}{R}_{\text{plat}}^2$$

where ?plat$M_{\text{plat}}$ and ?plat$R_{\text{plat}}$ are its mass and radius, respectively, modeled as a disk. Its angular momentum ?plat$L_{\text{plat}}$ is

$$L_{\text{plat}}=I_{\text{plat}}{\omega }_{\text{plat}}=\frac{1}{2}{M}_{\text{plat}}{R}_{\text{plat}}^2{\omega }_{\text{plat}}$$

where ?plat${\omega }_{\text{plat}}$ is its angular speed. This angular momentum is taken to be positive since it is counterclockwise.

Now, find your angular momentum. Your moment of inertia ?you$I_{\text{you}}$ is

$$I_{\text{you}}=M_{\text{you}}{R}_{\text{plat}}^2$$

since your radius of revolution is the same as the platform's radius and ?you$M_{\text{you}}$ is your given mass. Your angular speed ?′you${\omega }_{\text{you}}^{\prime }$ with respect to the platform is

$${\omega }_{\text{you}}^{\prime }=\frac{v_{\text{you}}}{R_{\text{plat}}}$$

where ?you$v_{\text{you}}$ is your given tangential speed with respect to the platform as you walk around the edge of the platform. Since you walk clockwise and the platform is rotating counterclockwise, your angular velocity ?you${\omega }_{\text{you}}$ with respect to the ground is

$${\omega }_{\text{you}}={\omega }_{\text{plat}}-{\omega }_{\text{you}}^{\prime }={\omega }_{\text{plat}}-\frac{v_{\text{you}}}{R_{\text{plat}}}$$

and is expressed here in terms of given quantities. Your angular momentum ?you$L_{\text{you}}$ is

$$L_{\text{you}}=I_{\text{you}}{\omega }_{\text{you}}=M_{\text{you}}{R}_{\text{plat}}^2({\omega }_{\text{plat}}-\frac{v_{\text{you}}}{R_{\text{plat}}})$$

You poodle's moment of inertia ?pood$I_{\text{pood}}$ is

$$I_{\text{pood}}=M_{\text{pood}}{\left(\frac{R_{\text{plat}}}{2}\right)}^2=\frac{1}{4}{M}_{\text{pood}}{R}_{\text{plat}}^2$$

Considerations similar to those for your angular velocity give the angular velocity ?pood${\omega }_{\text{pood}}$ of your poodle as

$${\omega }_{\text{pood}}={\omega }_{\text{plat}}-\frac{\left(v_{\text{you}}/2\right)}{\left(R_{\text{plat}}/2\right)}={\omega }_{\text{plat}}-\frac{v_{\text{you}}}{R_{\text{plat}}}$$

which happens be the same as yours. Your poodle's angular momentum ?pood$L_{\text{pood}}$ is

$$L_{\text{pood}}=I_{\text{pood}}{\omega }_{\text{pood}}=\frac{1}{4}{M}_{\text{pood}}{R}_{\text{plat}}^2({\omega }_{\text{plat}}-\frac{v_{\text{you}}}{R_{\text{plat}}})$$

Now for your mutt, who has wisely decided to sit still and enjoy the ride. Its moment of inertia ?mutt$I_{\text{mutt}}$ is

$$I_{\text{mutt}}=M_{\text{mutt}}{\left(\frac{3R_{\text{plat}}}{4}\right)}^2=\frac{9}{16}{M}_{\text{mutt}}{R}_{\text{plat}}^2$$

Its angular velocity is the same as the platform's, and its angular momentum ?mutt$L_{\text{mutt}}$ is

$$L_{\text{mutt}}=I_{\text{mutt}}{\omega }_{\text{plat}}=\frac{9}{16}{M}_{\text{mutt}}{R}_{\text{plat}}^2{\omega }_{\text{plat}}$$

Now, add all four angular momenta together, and perform some algebraic operations, to obtain the requested total angular momentum ?tot$L_{\text{tot}}$ expressed in terms of given quantities.

$$L_{\text{tot}}=\frac{1}{16}(8M_{\text{plat}}+16M_{\text{you}}+4M_{\text{pood}}+9M_{\text{mutt}})R_{\text{plat}}^2{\omega }_{\text{plat}}-\frac{1}{4}(4M_{\text{you}}+M_{\text{pood}})R_{\text{plat}}{v}_{\text{you}}$$

For the numerical answer, substitute the given values.

?tot=116(8⋅92.7+16⋅70.1+4⋅20.5+9⋅17.1)1.892⋅0.967−14(4⋅70.1+20.5)1.89⋅1.03=307 kg⋅m2/s