Chapter 12, Problem 22A

Students Art, Bart, Cis, and Dot sit on a rotating turntable at different distances from the center as indicated.

(A) Art, $m=60kg$, sits at $\frac{1}{4}r$.

(B) Bart, $m=25kg$, sits at $\frac{1}{2}r$.

(C) Cis, $m=50kg$, sits at $\frac{3}{4}r$.

(D) Dot, $m=20kg$, sits at r.

From greatest to least, rank the angular momenta of the four students.

To determine

The rank of the angular momenta of the four students from greatest to least.

Answer to Problem 22A

The rank of the angular momenta of the four students from greatest to least is $L_{\text{C}}>L_{\text{D}}>L_{\text{A}}>L_{\text{B}}$ .

Explanation of Solution

Formula used:

The expression for the angular momenta as follows:

  $L=mvr$

Here, $m$ is the mass, $v$ is the tangential speed, and $r$ is the radius of the curved path.

Calculation:

Find the angular momenta for case A as follows:

  $\begin{array}{l}{L}_{\text{A}}=m{v}_{\text{A}}{r}_{\text{A}}\\ L_{\text{A}}=\left(60\text{ kg}\right)v\left(\frac{r}{4}\right)\\ L_{\text{A}}=15vr\end{array}$

Find the angular momenta for case B as follows:

  $\begin{array}{l}{L}_{\text{B}}=m{v}_{\text{B}}{r}_{\text{B}}\\ L_{\text{B}}=\left(25\text{ kg}\right)v\left(\frac{r}{2}\right)\\ L_{\text{B}}=12.5vr\end{array}$

Find the angular momenta for case C as follows:

  $\begin{array}{l}{L}_{\text{C}}=m{v}_{\text{C}}{r}_{\text{C}}\\ L_{\text{C}}=\left(50\text{ kg}\right)v\left(\frac{3}{4}r\right)\\ L_{\text{C}}=37.5vr\end{array}$

Find the angular momenta for case D as follows:

  $\begin{array}{l}{L}_{\text{C}}=m{v}_{\text{C}}{r}_{\text{C}}\\ L_{\text{C}}=\left(20\text{ kg}\right)v\left(r\right)\\ L_{\text{C}}=20vr\end{array}$

By comparing the values, the angular momenta is $L_{\text{C}}>L_{\text{D}}>L_{\text{A}}>L_{\text{B}}$ .

Conclusion:

Thus, the rank of angular momenta of the four students from greatest to least is $L_{\text{C}}>L_{\text{D}}>L_{\text{A}}>L_{\text{B}}$ .