Question

mary and her younger brother alex decide to ride the 32 - foot - diameter carousel at the state fair. mary sits on one of the horses in the outer section at a distance of 13 feet from the center. alex decides to play it safe and chooses to sit in the inner section at a distance of 7 feet from the center.
(a) what is marys angular speed ( omega_{m} ) compared to that of alexs angular speed ( omega_{a} )? give your answer as a multiple of ( omega_{a} )
( omega_{m}= ) ( omega_{a} )
(b) what is marys tangential speed ( v_{m} ) compared to that of alexs tangential speed ( v_{a} )? give your answer as a multiple of ( v_{a} ).
( v_{m}= ) ( v_{a} )
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Explanation:

Step1: Analyze angular speed

In a carousel, all points have the same angular speed. So, \(\omega_M=\omega_A\).

Step2: Analyze tangential speed

The formula for tangential speed is \(v = r\omega\). For Mary, \(v_M=r_M\omega\) and for Alex, \(v_A = r_A\omega\). Then \(\frac{v_M}{v_A}=\frac{r_M}{r_A}\). Given \(r_M = 13\) feet and \(r_A=7\) feet. So \(\frac{v_M}{v_A}=\frac{13}{7}\).

Answer:

a. \(\omega_M = 1\omega_A\)
b. \(v_M=\frac{13}{7}v_A\)