Question

the circles radius in case a is three times that of case b. the period (time for one circle) in case a is twice that of case b. how does the speed in case a compare to the speed in case b? the speed in case a is... ... the speed of case b.

Explanation:

Step1: Recall the formula for speed in circular motion

The speed \(v\) in circular motion is given by \(v=\frac{2\pi r}{T}\), where \(r\) is the radius and \(T\) is the period.

Step2: Calculate the speed for Case A

For Case A, \(r_A = 3R\) and \(T_A=2T\). Substituting into the speed formula: \(v_A=\frac{2\pi(3R)}{2T}=\frac{3\pi R}{T}\)

Step3: Calculate the speed for Case B

For Case B, \(r_B = R\) and \(T_B=T\). Substituting into the speed formula: \(v_B=\frac{2\pi R}{T}\)

Step4: Find the ratio of \(v_A\) to \(v_B\)

\(\frac{v_A}{v_B}=\frac{\frac{3\pi R}{T}}{\frac{2\pi R}{T}}\). The \(\pi R\) and \(T\) terms cancel out, giving \(\frac{v_A}{v_B}=\frac{3}{2}\)

Answer:

The speed in Case A is \(\frac{3}{2}\) times the speed in Case B.