Question

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

case a

case b

the speed in case a is...

tap to select answer.

... the speed of case b.

Explanation:

Step1: Calculate the circumference for each case

The formula for the circumference of a circle is \(C = 2\pi r\).
For Case A: \(C_A=2\pi(2R) = 4\pi R\)
For Case B: \(C_B = 2\pi R\)

Step2: Calculate the speed for each case

The formula for speed is \(v=\frac{d}{t}\) (where \(d\) is distance and \(t\) is time). Here, \(d\) is the circumference and \(t\) is the period.
For Case A: \(v_A=\frac{C_A}{t_A}=\frac{4\pi R}{\frac{1}{2}T}=\frac{8\pi R}{T}\)
For Case B: \(v_B=\frac{C_B}{t_B}=\frac{2\pi R}{T}\)

Step3: Find the ratio of \(v_A\) to \(v_B\)

\(\frac{v_A}{v_B}=\frac{\frac{8\pi R}{T}}{\frac{2\pi R}{T}} = 4\)

Answer:

The speed in Case A is \(4\) times the speed of Case B.