Question

the circles radius in case a is twice that of case b. the period (time for one circle) is the same in each case. how does the speed in case a compare to the speed in case b?
case a
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, \(r = 2R\) and \(T\) is the same. So \(v_A=\frac{2\pi(2R)}{T}=\frac{4\pi R}{T}\).

Step3: Calculate the speed for Case B

For Case B, \(r = R\). So \(v_B=\frac{2\pi R}{T}\).

Step4: Compare \(v_A\) and \(v_B\)

Divide \(v_A\) by \(v_B\): \(\frac{v_A}{v_B}=\frac{\frac{4\pi R}{T}}{\frac{2\pi R}{T}} = 2\).

Answer:

The speed in Case A is twice the speed of Case B.