A point moves at a steady 3.0 m/s around a circle of radius 0.60 m. No stopwatch has timed a revolution yet.
Can the speed formula be run backward, to hand back the period or the radius?
The point's numbers are known: a steady 3.0 m/s on a circle of radius 0.60 m. The period is the unknown.
Plain arithmetic gets close. One revolution covers 2 × π × 0.60 = 3.8 m, and the point covers 3.0 m each second, so a revolution takes a little over one second.
You have seen the equation: v = 2πr/T. So far the unknown has always been v.
The same equation holds when the unknown is T or r. The work is making the unknown the subject.
To make T the subject: multiply both sides by T, to get vT = 2πr. Then divide both sides by v.
That leaves T = 2πr/v.
To make r the subject: start again from vT = 2πr. Then divide both sides by 2π.
That leaves r = vT/(2π).
Units stay honest: T comes out in seconds, and r comes out in meters.
A point moves at a steady 3.0 m/s around a circle of radius 0.60 m. Calculate the period.
Write down the values:
v = 3.0 m/s, r = 0.60 m, T = ?
Write down the equation:
v = 2πr/T
Make T the subject:
T = 2πr/v
Substitute in the values, and calculate:
T = (2 × π × 0.60) ÷ 3.0 = 1.3
The period is 1.3 s.
Elif runs at a steady 4.0 m/s around a circular trail, and each lap takes 47 s. Calculate the trail's radius.
Write down the values:
v = 4.0 m/s, T = 47 s, r = ?
Write down the equation:
v = 2πr/T
Make r the subject:
r = vT/(2π)
Substitute in the values, and calculate:
r = (4.0 × 47) ÷ (2 × π) = 30
The trail's radius is 30 m.
Fergus's drone flies a circular loop at a steady 6.0 m/s, one lap every 21 s. He wants the loop's radius. Which rearrangement of v = 2πr/T should he use?
The same drone: v = 6.0 m/s and T = 21 s. Substitute into your rearranged equation and calculate the loop's radius, in m, to the nearest 1 m. Answers within 1 m are accepted.
A remote-control car circles at a steady 2.0 m/s on a circle of radius 1.4 m. Enter the period in s, to the nearest 0.1 s. Answers within 0.1 s are accepted.
Henrik swings a ball in a horizontal circle at a steady 2.2 m/s, 0.90 m from his hand. Enter the time one revolution takes, in s, to the nearest 0.1 s. Answers within 0.1 s are accepted.
A motorcycle holds a steady 12 m/s around a flat circular course, and each lap takes 9.0 s. Enter the course's radius in m, to the nearest 1 m. Answers within 1 m are accepted.
A cable car rounds a circular turnaround at a steady 6.0 m/s, and one circuit takes 14 s. Which calculation gives the turnaround's radius in meters?