When a point on a rigid system rotates through an angle theta, it travels a curved distance called the arc length:
This is the definition of the radian — theta in radians is defined precisely as s/r. So s = r*theta is not really a formula to memorize; it is the definition of the unit rearranged.
A wheel of radius 0.4 m rotates through 270°. What arc length does a point on the rim travel?
The tangential velocity of a point on a rotating system is its instantaneous linear velocity — the speed at which it moves along its circular path, directed tangent to the circle at that point:
This comes directly from s = r*theta by differentiating both sides with respect to time: ds/dt = r * (d theta/dt), which gives v = r*omega.
A merry-go-round rotates at 0.5 rad/s. Child A sits 1.2 m from the center. Child B sits 2.4 m from the center. Find each child's linear speed and explain why B feels faster.
When a rigid system has angular acceleration alpha, every point on it also has a tangential acceleration — the rate at which its linear speed is changing:
This is the rotational analog of linear acceleration — it points tangent to the circular path and changes the magnitude of the velocity.
The central organizing principle of this lesson: for any rigid system, angular quantities are universal — the same for every point — while linear quantities scale with r.
Set omega, alpha, and two radii. Compare what happens at each point — omega and alpha are identical everywhere; v, a_T, and a_c scale with r.
| Quantity | Point A (r=0.5m) | Point B (r=1.5m) |
|---|---|---|
| omega (rad/s) | 6 ← same | same → 6 |
| alpha (rad/s²) | 2 ← same | same → 2 |
| v = r*omega (m/s) | 3.00 m/s | 9.00 m/s |
| a_T = r*alpha (m/s²) | 1.00 m/s² | 3.00 m/s² |
| a_c = r*omega² (m/s²) | 18.00 m/s² | 54.00 m/s² |
Notice: omega and alpha rows are always identical — rigid system rule. The yellow rows scale with r. Move radius B farther and watch v and a_T grow proportionally.
A 1.2 m rod rotates about one end with alpha = 4 rad/s² and omega = 3 rad/s at a given instant. Find the tangential and centripetal acceleration of the tip (r = 1.2 m) and the midpoint (r = 0.6 m).
The AP exam frequently asks you to sketch or interpret graphs of rotational and linear quantities for points on a rotating rigid system. The key is knowing which graphs are identical for all points and which differ by r.
The graph of omega vs. t is identical regardless of which point on the rigid system you pick. All points share the same omega. Slope = alpha (also the same for all).
The graph of v vs. t for a point at radius r has slope = a_T = r*alpha. A point at 2r has a line twice as steep. Same starting point if system starts from rest, but diverge as alpha acts.