Linear and rotational motion

Rotational motion and motion along a line are two views of the same thing. A rigid body turns with one angular velocity $\omega$, shared by every point on it. But how fast a particular point actually moves, and how it accelerates, depends on how far that point sits from the axis.

The bridge equations make the link. A point at radius $r$ has tangential speed $v = r\omega$, tangential acceleration $a_t = r\alpha$, and sweeps an arc $s = r\theta$. Going around in a circle, it also accelerates toward the axis with the centripetal acceleration $a_c = \omega^2 r$.

For students arriving from algebra-based Physics 1, these relations will look familiar. The Physics C work is using them cleanly: every linear quantity scales with $r$, the tangential and centripetal accelerations are different perpendicular pieces, and the angle is always in radians.

Every point on a rigid rotating body shares the same $\omega$, but not the same speed. The tangential speed of a point at radius $r$ is $v = r\omega$, directed along the circle. A point twice as far from the axis moves twice as fast.

The same factor of $r$ links the angle turned to the distance traveled along the arc: $s = r\theta$. If a wheel of radius $0.3$ m turns through $2$ rad, a rim point travels $s = 0.3 \times 2 = 0.6$ m.

The common slip is to treat the body as moving at one speed everywhere, as if being rigid meant a single $v$. It does not: rigidity fixes $\omega$, and $v = r\omega$ then grows straight out from the axis, from zero at the center to its largest value at the rim.

All points share $\omega$, but tangential speed is $v = r\omega$ and arc length is $s = r\theta$. Both grow with the radius.

A point on a rotating body can have two acceleration components at once, pointing in different directions. The tangential acceleration $a_t = r\alpha$ lies along the direction of motion and changes the point's speed. The centripetal acceleration $a_c = \omega^2 r$ points toward the axis and changes the direction of motion.

They are perpendicular, so the total linear acceleration has magnitude $a = \sqrt{a_t^2 + a_c^2}$. They also answer to different things: $a_t$ is nonzero only when $\alpha \neq 0$, while $a_c$ is present whenever the body is turning, no matter what $\alpha$ does.

That second fact is where points are lost. A body spinning at a steady rate has $\alpha = 0$, so $a_t = 0$, but it is still accelerating: $a_c = \omega^2 r$ keeps pulling each point toward the axis. Constant speed is not zero acceleration.

Tangential $a_t = r\alpha$ changes speed; centripetal $a_c = \omega^2 r$ changes direction. They are perpendicular, and $a_c$ is present whenever the body turns, even at $\alpha = 0$.

The bridge equations $s = r\theta$, $v = r\omega$, and $a_t = r\alpha$ are clean only because the angle is measured in radians. The radian is defined as arc length divided by radius, which is exactly what makes $s = r\theta$ true with no extra factor.

So before using any bridge equation, convert. Multiply degrees by $\pi/180$, and multiply revolutions by $2\pi$. One full turn is $2\pi$ rad, one revolution, or $360^\circ$. An angular speed given in rev/s or deg/s has to become rad/s first.

Two slips show up. One is leaving an angle in degrees and plugging it straight in, which is off by a factor of about $57$. The other is over-correcting, dividing an angle that is already in radians by $2\pi$. Check the units before you multiply.

$s = r\theta$, $v = r\omega$, and $a_t = r\alpha$ need the angle in radians. Convert degrees ($\times\, \pi/180$) or revolutions ($\times\, 2\pi$) first.

Take a disk spinning with $\omega = 5$ rad/s and $\alpha = 2$ rad/s$^2$. Look at a point $r = 0.40$ m from the axis. Its tangential speed is $v = r\omega = 0.40 \times 5 = 2.0$ m/s, along the circle.

Its two accelerations are $a_t = r\alpha = 0.40 \times 2 = 0.8$ m/s$^2$ along the motion, and $a_c = \omega^2 r = 25 \times 0.40 = 10$ m/s$^2$ toward the axis. The centripetal part is much larger here, and the two are perpendicular, so the total is $a = \sqrt{0.8^2 + 10^2} \approx 10.0$ m/s$^2$.

Move the point in to $r = 0.20$ m and every linear quantity halves: $v = 1.0$ m/s, $a_t = 0.4$ m/s$^2$, $a_c = 5.0$ m/s$^2$. The angular quantities $\omega$ and $\alpha$ do not change, only the radius does.

At a point, read off $v = r\omega$, $a_t = r\alpha$, and $a_c = \omega^2 r$. Halving the radius halves each one, while $\omega$ and $\alpha$ stay the same.

Crossing between angular and linear quantities is where these errors live, and every pitfall here is one the diagnostic is tuned to catch.