Chapter 8: Rotational Dynamics

1. Splitting Motion into "Straight-line" Pieces

Whenever an object moves in two dimensions (e.g., around a bend) we find it helpful to split its acceleration and the total force acting on it into two perpendicular pieces:

Name	Symbol	What it does
Tangential component	$a_t,\,F_t$	Changes the speed along the path
Radial (centripetal) component	$a_r,\,F_r$	Bends the path, keeping the object "curving"

Mathematically

\sum F_t = m\,a_t,\qquad \sum F_r = m\,a_r

If the path lies in a plane (most textbook problems do), the out-of-plane component of the force is zero: $\sum F_z = 0$ .

2. Uniform Circular Motion (speed stays constant)

When speed is constant, the tangential piece disappears: $a_t = 0,\;F_t = 0$ .
Only a centripetal force is needed to keep the object moving in a circle:

a_r = \frac{v^{2}}{r},\qquad F_r = m\,\frac{v^{2}}{r}

Common forces (tension, gravity, friction, normal force, etc.) supply this inward pull. If that pull vanished, the object would shoot off straight.

3. Non-Uniform Circular Motion (speed changes)

If the speed is changing, we need both pieces:

\sum F_r = m\,a_r,\qquad \sum F_t = m\,a_t

The tangential force points along the path (speed-up) or opposite (slow-down). Typical cause? Gravity in a vertical circle.

4. The Vertical Circle & "Critical Speed"

Picture a ball tied to a string whirling in a vertical circle. At the very top, gravity and the string tension $T$ both point downward. To keep the string taut (so the ball doesn't fall), the centripetal requirement at that instant is

m\,\frac{v^{2}_{\text{top}}}{r}=T+mg

The lowest possible (critical) speed that still satisfies this, when $T=0$ , is

v_c=\sqrt{r\,g}

If the ball is any slower, the string goes slack and the ball drops.

5. Worked Examples

Example 1 – Choosing the Right Free-Body Diagram

A car drives through the bottom of a circular valley at constant speed. What forces act on the car at the bottom of the valley, and what is their relationship?

Gravity pulls straight down.
The road's normal force pushes up — and must be bigger than gravity so that $N - mg = m\,a_r$ !

The correct free-body diagram therefore shows a long upward normal and a shorter downward weight.

Example 2 – Two Strings on a Rotating Block

A block is tied to a vertical rod by an upper and a lower string; the system spins steadily. The upper string's tension is $80\;\text{N}$ . What is the tension in the lower string, and what is the angular speed of the system?

Step 1. Diagram & axes Take $+r$ to be outward from the rod. The block experiences

Upper string tension $T_u = 80\;\text{N}$ (pulling upward & inward along its line)
Lower string tension $T_\ell$ (pulling downward & inward)
Weight $mg$ straight down

Break each tension into vertical and horizontal (radial) components.

Step 2. Vertical equilibrium Because the block doesn't rise or fall, the vertical forces cancel:

T_{u,y} + T_{\ell,y} - mg = 0

Step 3. Horizontal (radial) force The inward components add to provide the centripetal force:

T_{u,r} + T_{\ell,r} = m\,\frac{v^{2}}{r}

(Geometry from the picture supplies the needed sines/cosines.)

Result Solving gives

$T_\ell = 31.1\;\text{N}$
Angular speed $\omega = 45.1\;\text{rad/s} \approx 430\;\text{rev/min}$

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Example 3 – Ball on a String, 30° Below the Top

A $2.0\;\text{kg}$ ball on an $L=0.80\;\text{m}$ string is $30^\circ$ below the highest point; tension is $20\;\text{N}$ . What is the speed of the ball at this position, and what is its tangential acceleration?

1. Identify forces At that instant:

Tension $T=20\;\text{N}$ (toward the center)
Weight $mg$ (downward)

Resolve weight into radial and tangential directions; the radial direction points toward the circle's center.

2. Radial equation

T + mg\cos30^\circ = m\,\frac{v^{2}}{r}

Solve for $v$ . Using $g = 9.8\;\text{m/s}^2$ :

20 + (2.0)(9.8)\cos30^\circ = 2.0\,\frac{v^{2}}{0.80}

v = 3.8\;\text{m/s}

3. Tangential acceleration Tangential component of weight: $F_t = mg\sin30^\circ$ . Thus

a_t = \frac{F_t}{m}= g\sin30^\circ = (9.8)(0.5)=4.9\;\text{m/s}^2