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Physics · Ch 1 — Rotational Dynamics

Kinematics of Circular Motion

1.2.1

Kinematics of Circular Motion

To describe circular motion quantitatively we use ANGULAR versions of the usual translational kinematic quantities. Angular displacement θ\theta (in radians) plays the role distance/displacement s plays in translational motion; angular velocity ω⃗=dθ⃗dt\vec{\omega}=\frac{d\vec{\theta}}{dt} plays the role of linear velocity v⃗=ds⃗dt\vec{v}=\frac{d\vec{s}}{dt}; and angular acceleration α⃗=dω⃗dt\vec{\alpha}=\frac{d\vec{\omega}}{dt} plays the role of linear acceleration a⃗=dv⃗dt\vec{a}=\frac{d\vec{v}}{dt}. These angular quantities connect back to the familiar linear (tangential) ones through the radius vector r⃗\vec{r} (measured from the centre of the circular path to the particle): the tangential velocity is v⃗=ω⃗×r⃗\vec{v}=\vec{\omega}\times\vec{r}, with magnitude v=ωrv=\omega r.

The DIRECTION of ω⃗\vec{\omega} (and of α⃗\vec{\alpha}) is always along the axis of rotation -- never lying in the plane of the circle itself -- fixed by the right-hand thumb rule: curl the fingers of the right hand along the sense in which the particle is actually revolving, and the outstretched thumb then points along ω⃗\vec{\omega}.

Figure 1.1Fig. 1.1: Directions of angular velocity — two horizontal circular loops with the angular velocity vector along the rotation axis, upward for one sense of revolution and downward for the opposite sense (right-hand thumb rule)
Fig. 1.1 — Fig. 1.1: Directions of angular velocity — two horizontal circular loops with the angular velocity vector along the rotation axis, upward for one sense of revolution and downward for the opposite sense (right-hand thumb rule)

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A schematic showing a particle moving along a circular path in a given plane, together with the axis of rotation drawn perpendicular to that plane through the centre of the circle. The angular velocity vector ω⃗\vec{\omega} is drawn ALONG this axis (not in the plane of the circle itself), with its direction fixed by the right-hand thumb rule: if the fingers of the right hand are curled in the sense the particle is actually revolving (say anticlockwise as seen from one end of the axis), the outstretched thumb points along ω⃗\vec{\omega} from that same end. The figure's purpose is purely to establish this dot/cross-free 3-D convention -- that ω⃗\vec{\omega} is always perpendicular to t …

If the angular speed is INCREASING, α⃗\vec{\alpha} points along the same direction as ω⃗\vec{\omega}; if the speed is DECREASING, α⃗\vec{\alpha} points opposite to ω⃗\vec{\omega}. As long as α⃗\vec{\alpha} stays fixed along the (unchanging) axis of rotation, both ω⃗\vec{\omega} and α⃗\vec{\alpha} keep the same direction throughout, and we can then drop the vector notation and use plain scalar kinematics, writing the rotational analogues of the familiar ss-uu-vv-aa-tt equations exactly as summarised in Table 1.1. (If instead α⃗\vec{\alpha} had a component perpendicular to the axis, it would keep changing the DIRECTION of ω⃗\vec{\omega} without changing its magnitude -- continuously tilting the plane of rotation itself -- a more complicated situation this chapter does not pursue further.)

Figure dyk-1.2.1Angular acceleration tilting the plane of rotation — when α has a component perpendicular to the axis, it continuously changes the direction of ω and the plane of rotation itself tilts (Do you know? box sketch)
Fig. dyk-1.2.1 — Angular acceleration tilting the plane of rotation — when α has a component perpendicular to the axis, it continuously changes the direction of ω and the plane of rotation itself tilts (Do you know? box sketch)

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this sketch shows. A loop rotating with angular velocity ω₀ about a vertical axis, and the same loop a moment later after an angular acceleration α⃗\vec{\alpha} directed OFF the axis has acted: the axis itself has tilted, carrying the plane of rotation with it. As long as α⃗\vec{\alpha} stays perpendicular to ω⃗\vec{\omega}, only the DIRECTION of ω⃗\vec{\omega} changes — not its magnitude — so th …

If T is the period (time for one revolution) and n is the frequency (revolutions per second), n=1Tn=\frac{1}{T} and the angular speed of a particle in UNIFORM circular motion (UCM, where the SPEED stays constant and only the direction of velocity changes) is ω=2πT=2πn\omega=\frac{2\pi}{T}=2\pi n. For UCM, the direction of v⃗\vec{v} stays tangential at every instant, and the resulting acceleration -- called the centripetal or radial acceleration ar⃗=−ω2r⃗\vec{a_r}=-\omega^2\vec{r}, of constant magnitude ar=ω2r=v2ra_r=\omega^2r=\frac{v^2}{r} and always pointing towards the centre -- exists purely to keep bending the velocity's direction. …

Figure 1.2Fig. 1.2: Directions of linear velocity and acceleration for a particle P in circular motion — radius vector from centre C, tangential velocity at P, and the radial (centripetal) acceleration pointing towards the centre
Fig. 1.2 — Fig. 1.2: Directions of linear velocity and acceleration for a particle P in circular motion — radius vector from centre C, tangential velocity at P, and the radial (centripetal) acceleration pointing towards the centre

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A single point on a circular trajectory (radius r from the centre) is shown with its position vector r⃗\vec{r} drawn from the centre outward to that point, its linear (tangential) velocity vector v⃗\vec{v} drawn TANGENT to the circle at that point (perpendicular to r⃗\vec{r}, in the sense of motion), and the centripetal/radial acceleration vector ar⃗\vec{a_r} drawn pointing INWARD along −r⃗-\vec{r}, i.e. towards the centre of the circle. The figure visually establishes the mutually perpendicular triad at any instant of uniform circular motion: r⃗\vec{r} radially outward, v⃗\vec{v} tangential, and ar⃗\vec{a_r} radially inward, consistent with v⃗=ω⃗×r⃗\vec{v}=\vec{\omega}\times\vec{r} …

Figure 1.3Fig. 1.3: Direction of angular acceleration — for increasing speed the angular acceleration is along the angular velocity (both up); for decreasing speed it is opposite to the angular velocity
Fig. 1.3 — Fig. 1.3: Direction of angular acceleration — for increasing speed the angular acceleration is along the angular velocity (both up); for decreasing speed it is opposite to the angular velocity

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A diagram similar to Fig. 1.1, showing the same axis-of-rotation convention, but now for the angular acceleration vector α⃗=dω⃗/dt\vec{\alpha}=d\vec{\omega}/dt. It shows two cases side by side (or annotated): when the particle's speed is INCREASING, α⃗\vec{\alpha} points along the SAME direction as ω⃗\vec{\omega} (both arrows along the axis, same sense); when the speed is DECREASING, α⃗\vec{\alpha} points OPPOSITE to ω⃗\vec{\omega} (arrows along the axis but in opposite senses). No numeric values are shown; the figure exists purely to fix this sign/direction conventi …

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