Angular speed and tangential speed on a disc

Classical mechanics
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Tangential speed is the speed of an object undergoing circular motion, i.e., moving along a circular path.^[1] A point on the outside edge of a merry-go-round or turntable travels a greater distance in one complete rotation than a point nearer the center. Travelling a greater distance in the same time means a greater speed, and so linear speed is greater on the outer edge of a rotating object than it is closer to the axis. This speed along a circular path is known as tangential speed because the direction of motion is tangent to the circumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and both use units of m/s, km/h, and others.

Related quantities

Rotational speed (or rotational frequency) involves the number of revolutions per unit of time. All parts of a rigid merry-go-round or turntable turn about the axis of rotation in the same amount of time. Thus, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time. It is common to express rotational rates in revolutions per minute (RPM). When a direction is assigned to rotational speed, it is known as rotational velocity, a vector whose magnitude is the rotational speed. (Angular speed and angular velocity are related to the rotational speed and velocity by a factor of 2 $π$ , the number of radians turned in a full rotation.)

Tangential speed and rotational speed are related: the greater the "RPMs", the larger the speed in metres per second. Tangential speed is directly proportional to rotational speed at any fixed distance from the axis of rotation.^[1] However, tangential speed, unlike rotational speed, depends on radial distance (the distance from the axis). For a platform rotating with a fixed rotational speed, the tangential speed in the centre is zero. Towards the edge of the platform the tangential speed increases proportional to the distance from the axis.^[2] In equation form:

v\propto \!\,r\omega \,,

where $v$ is tangential speed and $ω$ (Greek letter omega) is rotational speed. One moves faster if the rate of rotation increases (a larger value for $ω$ ), and one also moves faster if movement farther from the axis occurs (a larger value for $r$ ). Move twice as far from the rotational axis at the centre and you move twice as fast. Move out three times as far, and you have three times as much tangential speed. In any kind of rotating system, tangential speed depends on how far you are from the axis of rotation.

When proper units are used for tangential speed $v$ , rotational speed $ω$ , and radial distance $r$ , the direct proportion of $v$ to both $r$ and $ω$ becomes the exact equation

v=r\omega \,.

This comes from the following: the linear (tangential) velocity of an object in rotation is the rate at which it covers the circumference's length:

v={\frac {2\pi r}{T))

The angular velocity ${\textstyle \omega }$ is defined as $2\pi /T$ , where T is the rotation period, hence $v=\omega r$ .

Thus, tangential speed will be directly proportional to $r$ when all parts of a system simultaneously have the same $ω$ , as for a wheel, disk, or rigid wand.

References

^ ^a ^b Hewitt 2007, p. 131
^ Hewitt 2007, p. 132

Hewitt, P.G. (2007). Conceptual Physics. Pearson Education. ISBN 978-81-317-1553-6. Retrieved 2023-07-20.
Richard P. Feynman, Robert B. Leighton, Matthew Sands. The Feynman Lectures on Physics, Volume I, Section 8–2. Addison-Wesley, Reading, Massachusetts (1963). ISBN 0-201-02116-1.

Classical mechanics SI units

Dimensions	1	L	L²	Dimensions	1	$θ$	$θ$ ²
Linear/translational quantities				Angular/rotational quantities
T	time: $t$ s	absement: $A$ m s		T	time: $t$ s
1		distance: $d$ , position: $r$ , $s$ , $x$ , displacement m	area: $A$ m²	1		angle: $θ$ , angular displacement: $θ$ rad	solid angle: $Ω$ rad², sr
T⁻¹	frequency: $f$ s⁻¹, Hz	speed: $v$ , velocity: $v$ m s⁻¹	kinematic viscosity: $ν$ , specific angular momentum: $h$ m² s⁻¹	T⁻¹	frequency: $f$ , rotational speed: $n$ , rotational velocity: $n$ s⁻¹, Hz	angular speed: $ω$ , angular velocity: $ω$ rad s⁻¹
T⁻²		acceleration: $a$ m s⁻²		T⁻²	rotational acceleration s⁻²	angular acceleration: $α$ rad s⁻²
T⁻³		jerk: $j$ m s⁻³		T⁻³		angular jerk: $ζ$ rad s⁻³
M	mass: $m$ kg	weighted position: $M ⟨ x ⟩ = \sum m x$		ML²	moment of inertia: $I$ kg m²
MT⁻¹	Mass flow rate: ${\dot {m))$ kg s⁻¹	momentum: $p$ , impulse: $J$ kg m s⁻¹, N s	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s	ML²T⁻¹		angular momentum: $L$ , angular impulse: $Δ L$ kg m² s⁻¹	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s
MT⁻²		force: $F$ , weight: $F g$ kg m s⁻², N	energy: $E$ , work: $W$ , Lagrangian: $L$ kg m² s⁻², J	ML²T⁻²		torque: $τ$ , moment: $M$ kg m² s⁻², N m	energy: $E$ , work: $W$ , Lagrangian: $L$ kg m² s⁻², J
MT⁻³		yank: $Y$ kg m s⁻³, N s⁻¹	power: $P$ kg m² s⁻³, W	ML²T⁻³		rotatum: $P$ kg m² s⁻³, N m s⁻¹	power: $P$ kg m²s⁻³, W

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