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'''Rotational mechanics''' is the area of [[mechanics]] concerned with objects that are rotating or following curved paths. An example is [[circular motion]]. In general the motion of any object can be thought of in terms of a combination of linear and rotational motions.
==Rotational Kinematics==
Rotational kinematics involve describing the motion of rotating objects without understanding the forces that cause them to undergo that motion. It has many similarites to linear motion. Instead of [[displacement]] (distance) in linear mechanics, rotational mechanics considers angular displacement which is the angle something has rotated by. Similarly, [[velocity]] (rate of change of displacment) becomes angular velocity, the rate of change of angular displacement over time. Normally the Greek letter θ is used to denote angular displacement, an ω (omega) is used to denote angular velocity and α to denote angular acceleration. Angular displacement has units of [[radians]] so angular velocity has units of radians per second. These quantities can be related to each other through:
<math>
\alpha = \frac{\mathrm{d} \omega}{\mathrm{d} t} = \frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2}
</math>
From these definitions, rotational versions of the [[SUVAT equations]] can be derived when the angular acceleration is constant. They are:
*<math>\omega = \omega_0 + \alpha t</math>
*<math>\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2</math>
*<math>\theta = \theta_0 + \omega - \frac{1}{2} \alpha t^2</math>
*<math>\theta = \theta_0 + \frac{1}{2}(u+v)t</math>
*<math>\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)</math>
where θ<sub>0</sub> is the inital angular displacement and ω<sub></sub> is the final angular velocity.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#drot Rotation equations on hyperhpysics]</ref>
==Vector Nature of Angular Velocity==
In order to describe which axis an object is rotating around, angular velocity is a [[vector]] quantity. It is defined as having a magnitude equal to the angular speed of the object and points along the axis of rotation defined by the [[right hand rule]]. In this way, angular displacement and angular accleration are also vector quantities.
Angular velocity is an example of a pseudovector.<ref>[http://mathworld.wolfram.com/Pseudovector.html mathworld.wolfram.com]</ref> This means that although it often behaves like a vector, it does not always behave as a true vector would. An example of this is mirror symmetry. If a vector is reflected in a mirror, it will generally point in a different direction. However, if a rotating objecct is reflected in a mirror, it rotates in the same way and its angular velocity vector points in the same direction as before.
==Relation to Linear Mechanics==
Suppose an object is rotating about an axis ar some distance r. Suppose it moves a small distance ds, which is equal to r dθ for a small angular displacement dθ. Dividing both sides by dt to produce:
<math>
\frac{\mathrm{d} s}{\mathrm{d} t} = r \frac{\mathrm{d} \theta}{\mathrm{d} t}
</math>
which is simply v=rω. This equation relates the speed of a particle to its angular speed.
Furthermore, the acceleration of the particle can be written in terms of a radial component (directed towards the axis of rotation) and a tangential component (perpendicular to the radial component). Differentiating v=rω again produces a relation between the tangential acceleration and angular acceleration: a<sub>tan</sub>=rα. The radial component is the centripetal acceleration and causes the particle to rotate about an axis. It can be expressed as: a<sub>rad</sub>=ω<sup>2</sup>r.
==Angular Momentum==
A key quantity in linear mechancis is [[momentum (physics)|momentum]]. Unsuprisingly, there is a rotational equivalent known as "angular momentum" and this too is a pseudovector. It is defined as the [[cross product]] of the position vector of a particle with its linear momentum:
<math>
\vec{J} = \vec{r} \times \vec{p}
</math>
Sometimes it may be written using the letter L. Like linear momentum, it has a [[conservation of angular momentum|conservation law]]: the sum of the angular momenta of particles in a system will remain constant so long as no external torques act on that system.
==Rotational Dynamics==
Rotational dynamics involves describing the motion of rotating objects by understanding the [[force]]s that act upon them. The rotational equivalent of a [[force]] is a [[torque]]. A torque acting on an object produces an angular acceleration. Torque is comonly denoted as a τ. It is defined as τ=Fl where F is the force and l is the perpendicular distance between the force and the axis of rotation. This can be extended to a vector using the [[cross product]] as:
<math>
\vec{\tau} = \vec{r} \times \vec{F}
</math>
with <math>\vec{r}</math> is the position vector of the object.
The [[moment of inertia]] is a rotational equivalent to [[mass (science)|mass]] in linear mechanics. It is a measure of the resistance to acceleration by a torque about a particular axis and defined as:
<math>I = \sum_{n=1}^{n} m_{n} r_{n}^2</math>
A rotational version of [[Newton's Laws of Motion|Newton's second law]] can now be expressed as:
<math>
\sum \vec{tau} = I \vec{\alpha}
</math>
More generally, it is expressed in terms of angular momentum:
<math>
\sum \vec{\tau} = \frac{\vec{J}}{\mathrm{d} t}
</math>
==References==
<references/>
==See Also==
*[[Angular impulse]]
*[[Torque]]
*[[Centripetal force]]
*[[SUVAT equations]]
[[Category:Physics]]
[[Category:Mechanics]]
==Rotational Kinematics==
Rotational kinematics involve describing the motion of rotating objects without understanding the forces that cause them to undergo that motion. It has many similarites to linear motion. Instead of [[displacement]] (distance) in linear mechanics, rotational mechanics considers angular displacement which is the angle something has rotated by. Similarly, [[velocity]] (rate of change of displacment) becomes angular velocity, the rate of change of angular displacement over time. Normally the Greek letter θ is used to denote angular displacement, an ω (omega) is used to denote angular velocity and α to denote angular acceleration. Angular displacement has units of [[radians]] so angular velocity has units of radians per second. These quantities can be related to each other through:
<math>
\alpha = \frac{\mathrm{d} \omega}{\mathrm{d} t} = \frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2}
</math>
From these definitions, rotational versions of the [[SUVAT equations]] can be derived when the angular acceleration is constant. They are:
*<math>\omega = \omega_0 + \alpha t</math>
*<math>\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2</math>
*<math>\theta = \theta_0 + \omega - \frac{1}{2} \alpha t^2</math>
*<math>\theta = \theta_0 + \frac{1}{2}(u+v)t</math>
*<math>\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)</math>
where θ<sub>0</sub> is the inital angular displacement and ω<sub></sub> is the final angular velocity.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#drot Rotation equations on hyperhpysics]</ref>
==Vector Nature of Angular Velocity==
In order to describe which axis an object is rotating around, angular velocity is a [[vector]] quantity. It is defined as having a magnitude equal to the angular speed of the object and points along the axis of rotation defined by the [[right hand rule]]. In this way, angular displacement and angular accleration are also vector quantities.
Angular velocity is an example of a pseudovector.<ref>[http://mathworld.wolfram.com/Pseudovector.html mathworld.wolfram.com]</ref> This means that although it often behaves like a vector, it does not always behave as a true vector would. An example of this is mirror symmetry. If a vector is reflected in a mirror, it will generally point in a different direction. However, if a rotating objecct is reflected in a mirror, it rotates in the same way and its angular velocity vector points in the same direction as before.
==Relation to Linear Mechanics==
Suppose an object is rotating about an axis ar some distance r. Suppose it moves a small distance ds, which is equal to r dθ for a small angular displacement dθ. Dividing both sides by dt to produce:
<math>
\frac{\mathrm{d} s}{\mathrm{d} t} = r \frac{\mathrm{d} \theta}{\mathrm{d} t}
</math>
which is simply v=rω. This equation relates the speed of a particle to its angular speed.
Furthermore, the acceleration of the particle can be written in terms of a radial component (directed towards the axis of rotation) and a tangential component (perpendicular to the radial component). Differentiating v=rω again produces a relation between the tangential acceleration and angular acceleration: a<sub>tan</sub>=rα. The radial component is the centripetal acceleration and causes the particle to rotate about an axis. It can be expressed as: a<sub>rad</sub>=ω<sup>2</sup>r.
==Angular Momentum==
A key quantity in linear mechancis is [[momentum (physics)|momentum]]. Unsuprisingly, there is a rotational equivalent known as "angular momentum" and this too is a pseudovector. It is defined as the [[cross product]] of the position vector of a particle with its linear momentum:
<math>
\vec{J} = \vec{r} \times \vec{p}
</math>
Sometimes it may be written using the letter L. Like linear momentum, it has a [[conservation of angular momentum|conservation law]]: the sum of the angular momenta of particles in a system will remain constant so long as no external torques act on that system.
==Rotational Dynamics==
Rotational dynamics involves describing the motion of rotating objects by understanding the [[force]]s that act upon them. The rotational equivalent of a [[force]] is a [[torque]]. A torque acting on an object produces an angular acceleration. Torque is comonly denoted as a τ. It is defined as τ=Fl where F is the force and l is the perpendicular distance between the force and the axis of rotation. This can be extended to a vector using the [[cross product]] as:
<math>
\vec{\tau} = \vec{r} \times \vec{F}
</math>
with <math>\vec{r}</math> is the position vector of the object.
The [[moment of inertia]] is a rotational equivalent to [[mass (science)|mass]] in linear mechanics. It is a measure of the resistance to acceleration by a torque about a particular axis and defined as:
<math>I = \sum_{n=1}^{n} m_{n} r_{n}^2</math>
A rotational version of [[Newton's Laws of Motion|Newton's second law]] can now be expressed as:
<math>
\sum \vec{tau} = I \vec{\alpha}
</math>
More generally, it is expressed in terms of angular momentum:
<math>
\sum \vec{\tau} = \frac{\vec{J}}{\mathrm{d} t}
</math>
==References==
<references/>
==See Also==
*[[Angular impulse]]
*[[Torque]]
*[[Centripetal force]]
*[[SUVAT equations]]
[[Category:Physics]]
[[Category:Mechanics]]