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Rotational mechanics (view source)
Revision as of 17:40, September 5, 2017
, 17:40, September 5, 2017Added section on rotational kinetic energy
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:
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>
:<math>
\alpha = \frac{\mathrm{d} \omega}{\mathrm{d} t} = \frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2}
\alpha = \frac{\mathrm{d} \omega}{\mathrm{d} t} = \frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2}
</math>
</math>
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:
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>
:<math>
\frac{\mathrm{d} s}{\mathrm{d} t} = r \frac{\mathrm{d} \theta}{\mathrm{d} t}
\frac{\mathrm{d} s}{\mathrm{d} t} = r \frac{\mathrm{d} \theta}{\mathrm{d} t}
</math>
</math>
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:
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>
:<math>
\vec{J} = \vec{r} \times \vec{p}
\vec{J} = \vec{r} \times \vec{p}
</math>
</math>
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:
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>
:<math>
\vec{\tau} = \vec{r} \times \vec{F}
\vec{\tau} = \vec{r} \times \vec{F}
</math>
</math>
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:
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>
:<math>I = \sum_{i=1}^{n} m_{i} r_{i}^2</math>
A rotational version of [[Newton's Laws of Motion|Newton's second law]] can now be expressed as:
A rotational version of [[Newton's Laws of Motion|Newton's second law]] can now be expressed as:
<math>
:<math>
\sum \vec{\tau} = I \vec{\alpha}
\sum \vec{\tau} = I \vec{\alpha}
</math>
</math>
More generally, it is expressed in terms of angular momentum:
More generally, it is expressed in terms of angular momentum:
<math>
:<math>
\sum \vec{\tau} = \frac{\vec{J}}{\mathrm{d} t}
\sum \vec{\tau} = \frac{\mathrm{d} \vec{J}}{\mathrm{d} t}
</math>
</math>
==Energy in Rotating Systems==
A rotational equivalent of [[kinetic energy]] can be found for rotating objects. It follows a similar form to linear kinetic energy, namely:
:<math>
E = \frac{1}{2} I \omega^2
</math>
The work energy principle can also be extended to rotational systems, where instead of it being [[force]] multiplied by [[displacement]], it is torque multiplied by angular displacement:
:<math>
W = \tau \, \Delta \theta = \frac{1}{2}I\left( \omega^2_f - \omega^2_i \right)
</math>
where ω<sub>f</sub> and ω<sub>i</sub> are the final and initial angular velocities and W is the work done.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/rke.html Rotational kinetic energy on Hyperphysics]</ref>
===Derivation===
Suppose a mass is rotating around some axis at an angular speed ω. Now consider a point particle, with [[mass (science)|mass]] dm, that is part of this mass. If it moves at some speed v as it rotates around the axis, it has kinetic energy:
:<math>
dE = \frac{1}{2} dm \, v^2
</math>
To find the total kinetic energy, we must [[integration|integrate]] over the mass:
:<math>
E = \int \frac{1}{2} v^2 \, dm
</math>
Noting that v=ωr and that ω is constant for all masses, and so may be taken outside of the integral, the kinetic energy of mass is:
:<math>
E = \frac{1}{2} \left( \int r^2 \, dm \right) \omega^2
</math>
As the term in brackets is the [[moment of inertia]], I, the equation of the form above is derived.
==References==
==References==
{{reflist}}
==See Also==
==See Also==