Rotational Kinematics

Claimed by Aditya Kuntamukkula 2017

The Main Idea

Rotational motion is defined as an object moving around an axis in contrast to translational motion which involves the object moving in a straight trajectory.

A Mathematical Model

Rotation can be characterized by its angular velocity and angular acceleration. The equations are listed below.

Angular velocity:

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}} }[/math] ,

where [math]\displaystyle{ {\boldsymbol{v}} }[/math] is the velocity of the object and [math]\displaystyle{ {\boldsymbol{r}} }[/math] is the radius of the circle of motion. It can also be represented as the change in angle over the distance traveled in the formula shown below:

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}} }[/math] , where [math]\displaystyle{ {\boldsymbol{d\theta}} }[/math] is the change in angle and [math]\displaystyle{ {\boldsymbol{dt}} }[/math] is the change in time.

Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)

Angular acceleration is equal to alpha:

[math]\displaystyle{ \boldsymbol{{\alpha}} = \frac{\boldsymbol{a_t}}{\boldsymbol{r}} }[/math] ,

where [math]\displaystyle{ {\boldsymbol{a_t}} }[/math] is the tangential acceleration of the object and [math]\displaystyle{ {\boldsymbol{r}} }[/math] is the radius of the circle of motion.

Rotational Kinetic Energy: An object with a center of mass at rest can still have rotational kinetic energy. For example, if a disk is suspended in the air and spun, it has no translational kinetic energy. The position of the disk does not change. However, since it is spinning (rotating), it still has kinetic energy. To account for this, we can relate angular velocity with the moment of inertia of the object to find a value for the rotational kinetic energy.

Rotational Kinetic Energy:

[math]\displaystyle{ {KE}_{rot} = \frac{{1}}{{2}}{I}_{cm}{ω^2} }[/math]

Relation to Work and Energy Principle:

The energy principle states:

[math]\displaystyle{ {E}_{f} = {E}_{i} + W }[/math]

We can apply the energy principle to rotational kinetic energy as well to find changes in kinetic energy and work done on the system.

A Computational Model

As shown in this Earth- spacecraft model, [1], rotation can be modeled through VPython and other computer languages.

from __future__ import division from visual import * from visual.graph import * #Invoke graphing routines scene.y = 400 #Move orbit window below graph

scene.width =1024 scene.height = 760

1. CONSTANTS

G = 6.7e-11 mEarth = 6e24 mMoon = 0 mcraft = 15000 deltat = 5000 pscale = 0.5 fscale = 30000 dpscale = .5

1. OBJECTS AND INITIAL VALUES

Earth = sphere(pos=vector(0,0,0), radius=6.4e6, color=color.cyan) Moon = sphere(pos = vector(4e8,0,0), radius = 1.75e6, color = color.white) scene.range=11*Earth.radius parr = arrow(color=color.green) farr = arrow(color = color.cyan) dparr = arrow(color = color.red) Fnet_tangent_arrow = arrow(color = color.yellow) Fnet_perp_arrow = arrow(color = color.magenta)

1. Choose an exaggeratedly large radius for the
2. space craft so that you can see it!

craft = sphere(pos=vector(-71680000,-3008000,0),radius = Earth.radius/3, color=color.yellow) vcraft = vector(0,sqrt((G * mEarth)/(mag(craft.pos - Earth.pos))),0) pcraft = mcraft*vcraft vMoon = vector(0,sqrt((G * mEarth)/(mag(Moon.pos - Earth.pos))),0) trail1 = curve(color=craft.color) ## craft trail: starts with no points trail2 = curve(color=Moon.color) t = 0 scene.autoscale = 0 ## do not allow camera to zoom in or out momentum_Moon = mMoon * vMoon

1. CALCULATIONS

print("p=",pcraft) while t < 200000: #time of model:

   rate(10)       ## slow down motion to make animation look nicer
   ## you must add statements for the iterative update of 
   ## gravitational force, momentum, and position
   R1 = craft.pos - Earth.pos
   R2 = craft.pos - Moon.pos
   
   Rmag1 = mag(R1)
   Rmag2 = mag(R2)
   
   Rhat1 = norm(R1)
   Rhat2 = norm(R2)
   
   Fmag1 = (G*(mEarth)*(mcraft))/(Rmag1**2)
   Fmag2 = (G*(mMoon)*(mcraft))/(Rmag2**2)
   
   Fnet1 = -Fmag1 * Rhat1
   Fnet2 = -Fmag2 * Rhat2
   
   Fnet = Fnet1 + Fnet2
   
   vcraft = vcraft + (Fnet*deltat)/mcraft
   p_initial = pcraft
   pcraft = pcraft + Fnet*deltat
   p_final = pcraft
   craft.pos = craft.pos + vcraft*deltat
   parr.pos = craft.pos
   farr.pos = craft.pos
   parr.axis = pcraft * pscale
   farr.axis = Fnet * fscale
   pcraft_i = pcraft + vector(0,0,0)
   deltap = pcraft - pcraft_i
   dparr.pos = craft.pos
   dparr.axis = Fnet * fscale

   Fnet_tangent = ((mag(p_final) - mag(p_initial))/deltat) * norm(pcraft)
   Fnet_tangent_arrow.pos = craft.pos
   Fnet_tangent_arrow.axis = Fnet_tangent * 90000

   Fnet_perp = Fnet - Fnet_tangent
   Fnet_perp_arrow.pos = craft.pos
   Fnet_perp_arrow.axis = Fnet_perp * 10000

   scene.center = craft.pos
   scene.range = craft.radius * 60

   ## check to see if the spacecraft has crashed on the Earth.
   ## if so, get out of the calculation loop

   if Rmag1 < Earth.radius:
       break
   if Rmag2 < Moon.radius:
       break

   trail1.append(pos=craft.pos) ## this adds the new position of the spacecraft to the trail
   trail2.append(pos = Moon.pos)
   t = t+deltat

print("Fnet=",Fnet) print('Calculations finished after ',t,'seconds') print("craft.pos=",craft.pos) print("vcraft=",vcraft) print("F_perp=",Fnet_perp)

Examples

Simple

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}} }[/math]

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

Medium

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

[math]\displaystyle{ \boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}} }[/math]

[math]\displaystyle{ \boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}} }[/math]

[math]\displaystyle{ \boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}} }[/math]

[math]\displaystyle{ \boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}} }[/math]

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

Difficult

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}} }[/math]

[math]\displaystyle{ \boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}} }[/math]

[math]\displaystyle{ \boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}} }[/math]

Connectedness

Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

See also

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

Further reading

Books, Articles or other print media on this topic

External links

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

References

[1] Biomechanics, Basic. “It Is Important When Learning about (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web