Rotational motion

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Rotational motion is similar to circular motion, except the object involved is a rigid body in which all points rotate around the center of mass of the object and not around a fixed point.

Rotational motion can be pure rotational motion or a combination of translation and rotation.

Pure rotational motion is circular movement in which all points in the body move in circles, and that the centers of these circles all lie on a line called the axis of rotation. Pure Rotation is caused by an arrangement called a 'force couple'. This is where two equal and opposite forces act on the object from an equal [perpendicular]distance apart.

Translation and Rotation is caused by a single force which does not pass directly through the center of mass. The center of mass will move in a straight line, meanwhile, all points of the object will rotate about the center of mass.

1 Angular Quantities
2 See also
3 References

[edit] Angular Quantities

For translational motion, we use displacement, velocity and acceleration to describe an object's motion. These depend on the distance from the centre of rotation and therefore cannot be used to describe rotational motion.

[edit] Angular Displacement

The Angle through which the objects rotates is called angular displacement

Angular Displacement is measured in radians rather than degrees. This is because it provides a very simple relationship between distance traveled around the circle and the distance r from the centre.

$\theta=\frac{s}{r}$

For example if an object rotates 360° around a circle radius r the angular displacement is given by the distance traveled the circumference which is 2πr. Divided by the radius in: θ = 2πr/r which easily simplifies to 2π. Therefore 1 revolution is 2π rad.

[edit] Angular Velocity

To describe how quickly an object is rotating, angular velocity is used. Angular velocity is measured in $r a d / s$ and is symbolized by the Greek letter omega ( $ω$ ).

$\omega = \frac{d\theta}{dt}$

When an object rotates it also has a translational speed at every point on the object, which depends on the distance from the centre of rotation. The Angular Velocity is given by:

$\omega=\frac{\theta}{t}$ and since $\theta = \frac{s}{r}$ ,

$\omega=\frac{s}{rt}$ and since $v=\frac{s}{t}$ ,

$\omega=\frac{v}{r}$ or rearranged to give $v=r\omega\,\!$ .

[edit] Angular acceleration

When the angular velocity is changing this is called angular acceleration, it has symbol α (the Greek letter alpha) and is measured in rad/s². Angular acceleration = (change in angular velocity)/(change in time)

$\alpha = \frac{\Delta \omega}{\Delta t}$

If the limit of this as Δt approaches 0 is taken, this equation becomes the more general:

$\alpha = \frac{d\omega}{dt}$

Thus, angular acceleration is the first derivative of angular velocity, just as acceleration is the first derivative of velocity.

The translational acceleration of a point on the object rotating is given by a = rα where r is the radius or distance from centre of rotation. This is also the tangential component of acceleration: it is tangential to the direction of motion of the point. If this component is 0, then the magnitude of the velocity of the points remains constant (as in uniform circular motion). The radial acceleration (perpendicular to direction of motion) is given by a = v²/r = ω²r.

The direction of the net acceleration of the object is always directed towards the center of the rotational motion.

For problems with uniform angular acceleration just as in translational motion there are 4 equations that relate the 5 variables:

angular acceleration
initial angular velocity
final angular velocity
angular displacement
time taken

The equations can be easily derived from the kinematic equations and are:

$\omega_f = \omega_i + \alpha t\;\!$

$\theta = \omega_i t + \begin{matrix}\frac{1}{2}\end{matrix} \alpha t^2$

$\omega_f^2 = \omega_i^2 + 2 \alpha\theta$

$\theta = \begin{matrix}\frac{1}{2}\end{matrix} \left(\omega_f + \omega_i\right) t$

[edit] Rotational Inertia

Increasing the mass increases the rotational inertia of an object. But the distribution of the mass is more important, ie distributing the mass further from the centre of rotation increases rotational inertia by a greater degree. Rotational Inertia is measured in kilogram metre² (kg m²)

[edit] Torque

Torque is the turning effect of a force applied at a perpendicular distance from the centre of rotation of a rotating object. T=F*r A net torque acting upon an object will produce angular acceleration of the object. Torque = rotational Inertia (I) times angular acceleration ( $α$ )

[edit] Angular Momentum

L is a measure of the difficulty of bringing a rotating object to rest.

L = I ω

[edit] See also

Rotation around a fixed axis

[edit] References

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