Rotation around a fixed axis

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Rotational motion can occur around more than one axis at once, and can involve phenomena such as wobbling and precession. Rotation around a fixed axis is a special case of rotational motion, which does not involve those phenomena. The kinematics and dynamics of rotation around a fixed axis of a rigid object are mathematically much simpler than those for rotation of a rigid body; they are entirely analogous to those of linear motion along a single fixed direction, which is not true for rotation of a rigid body. The expressions for the kinetic energy of the object, and for the forces on the parts of the object, are also simpler for rotation around a fixed axis, than for general rotational motion. For these reasons, rotation around a fixed axis is typically taught in introductory physics courses after students have mastered linear motion; the full generality of rotational motion is not usually taught in introductory physics classes.

In the beginning study of linear motion, objects are treated as point particles without structure; it does not matter where a force is applied, only that it is applied. However, in practice, the point of application of force does matter. In tennis, for example, if a tennis ball is struck with a strong horizontal force acting through its center of mass, it may travel a long distance before hitting the ground, far out of bounds. Instead, the same force applied in an upward, glancing stroke will yield topspin to the ball, which can cause it to land in the opponent’s court.

The concepts of rotational equilibrium and rotational dynamics are also important in other disciplines. For example, students of architecture benefit from understanding the forces that act on buildings and biology students should understand the forces at work in muscles, bones, and joints. These forces create torques, which tell us how the forces affect an object's equilibrium and rate of rotation. ^[1]

An object remains in a state of uniform rotational motion unless acted on by a net torque. This principle is analogous to Newton’s first law of motion. Further, the angular acceleration of an object is proportional to the net torque acting on it, which is the analog of Newton’s Second Law of motion. A net torque acting on an object causes a change in its rotational energy.

Finally, torques applied to an object through a given time interval can change the object's angular momentum. If there are no external torques, angular momentum is conserved, a property that explains some of the mysterious and formidable properties of pulsars—remnants of supernova explosions that rotate at equatorial speeds approaching that of light.

1 Translation and Rotation
2 Rotational Variables
3 Examples and applications
- 3.1 Constant angular speed
- 3.2 Centripetal force
4 References
5 See also

[edit] Translation and Rotation

See also: Rigid body

A rigid body is an object of finite extent in which all the distances between the component particles are constant. No truly rigid body exists; external forces can deform any solid. For our purposes, then, a rigid body is a solid which requires large forces to deform it appreciably.

A change in the position of a particle in three-dimensional space can be completely specified by three coordinates. A change in the position of a rigid body is more complicated to describe. It can be regarded as a combination of two distinct types of motion: translational motion and rotational motion.

Purely translational motion occurs when every particle of the body has the same instantaneous velocity as every other particle; then the path traced out by any particle is exactly parallel to the path traced out by every other particle in the body. Under translational motion, the change in the position of a rigid body is specified completely by three coordinates such as x, y, and z giving the displacement of any point, such as the center of mass, fixed to the rigid body.

Purely rotational motion occurs if every particle in the body moves in a circle about a single line. This line is called the axis of rotation. Then the radius vectors from the axis to all particles undergo the same angular displacement in the same time. The axis of rotation need not go through the body. In general, any rotation can be specified completely by the three angular displacements with respect to the rectangular-coordinate axes x, y, and z. Any change in the position of the rigid body is thus completely described by three translational and three rotational coordinates.

Any displacement of a rigid body may be arrived at by first subjecting the body to a displacement followed by a rotation, or conversely, to a rotation followed by a displacement. We already know that for any collection of particles—whether at rest with respect to one another, as in a rigid body, or in relative motion, like the exploding fragments of a shell, the acceleration of the center of mass is given by;

$\Sigma F_{net}=M a_{cm}\;\!$

where M is the total mass of the system and a_cm is the acceleration of the center of mass. There remains the matter of describing the rotation of the body about the center of mass and relating it to the external forces acting on the body. The kinematics and dynamics of rotational motion around a single axis resemble the kinematics and dynamics of translational motion; rotational motion around a single axis even has a work-energy theorem analogous to that of particle dynamics.

[edit] Rotational Variables

[edit] Angular Position

Top view of a rotating system

The Figure shows a reference line, fixed in the body, perpendicular to the rotation axis and rotating with the body. The angular position of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position. From geometry, we know that θ is given by

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

Here s is the length of a circular arc that extends from the x axis (the zero angular position) to the reference line, and r is the radius of the circle.

An angle defined in this way is measured in radians (rad) rather than in revolutions (Rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension. Because the circumference of a circle of radius r is $2π r$ , there are $2π$ radians in a complete circle:

$1\; rev=360^{o}=\frac{2\pi r}{r}=2\pi \; rad$

and thus

$1\; rad=57.3^{o}=0.159\; rev$

We do not reset $θ$ to zero with each complete rotation of the reference line about the rotation axis. If the reference line completes two revolutions from the zero angular position, then the angular position $θ$ of the line is $\theta=4\pi\;rad$ . ^[1]

[edit] Angular Displacement

Main article: Angular displacement

Diagram of angular displacement.

If the object in the figure rotates about the rotation axis as shown in the Figure, changing the angular position of the reference line from $θ 1$ to $θ 2$ , the body undergoes an angular displacement $Δθ$ given by

$\Delta \theta = \theta_{2} - \theta_{1} \!$

This definition of angular displacement holds not only for the rigid body as a whole, but also for every particle within that body, because the particles are all locked together.

If a body is in translational motion along an x axis, its displacement $Δ x$ is either positive or negative, depending on whether the body is moving in the positive or negative direction of the axis. Similarly, the angular displacement $Δθ$ of a rotating body is either positive or negative, according to the following rule:

An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative.^[1]

[edit] Angular Velocity

Main article: Angular velocity

The average angular velocity is defined as the ratio of angular displacement to the time in which it occurs. Its sign indicates the direction of rotation.

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

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

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

When an object rotates it also has a translational speed v at every point on the object, which depends on the distance from the centre of rotation. Consider an object with a constant angular velocity:

$\omega=\frac{d\theta}{dt}$ . Since $\theta = \frac{s}{r}$ , where s is the arc length and r the radius (which is constant), then

$\omega=\frac{ds}{rdt}$ . Substituting the definition $v = \frac{ds}{dt}$ , gives

$\omega=\frac{v}{r}$ , which can be rearranged to give $v=r\omega\,\!$ . This formula also holds if

ω

is not constant.

The angular velocity is sometimes called the angular frequency. It can be deduced from the frequency, the number of rotations in a given time.

[edit] Angular acceleration

Main article: Angular acceleration

When the angular velocity is changing, there is an angular acceleration, symbolized by α (the Greek letter alpha) and measured in rad/s². The average angular acceleration is the change in angular velocity divided by the time in which it occurs.

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

If we take the limit of this as Δt approaches 0, this equation becomes:

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

Thus, the angular acceleration is the first derivative of the 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 the 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 we have uniform circular motion, and the velocity of the points remains constant. The radial acceleration (perpendicular to direction of motion) is given by a = v²/r = ω²r and is 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:

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

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

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

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

The angular acceleration is caused by the torque, which can have a positive or negative value in accordance with the convention of positive and negative angular frequency. The ratio of torque and angular acceleration (how difficult it is to start, stop, or otherwise change rotation) is given by the moment of inertia.

[edit] Moment of Inertia

Main article: Moment of inertia

Increasing the mass increases the moment of inertia, symbolized by $I$ , which is sometimes called the rotational inertia of an object. But the distribution of the mass is more important, i.e. distributing the mass further from the centre of rotation increases the moment of inertia by a greater degree. The moment of inertia is measured in kilogram metre² (kg m²)

The energy required or released during rotation is the torque times the rotation angle; the energy stored in a rotating object is one half of the moment of inertia times the square of the angular velocity. The power required for angular acceleration is the torque times the angular velocity.

[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 an angular acceleration of the object. The sign of the torque is the same as the sign of the angular acceleration it will produce. Torque = rotational Inertia (I) times angular acceleration ( $α$ ). This equation shows that, if an object has an angular acceleration, there is a torque on it.

[edit] Angular Momentum

Main article: Angular momentum

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

L = I ω

Angular momentum is a conserved quantity. In an isolated system it remains the same.

[edit] The quantities considered as vectors

See also: Vector (spatial)

The development above is a special case of general rotational motion. In the general case, angular displacement, angular velocity, angular acceleration and torque are considered to be vectors.

An angular displacement is considered to be a vector pointing along the axis, of magnitude equal to that of $Δθ$ . A right-hand rule is used to find which way it points along the axis; if the fingers of the right hand are curled to point in the way that the object rotated, then the thumb of the right hand can be pointed in the direction of the vector.

The angular velocity vector also points along the axis of rotation in the same way as the angular displacements it causes. If a disk spins counterclockwise as seen from above, its angular velocity vector points upwards. Similarly, the angular acceleration points along the axis of rotation in the same direction that the angular velocity would point if the angular acceleration were maintained for a long time.

The torque vector points along the axis around which the torque tends to cause rotation. To maintain rotation around a fixed axis, the total torque vector has to be along the axis, so that it only changes the magnitude and not the direction of the angular velocity vector. In the case of a hinge, only the component of the torque vector along the axis has effect on the rotation, other forces and torques are compensated by the structure.

[edit] Examples and applications

[edit] Constant angular speed

Main article: Uniform circular motion

See also: Circular motion

The simplest case of rotation around a fixed axis is that of constant angular speed. Then the total torque is zero. For the example of the Earth rotating around its axis, there is very little friction. For a fan, the motor applies a torque to compensate for friction. The angle of rotation is a linear function of time, which modulo 360° is a periodic function.

An example of this is the two-body problem with circular orbits.

[edit] Centripetal force

Main article: Centripetal force

See also: Centrifugal force and Fictitious force

Internal tensile stress provides the centripetal force that keeps a spinning object together. A rigid body model neglects the accompanying strain. If the body is not rigid this strain will cause it to change shape. This is expressed as the object changing shape due to the "centrifugal force".

Celestial bodies rotating about each other often have elliptic orbits. The special case of a circular orbits is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion. The centripetal force is provided by gravity, see also two-body problem. This usually also applies for a spinning celestial body, so it need not be solid to keep together, unless the angular speed is too high in relation to its density. (It will, however, tend to become oblate.) For example, a spinning celestial body of water must take at least 3 hours and 18 minutes to rotate, regardless of size, or the water will separate. If the density of the fluid is higher the time can be less. See orbital period.

[edit] References

^ ^a ^b ^c Fundamentals of Physics Extended 7^th Edition by Halliday, Resnick and Walker. ISBN 0-471-23231-9

2. Concepts of Physics Volume 1,1st edition Seventh reprint by Harish Chandra Verma ISBN 81-7709-187-5

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Rotation around a fixed axis

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