Rotation around a fixed axis – Pulse Valve manufacturer – Solenoid Valve manufac
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 particleshether 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
where M is the total mass of the system and acm 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.
Kinematics
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 is given by
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 has no dimensions. Because the circumference of a circle of radius r is 2r, there are 2 radians in a complete circle:
Thus
We do not reset 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 of the line is .
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 1 to 2, the body undergoes an angular displacement given by
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 the x-axis, its displacement 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 of a rotating body is either positive or negative, according to the following convention: an angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative.
Angular speed and angular velocity
Main articles: Angular speed and angular velocity
The average angular speed is defined as the ratio of angular displacement to the time in which it occurs:
The sign of indicates the direction of rotation.
With differential calculus the instantaneous angular velocity (t) can be defined:
Angular velocity is the first derivative of angular position, just as velocity is the first derivative of position.
The angular velocity of a particle can be related to its translational velocity v, which depends on the distance from the centre of rotation. Since = s/r and r is constant,
Thus v = r.
The angular velocity is sometimes called the angular frequency. It can be deduced from the frequency, the number of rotations in a given time.
Angular acceleration
Main article: Angular acceleration
A changing angular velocity indicates the presence of an angular acceleration, measured in rad s2. The average angular acceleration over a time interval t is given by
The instantaneous acceleration (t) is given by
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
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, the motion is uniform circular motion, and the velocity changes in direction only.
The radial acceleration (perpendicular to direction of motion) is given by
and is directed towards the center of the rotational motion.
Equations of kinematics
The five quantities angular displacement, initial angular velocity, final angular velocity, angular acceleration, and time can be related by four equations of kinematics:
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.
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 metre (kg m)
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.
Dynamics
Torque
Main article: Torque
Torque is the twisting effect of a force F applied to a rotating object which is at position r from its axis of rotation. Mathematically,
where denotes the cross product. A net torque acting upon an object will produce an angular acceleration of the object according to
just as F = ma in linear dynamics.
Angular Momentum
Main article: Angular momentum
The angular momentum L is a measure of the difficulty of bringing a rotating object to rest. It is given by
Angular momentum is related to angular velocity by
just as p = mv in linear dynamics.
Torque and angular momentum are related according to
just as F = dp/dt in linear dynamics. In the absence of an external torque, the angular momentum of a body remains constant. The conservation of angular momentum is notably demonstrated in figure skating: when pulling the arms closer to the body during a spin, the moment of inertia is decreased, and so the angular velocity is increased.
Kinetic energy
The kinetic energy Krot due to the rotation of the body is given by
just as Ktrans = 12mv2 in linear dynamics.
Vector expression
See also: Vector (geometry)
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 . 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.
Examples and applications
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.
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.
See also
Fictitious force
Centrifugal force
Centripetal force
artificial gravity by rotation
axle
carousel, Ferris wheel
centrifuge
circular motion
Coriolis effect
flywheel
gyration
revolutions per minute
revolving door
rigid body angular momentum
rotational speed
rotational symmetry
spin
References
^ a b c Halliday, David; Resnick, Robert; Walker, Jearl. Fundamentals of Physics Extended (7th ed.). ISBN 0471232319.
Further reading
Concepts of Physics Volume 1, 1st edition Seventh reprint by Harish Chandra Verma ISBN 81-7709-187-5
Categories: Celestial mechanics | Euclidean symmetriesHidden categories: Articles needing cleanup from August 2007 | All pages needing cleanup
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