Rotation around a fixed axis
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Rotation around a fixed axis or axial rotation is a special case of
This concept assumes that the rotation is also stable, such that no
Translation and rotation
A rigid body is an object of a 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 circular motion.
Purely
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
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
Kinematics
Angular displacement
Given a particle that moves along the circumference of a circle of radius , having moved an arc length , its angular position is relative to its initial position, where .
In mathematics and physics it is conventional to treat the radian, a unit of plane angle, as 1, often omitting it. Units are converted as follows:
An angular displacement is a change in angular position:
Angular velocity
Change in angular displacement per unit time is called angular velocity with direction along the axis of rotation. The symbol for angular velocity is and the units are typically rad s−1. Angular speed is the magnitude of angular velocity.
The instantaneous angular velocity is given by
Using the formula for angular position and letting , we have also
where is the translational speed of the particle.Angular velocity and frequency are related by
Angular acceleration
A changing angular velocity indicates the presence of an angular acceleration in rigid body, typically measured in rad s−2. The average angular acceleration over a time interval Δt is given by
The instantaneous acceleration α(t) is given by
Thus, the angular acceleration is the rate of change of the angular velocity, just as acceleration is the rate of change of velocity.
The translational acceleration of a point on the object rotating is given by
The radial acceleration (perpendicular to direction of motion) is given by
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 relationship between torque and angular acceleration (how difficult it is to start, stop, or otherwise change rotation) is given by the moment of inertia: .
Equations of kinematics
When the angular acceleration is constant, the five quantities angular displacement , initial angular velocity , final angular velocity , angular acceleration , and time can be related by four equations of kinematics:
Dynamics
Moment of inertia
The moment of inertia of an object, symbolized by , is a measure of the object's resistance to changes to its rotation. The moment of inertia is measured in kilogram metre² (kg m2). It depends on the object's mass: increasing the mass of an object increases the moment of inertia. It also depends on the distribution of the mass: distributing the mass further from the center of rotation increases the moment of inertia by a greater degree. For a single particle of mass a distance from the axis of rotation, the moment of inertia is given by
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.The work done by a torque acting on an object equals the magnitude of the torque times the angle through which the torque is applied:
The power of a torque is equal to the work done by the torque per unit time, hence:
Angular momentum
The angular momentum is a measure of the difficulty of bringing a rotating object to rest. It is given by
where the sum is taken over all particles in the object.Angular momentum is the product of moment of inertia and angular velocity:
just as p = mv in linear dynamics.The analog of linear momentum in rotational motion is angular momentum. The greater the angular momentum of the spinning object such as a top, the greater its tendency to continue to spin.
The angular momentum of a rotating body is proportional to its mass and to how rapidly it is turning. In addition, the angular momentum depends on how the mass is distributed relative to the axis of rotation: the further away the mass is located from the axis of rotation, the greater the angular momentum. A flat disk such as a record turntable has less angular momentum than a hollow cylinder of the same mass and velocity of rotation.
Like linear momentum, angular momentum is vector quantity, and its conservation implies that the direction of the spin axis tends to remain unchanged. For this reason, the spinning top remains upright whereas a stationary one falls over immediately.
The angular momentum equation can be used to relate the moment of the resultant force on a body about an axis (sometimes called torque), and the rate of rotation about that axis.
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 due to the rotation of the body is given by
Kinetic energy is the energy of motion. The amount of translational kinetic energy found in two variables: the mass of the object () and the speed of the object () as shown in the equation above. Kinetic energy must always be either zero or a positive value. While velocity can have either a positive or negative value, velocity squared will always be positive.[1]
Vector expression
The above development is a special case of general rotational motion. In the general case, angular displacement, angular velocity, angular acceleration, and torque are considered to be
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 has rotated, then the thumb of the right hand points in the direction of the vector.
The
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 an effect on the rotation, other forces and torques are compensated by the structure.
Mathematical representation
In
By Rodrigues' rotation formula, the angle and axis determine a transformation that rotates three-dimensional vectors. The rotation occurs in the sense prescribed by the right-hand rule.
The rotation axis is sometimes called the Euler axis. The axis–angle representation is predicated on Euler's rotation theorem, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis.
It is one of many rotation formalisms in three dimensions.Examples and applications
Constant angular speed
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
An example of this is the two-body problem with circular orbits.
Centripetal force
Internal
Celestial bodies rotating about each other often have
Plane of rotation
In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.
The main use for planes of rotation is in describing more complex rotations in
Planes of rotation are not used much in
See also
- Anatomical terms of motion
- Artificial gravity by rotation
- Axle
- Axial precession
- Axial tilt
- Axis–angle representation
- Carousel, Ferris wheel
- Center pin
- Centrifugal force
- Centrifuge
- Centripetal force
- Circular motion
- Coriolis effect
- Fictitious force
- Flywheel
- Gyration
- Instant centre of rotation
- Linear-rotational analogs
- Optical axis
- Revolutions per minute
- Revolving door
- Rigid body angular momentum
- Rotation matrix
- Rotational speed
- Rotational symmetry
- Run-out
References
- ^ "What is Kinetic Energy". Khan Academy. Retrieved 2017-08-02.
- ^ "Multi Spindle Machines - An In-Depth Overview". Davenport Machine. Retrieved 2017-08-02.
- ISBN 9780387799469.
- ^ Lounesto (2001) pp. 222–223
- ISBN 0-471-23231-9
- Concepts of Physics Volume 1, by ISBN 81-7709-187-5