Talk:Rotational motion/Rotational motion

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In of linear motion, objects were treated as point particles without structure; it did not matter where a force was applied, only whether it was applied or not. The reality is that 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 stoke will yield topspin to the ball, which can cause it to land in the opponent’s court.

The concept 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 objects equilibrium and rate of rotation. ^[1]

We will find that an object remains in a state of uniform rotational motion unless acted on by a net torque. This principle is the equivalent to Newton’s first law. 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. 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 objects angular momentum. In the absence of 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.

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Rotational Variables

Angular Position

Top view of a rotating system

The Figure shows a reference line, foxed 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 posisition. 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 or radius r is $2\pi r$ , there are $2\pi$ radians in a complete cirle:

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 $\theta$ 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 $\theta$ of the line is $\theta =4\pi \;rad$ . ^[1]

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 $\theta _{1}$ to $\theta _{2}$ , the body undergoes an angular displacement $\Delta \theta$ given by

\Delta \theta =\theta _{2}-\theta _{1}\!

This definition of angular displacement holds not only for the rigid body as a hole 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 $\Delta x$ 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 $\Delta \theta$ 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]

Angular Velocity

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

\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 \,\!

.

Angular velocity is defined as the rotio of angular displacement in a given time

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

Rotational Inertia

Increasing the mass increases the rotational inertia, sometimes known as the moment of 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)

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 ( $\alpha$ )

Angular Momentum

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

L=I\omega

References

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

In the study of linear motion, objects were treated as point particles without structure; it did not matter where a force was applied, only whether it was applied or not. The reality is that 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 stoke will yield topspin to the ball, which can cause it to land in the opponent’s court.

The concept 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 objects equilibrium and rate of rotation. ^[1]

We will find that an object remains in a state of uniform rotational motion unless acted on by a net torque. This principle is the equivalent to Newton’s first law. 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. 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 objects angular momentum. In the absence of 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.

Translation and Rotation

A rigid body is defined as an object in which the relative separations of the component particles are unchanged, the displacement of any one particle with respect to any other particle in the body being fixed. No truly rigid body exists: external forces can deform any solid. For purposed, then, a rigid body is a solid in which the internal forces between the fundamental particles change so drastically with a variation in their relative separation that large forces are required to deform it appreciably.

Up to this point, we have studied the kinematics and dynamics of a particle whose position in three-dimensional space is completely specified by three coordinates. To describe a change in the position of a body of finite extent, such as a rigid body, is much more complicated. For convenience, it is regarded as a combination of two distinct types of motion: translational motion and rotational motion.

Purely translational motion occurs if every particle of the body has the same instantaneous velocity as every other particle, the path traced out by some particle being exactly the same as 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 some one 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 be with 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 displacing the body translationally, without rotation; or conversely, first a rotation and then a displacement. We already know that for any collection of 我靠particles—whether at rest with respect to one another, as in a rigid body, or in relative motion, as in the exploding fragments of a shell—the motion of the center of mass is completely determined by the resultant of the external forces acting on the system of particles is

\Sigma F_{net}=Ma_{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 rotation of the body about the center of mass and relating it to the external forces acting on the body. We shall find that the kinematics of rotation motion have many similarities the kinematics of translational motion; moreover, the dynamics of rotational motion involve forms of Newton’s second law of motion and of the work-energy theorem that are altogether analogous to those used in particle dynamics.

Rotational Variables

Angular Position

Top view of a rotating system

The Figure shows a reference line, foxed 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 posisition. 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 or radius r is $2\pi r$ , there are $2\pi$ radians in a complete cirle:

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 $\theta$ 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 $\theta$ of the line is $\theta =4\pi \;rad$ . ^[1]

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 refence line from $\theta _{1}$ to $\theta _{2}$ , the body undergoes an angular displacement $\Delta \theta$ given by

\Delta \theta =\theta _{2}-\theta _{1}\!

This definition of angular diplacemetn holds not only for the rigid body as a hole 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 axid, its diplacement $\Delta x$ is either positive or negative, depending on whether the body is moving in the positive or negative direction of the axis. Similarly, the angualr displacement $\Delta \theta$ 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]

Speed of rotation, angular acceleration, and torque

The speed of rotation is given by the angular frequency (rad/s) or frequency / rotational speed / revolutions per minute (turns/s, turns/min), or period (seconds, days, etc.).

With one direction of rotation considered positive, the sign of the angular frequency indicates the direction of rotation.

The time-rate of change of angular frequency is the scalar version of angular acceleration (rad/s²). This change is caused by the scalar version of 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 heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia.

The energy required for / 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 frequency. The power required for angular acceleration is the torque times the angular frequency.

Vectors

According to the right-hand rule, moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like most screws.

The angular velocity vector also describes the direction of the axis of rotation. In the case of a fixed axis this direction is along that axis and the rotation process is described by a scalar, the angular frequency, as a function of time.

Similarly the torque vector also describes around which axis it tends to cause rotation, or in other words, the direction in which it tends to change the angular velocity vector. To maintain rotation around the fixed axis the total force has to be zero and 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.

Centripetal force

In the case of a spinning object, internal tensile stress provides the centripetal force that keeps the object together.

A rigid body model neglects the accompanying strain.

If the body is not rigid this strain constitutes a change of shape. This is also expressed as 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. 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 oblate for oblateness due to rotation, in particular of a fluid celestial body.

Constant angular speed

The simplest case of rotation around a fixed axis is that of constant angular speed. The total torque is zero: in e.g. the case of the rotation of the Earth around its axis there is very little friction, in the case of e.g. 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.

References

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

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[Halliday-1] Fundamentals of Physics Extended 7^th Edition by Halliday, Resnick and Walker. ISBN 0-471-23231-9

[Halliday-2] Fundamentals of Physics Extended 7^th Edition by Halliday, Resnick and Walker. ISBN 0-471-23231-9

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