What does angular velocity depend on? Angular velocity
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Angular velocity.
Each point of a body rotating around a fixed axis passing through point O moves in a circle, and different points pass different paths in time Δt. So, AA 1 > BB 1 (Fig. 1.62), therefore, the speed modulus of point A is greater than the speed modulus of point B. But the radius vectors that determine the position of points A and B rotate in time Δt by the same angle Δφ.
Angle φ - the angle between the OX axis and the radius vector that determines the position of point A (see Fig. 1.62).
Let the body rotate uniformly, i.e., for any equal time intervals, the radius vectors rotate through the same angles.
The greater the angle of rotation of the radius vector, which determines the position of some point of a rigid body, for a certain period of time, the faster the body rotates and the greater its angular velocity.
The angular velocity of the body with uniform rotation called a value equal to the ratio of the angle of rotation of the body υφ to the time interval υt, during which this rotation occurred.
We will denote the angular velocity by the Greek letter ω (omega). Then by definition
Angular velocity in SI is expressed in radians per second (rad/s). For example, the angular velocity of the Earth's rotation around its axis is 0.0000727 rad/s, and that of a grinding wheel is about 140 rad/s.
Angular velocity can be related to rotational speed.
Rotation frequency- the number of complete revolutions per unit of time (in SI for 1 s).
If a body makes ν (Greek letter "nu") revolutions in 1 s, then the time for one revolution is 1/ν seconds.
The time it takes for a body to complete one revolution is called rotation period and are labeled T.
If φ 0 ≠ 0, then φ - φ 0 = ωt, or φ = φ 0 ± ωt.
The radian is equal to the central angle based on an arc whose length is equal to the radius of the circle, 1 rad \u003d 57 ° 17 "48". In radian measure, the angle is equal to the ratio of the length of the arc of a circle to its radius: φ = l/R.
The angular velocity takes on positive values if the angle between the radius vector that determines the position of one of the points of the rigid body and the OX axis increases (Fig. 1.63, a), and negative when it decreases (Fig. 1.63, b).
Thus, we can find the position of the points of a rotating body at any time.
Relationship between linear and angular velocities.
The speed of a point moving in a circle is often called linear speed to emphasize its difference from angular velocity.
We have already noted that during the rotation of an absolutely rigid body, its different points have unequal linear velocities, but the angular velocity for all points is the same.
Let's establish a connection between the linear velocity of any point of a rotating body and its angular velocity. A point lying on a circle with radius R travels 2πR in one revolution. Since the time of one revolution of the body is the period T, then the module of the linear velocity of the point can be found as follows:
Since ω = 2πν, then
The module of centripetal acceleration of a point of a body moving uniformly in a circle can be expressed in terms of the angular velocity of the body and the radius of the circle:
Hence,
and cs = ω 2 R.
Let's write down all possible calculation formulas for centripetal acceleration:
We have considered two simple motions of an absolutely rigid body - translational and rotational. However, any complex motion of an absolutely rigid body can be represented as the sum of two independent motions: translational and rotational.
Based on the law of independence of motions, one can describe the complex motion of an absolutely rigid body.
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In three-dimensional space, the angular velocity vector is equal in magnitude to the angle of rotation of a point around the center of rotation per unit time:
ω = d φ d t , (\displaystyle \omega =(\frac (d\varphi )(dt)),)and is directed along the axis of rotation according to the gimlet rule, that is, in the direction in which the gimlet or screw with a right-hand thread would be screwed in if it rotated in this direction. Another mnemonic approach for remembering the relationship between the direction of rotation and the direction of the angular velocity vector is that, to a conditional observer at the end of the angular velocity vector emerging from the center of rotation, the rotation itself appears to be occurring. against hour hand.
The angular velocity is an axial vector (pseudovector). When reflecting the axes of the coordinate system, the components of an ordinary vector (for example, the radius vector of a point) change sign. At the same time, the components of the pseudovector (in particular, the angular velocity) remain the same under such a coordinate transformation.
Tensor representation
Units
unit of measurement angular velocity, adopted in the International System of Units (SI) and in the CGS and MKGSS systems, - radians per second (Russian designation: rad/s, international: rad/s) . The technique also uses revolutions per second, much less often - degrees, minutes, seconds, arcs per second, degrees per second. Revolutions per minute are often used in technology - this has been going on since the times when the rotational speed of low-speed steam engines was determined simply by eye, counting the number of revolutions per unit of time.
Properties
The instantaneous velocity vector of any point of an absolutely rigid body rotating with an angular velocity is determined by the formula:
v → = [ ω → , r → ] , (\displaystyle (\vec (v))=[\ (\vec (\omega )),(\vec (r))\ ],)where is the radius vector to the given point from the origin located on the axis of rotation of the body, and square brackets denote the vector product . Linear velocity (coinciding with the modulus of the velocity vector) of a point at a certain distance (radius) r (\displaystyle r) from the axis of rotation can be calculated as follows: v = r ω . (\displaystyle v=r\omega .) If, instead of radians, other units of measurement of angles are used, then in the last two formulas a multiplier will appear that is not equal to one.
- In the case of plane rotation, that is, when all the velocity vectors of the points of the body always lie in the same plane (“plane of rotation”), the angular velocity of the body is always perpendicular to this plane, and in fact - if the plane of rotation is known in advance - it can be replaced by a scalar - a projection onto axis of rotation, that is, on a straight line, orthogonal to the plane of rotation. In this case, the kinematics of rotation is greatly simplified. However, in the general case, angular velocity can change direction over time in three-dimensional space, and such a simplified picture does not work.
- Motion with a constant angular velocity vector is called uniform rotational motion (in this case, the angular acceleration is zero). Uniform rotation is a special case of flat rotation.
- The derivative of the angular velocity with respect to time is the angular acceleration.
- The angular velocity (considered as a free vector) is the same in all inertial frames of reference that differ in the position of the reference point and the speed of its movement, but moving uniformly rectilinearly and translationally relative to each other. However, in these inertial frames of reference, the position of the axis or center of rotation of one and the same specific body at the same moment of time may differ (that is, there will be a different “point of application” of the angular velocity).
- In the case of a point moving in three-dimensional space, you can write an expression for the angular velocity of this point relative to the selected origin coordinates:
- In the case of uniform rotational motion (that is, motion with a constant angular velocity vector) of an absolutely rigid body, the Cartesian coordinates of the points of the body rotating in this way make
The distance and the time it takes to overcome this distance are connected by a physical concept - speed. And a person, as a rule, does not raise questions about the definition of this value. Everyone understands that driving a car at a speed of 100 km / h means driving 100 kilometers in one hour.
But what if the body is rotating? For example, an ordinary household fan makes about a dozen revolutions per second. And at the same time, the speed of rotation of the blades is such that they can easily be stopped by hand without harm to themselves. The Earth around its star - the Sun - makes one revolution in a whole year, which is more than 30 million seconds, but the speed of its movement in a circumstellar orbit is about 30 kilometers per second!
How to connect the usual speed with the speed of rotation, what does the formula for angular speed look like?
The concept of angular velocity
The concept of angular velocity is used in the study of the laws of rotation. It applies to all rotating bodies. Whether it is the rotation of some mass around another, as in the case of the Earth and the Sun, or the rotation of the body itself around the polar axis (the daily rotation of our planet).
The difference between angular velocity and linear velocity is that it captures the change in angle, not distance, per unit time. In physics, the angular velocity is usually denoted by the letter of the Greek alphabet "omega" - ω.
The classical formula for the angular velocity of rotation is considered as follows.
Imagine that a physical body rotates around some center A at a constant speed. Its position in space relative to the center is determined by the angle φ. At some point in time t1, the body under consideration is at point B. The angle of deviation of the body from the initial φ1.
Then the body moves to point C. It is there at time t2. Time taken for this move:
The position of the body in space also changes. Now the deflection angle is φ2. The change in the angle over the period of time ∆t was:
∆φ = φ2 - φ1.
Now the formula for angular velocity is formulated as follows: angular velocity is defined as the ratio of the change in angle ∆φ over time ∆t.
Angular velocity units
The linear speed of the body is measured in different quantities. The movement of vehicles on the roads is usually indicated in kilometers per hour, sea vessels make knots - nautical miles per hour. If we consider the movement of cosmic bodies, then kilometers per second most often appear here.
The angular velocity, depending on the magnitude and on the object that rotates, is also measured in different units.
Radians per second (rad/s) is the classic measure of speed in the International System of Units (SI). They show how many radians (in one full revolution 2 ∙ 3.14 radians) the body manages to turn in one second.
Revolutions per minute (RPM) is the most common unit for designating rotational speeds in engineering. The shafts of engines, both electric and automobile, give out exactly (just look at the tachometer in your car) revolutions per minute.
Revolutions per second (rps) - used less often, primarily for educational purposes.
Period of circulation
Sometimes it is more convenient to use another concept to determine the speed of rotation. The period of revolution is usually called the time during which a certain body makes a revolution of 360 ° (full circle) around the center of rotation. The formula for the angular velocity, expressed in terms of the period of revolution, takes the form:
To express the speed of rotation of bodies by the period of revolution is justified in cases where the body rotates relatively slowly. Let us return to the consideration of the movement of our planet around the star.
The angular velocity formula allows you to calculate it, knowing the period of revolution:
ω \u003d 2P / 31536000 \u003d 0.000000199238499086111 rad / s.
Looking at the result obtained, one can understand why, considering the rotation of celestial bodies, it is more convenient to use the period of revolution. A person sees clear numbers in front of him and clearly imagines their scale.
Relationship between angular and linear velocities
In some problems, linear and angular velocity must be determined. The transformation formula is simple: the linear velocity of a body is equal to the product of the angular velocity and the radius of rotation. As shown in the picture.
The expression "works" in the reverse order, with its help the angular velocity is also determined. The formula through linear speed is obtained by simple arithmetic manipulations.
Usually, when talking about movement, we imagine an object that moves in a straight line. The speed of such movement is usually called linear, and the calculation of its average value is simple: it is enough to find the ratio of the distance traveled to the time during which it was overcome by the body. If the object moves in a circle, then in this case it is not linear, but what is this value and how is it calculated? This is exactly what will be discussed in this article.
Angular velocity: concept and formula
When moving along a circle, the speed of its movement can be characterized by the value of the angle of rotation of the radius that connects the moving object with the center of this circle. It is clear that this value is constantly changing depending on time. The speed with which this process occurs is nothing but the angular velocity. In other words, this is the ratio of the magnitude of the deviation of the radius vector of the object to the period of time that it took the object to complete such a turn. The angular velocity formula (1) can be written in the following form:
w = φ / t, where:
φ - angle of rotation of the radius,
t is the rotation time period.
Quantity units
In the International System of Common Units (SI), it is customary to use radians to characterize turns. Therefore, 1 rad/s is the basic unit that is used in angular velocity calculations. At the same time, no one forbids the use of degrees (recall that one radian is equal to 180 / pi, or 57˚18 '). Also, the angular velocity can be expressed in revolutions per minute or per second. If the movement along the circle occurs uniformly, then this value can be found by the formula (2):
where n is the rotational speed.
Otherwise, just as it is done for normal speed, the average, or instantaneous angular speed, is calculated. It should be noted that the quantity under consideration is a vector one. To determine its direction is usually used which is often used in physics. The angular velocity vector is directed in the same direction as the screw with a right-hand thread. In other words, it is directed along the axis around which the body rotates, in the direction from which the rotation is seen to occur counterclockwise.
Calculation examples
Suppose we need to determine what the linear and angular speed of the wheel is, if it is known that its diameter is one meter, and the angle of rotation changes in accordance with the law φ=7t. Let's use our first formula:
w \u003d φ / t \u003d 7t / t \u003d 7 s -1.
This will be the desired angular velocity. Now let's move on to finding the usual speed of movement. As you know, v = s / t. Given that s in our case is the wheels (l = 2π * r), and 2π is one full turn, we get the following:
v = 2π*r / t = w * r = 7 * 0.5 = 3.5 m/s
Here's another thread on the subject. It is known that at the equator it is equal to 6370 kilometers. It is required to determine the linear and angular speed of movement of points located on this parallel, which occurs as a result of the rotation of our planet around its axis. In this case, we need the second formula:
w \u003d 2π * n \u003d 2 * 3.14 * (1 / (24 * 3600)) \u003d 7.268 * 10 -5 rad / s.
It remains to find out what the linear speed is equal to: v \u003d w * r \u003d 7.268 * 10 -5 * 6370 * 1000 \u003d 463 m / s.
Angular velocity- vector physical quantity characterizing the speed of rotation of the body. The angular velocity vector is equal in magnitude to the angle of rotation of the body per unit time:
,and is directed along the axis of rotation according to the gimlet rule, that is, in the direction in which the gimlet with a right-hand thread would be screwed in if it rotated in the same direction.
unit of measurement angular velocity, adopted in the SI and CGS systems - radians per second. (Note: radian, like any units of angle measurement, is physically dimensionless, so the physical dimension of angular velocity is simply ). The technique also uses revolutions per second, much less often - degrees per second, degrees per second. Perhaps, revolutions per minute are used most often in technology - this has been going on since the times when the rotational speed of low-speed steam engines was determined simply “manually”, counting the number of revolutions per unit of time.
The (instantaneous) velocity vector of any point of an (absolutely) rigid body rotating at an angular velocity is given by:
where is the radius vector to the given point from the origin located on the axis of rotation of the body, and square brackets denote the cross product . The linear velocity (coinciding with the modulus of the velocity vector) of a point at a certain distance (radius) from the axis of rotation can be calculated as follows: If other units of angles are used instead of radians, then a multiplier not equal to one will appear in the last two formulas.
- In the case of planar rotation, i.e. when all the velocity vectors of the points of the body lie (always) in the same plane ("plane of rotation"), the angular velocity of the body is always perpendicular to this plane, and in fact - if the plane of rotation is known in advance - can be replaced by a scalar - projection onto an axis orthogonal to the plane of rotation. In this case, the kinematics of rotation is greatly simplified, however, in the general case, the angular velocity can change direction over time in three-dimensional space, and such a simplified picture does not work.
- The derivative of the angular velocity with respect to time is the angular acceleration.
- Motion with a constant angular velocity vector is called uniform rotational motion (in this case, the angular acceleration is zero).
- The angular velocity (considered as a free vector) is the same in all inertial frames of reference, however, in different inertial frames of reference, the axis or center of rotation of the same specific body at the same moment of time may differ (that is, there will be a different "point of application" of the angular speed).
- In the case of the movement of a single point in three-dimensional space, you can write an expression for the angular velocity of this point relative to the selected origin:
- In the case of uniform rotational motion (that is, motion with a constant angular velocity vector), the Cartesian coordinates of the points of a body rotating in this way perform harmonic oscillations with an angular (cyclic) frequency equal to the modulus of the angular velocity vector.
Connection with finite rotation in space
. . .see also
Literature
- Lur'e A. I. Analytical mechanics\\ A. I. Lur'e. - M.: GIFML, 1961. - S. 100-136
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- Kilowatt hour
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