Rotational motion around a fixed axis is another special case of rigid body motion.
Rotational movement of a rigid body around a fixed axis
it is called such a movement in which all points of the body describe circles, the centers of which are on the same straight line, called the axis of rotation, while the planes to which these circles belong are perpendicular rotation axis
(Fig.2.4).
In technology, this type of motion occurs very often: for example, the rotation of the shafts of engines and generators, turbines and aircraft propellers.
Angular velocity
. Each point of a body rotating around an axis passing through the point ABOUT, moves in a circle, and different points travel different paths over time. So, , therefore the modulus of the point velocity A more than a point IN (Fig.2.5). But the radii of the circles rotate through the same angle over time. Angle - the angle between the axis OH and radius vector, which determines the position of point A (see Fig. 2.5).
Let the body rotate uniformly, i.e., rotate through equal angles at any equal intervals of time. The speed of rotation of a body depends on the angle of rotation of the radius vector, which determines the position of one of the points of the rigid body for a given period of time; it is characterized angular velocity
.
For example, if one body rotates through an angle every second, and the other through an angle, then we say that the first body rotates 2 times faster than the second.
Angular velocity of a body during uniform rotation
is a quantity equal to the ratio of the angle of rotation of the body to the period of time during which this rotation occurred.
We will denote the angular velocity by the Greek letter ω
(omega). Then by definition
Angular velocity 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 disk is about 140 rad/s 1 .
Angular velocity can be expressed through rotation speed
, i.e. the number of full revolutions in 1s. If a body makes (Greek letter “nu”) revolutions in 1s, then the time of one revolution is equal to seconds. This time is called rotation period
and denoted by the letter T. Thus, the relationship between frequency and rotation period can be represented as:
A complete rotation of the body corresponds to an angle. Therefore, according to formula (2.1)
If during uniform rotation the angular velocity is known and at the initial moment of time the angle of rotation is , then the angle of rotation of the body during time t according to equation (2.1) is equal to:
If , then , or 
.
Angular velocity takes positive values if the angle between the radius vector, which determines the position of one of the points of the rigid body, and the axis OH increases, and negative when it decreases.
Thus, we can describe 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 when a rigid body rotates, its different points have unequal linear velocities, but the angular velocity is the same for all points.
There is a relationship between the linear speed of any point of a rotating body and its angular speed. Let's install it. A point lying on a circle of radius R, will cover the distance in one revolution. Since the time of one revolution of a body is a period T, then the modulus of the linear velocity of the point can be found as follows:
A rigid body can participate in two types of motion: translational and rotation. During the translational motion of a body, all its points make identical movements in equal periods of time, as a result of such movement, the speeds and accelerations of all points at each moment of time are the same. This means that it is enough to determine the law of motion of one point of the body to characterize the translational motion of the entire body.
If the body rotates, then all points of the rigid body move in circles with centers belonging to the straight line. This straight line is called the axis of rotation.
Any motion of a rigid body can be represented as a combination of translational motion and rotation. Let's consider plane motion. In this case, we decompose the elementary movement of some selected point of the body ($d\overline(s)$) into two movements: $d(\overline(s))_p$ - translational movement and $d(\overline(s))_v$ - rotational movement, with:
where $d(\overline(s))_p$ is the same for all points of the body. $d(\overline(s))_v-$ movement, which is carried out when a body is rotated by the same angle $d\varphi $ but relative to different axes.
Speed of complex motion of a rigid body
Let's divide both parts of expression (1) by a time interval equal to $dt$, we get:
\[\overline(v)=\frac(d\overline(s))(dt)=\frac(d(\overline(s))_p)(dt)+\frac(d(\overline(s)) _v)(dt)=(\overline(v))_0+\overline(v")\left(2\right),\]
where $(\overline(v))_0$ is the speed of translational motion of points of a rigid body (equal for all points); $\overline(v")$ - the speed caused by rotation differs for different points of the body.
The plane motion of a rigid body can be represented as the sum of two motions: translational with speed $(\overline(v))_0$ and rotation with angular speed $\overline(\omega )$.
The linear speed $\overline(v")$ of a point with a radius vector $\overline(r)$, which arises as a result of rotation of the body (linear speed of rotation of the point), is equal to:
\[\overline(v")=\left[\overline(\omega )\overline(r)\right]\left(3\right),\]
in expression (3) we mean the vector product. The linear rotation speed is found as:
where $\alpha $ is the angle between the direction of the angular velocity vector and the radius vector of the point (Fig. 1).
The speed of this point during complex movement is represented by the formula:
\[\overline(v)=(\overline(v))_0+\left[\overline(\omega )\overline(r)\right]\left(5\right).\]
There may be points in the body that participate in translational movement and rotation and at the same time remain motionless. Given $(\overline(v))_0\ $ and $\overline(\omega )$, one can find a radius vector ($\overline(r)$) such that $\overline(v)=0.$
Linear speed of a point moving around a circle
The movement of a material point along a circle is sometimes called the rotation of the point. The speed of movement of a material point in a circle is called linear speed in order to emphasize its difference from angular speed. When a point moves uniformly around a circle, we can write:
where $R$ is the radius of the circle; $s=\Delta \varphi R$ is the path a point travels in time $\Delta t$, equal to the length of the circular arc. Expression:
valid for uniform and uneven movement of a point around a circle.
With uniform motion in a circle, the motion can be characterized using the period of revolution of the point T, then:
Examples of problems on linear rotation speed
Example 1
Exercise. What is the linear speed of points lying on the surface of the Earth at the latitude of Moscow ($\alpha =56()^\circ $)?
Solution. Let's make a drawing.
Let's consider the motion of point A, which moves along a circle of radius $r$ in Fig. 2. The radius of this circle is related to the radius of the Earth ($R$) and the latitude of the area, which is indicated by the angle $\alpha$:
Let us take the radius of the Earth to be equal to $6.3\cdot (10)^6m.$ The period of revolution of the Earth around its axis is T= 86164 s. Let's calculate the linear speed of rotation of points at the indicated latitude:
Answer.$v=257\ \frac(m)(s)$
Example 2
Exercise. The helicopter rotor has a rotation frequency equal to $n$. The forward motion speed of the helicopter is $u$. What is the linear speed of one end of the propeller if its radius is $R$?
Solution. The speed of movement of the screw point during complex movement is equal to:
\[\overline(v)=(\overline(v))_0+\overline(v")\left(2.1\right),\]
where $(\overline(v))_0$ is the forward motion speed of the helicopter; $\overline(v")$ - linear speed of rotation of the screw end point.
In our case, according to the problem conditions:
\[\left|(\overline(v))_0\right|=u;;\ (\overline(v))_0\bot \overline(v"),\]
where $\overline(v")=\left[\overline(\omega )\overline(R)\right];;\ \left|\overline(v")\right|=\omega R.$
We find the speed of movement of the end of the screw as:
where $\omega =2\pi n.$
Answer.$v=\sqrt(u^2+(4(\pi )^2n^2R)^2)\ $
« Physics - 10th grade"
Angular velocity.
Each point of a body rotating around a fixed axis passing through point O moves in a circle, and different points travel different paths during time Δt. So, AA 1 > BB 1 (Fig. 1.62), therefore the modulus of the velocity of point A is greater than the modulus of the velocity of point B. But the radius vectors that determine the position of points A and B rotate during the time Δt by the same angle Δφ.
Angle φ is 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 periods of time, the radius vectors rotate through equal angles.
The greater the angle of rotation of the radius vector, which determines the position of some point of a rigid body, over a certain period of time, the faster the body rotates and the greater its angular velocity.
Angular velocity of a body during uniform rotation is a quantity equal to the ratio of the angle of rotation of the body υφ to the time period υt during which this rotation occurred.
We will denote 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 the grinding disk 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 of one revolution is equal to 1/ν seconds.
The time it takes a body to complete one complete revolution is called rotation period and is designated by the letter T.
If φ 0 ≠ 0, then φ - φ 0 = ωt, or φ = φ 0 ± ωt.
A radian is equal to the central angle subtended by an arc whose length is equal to the radius of the circle, 1 rad = 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 positive values if the angle between the radius vector, which determines the position of one of the points of the rigid body, and the OX axis increases (Fig. 1.63, a), and negative values 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 when an absolutely rigid body rotates, its different points have unequal linear velocities, but the angular velocity is the same for all points.
Let us establish a connection between the linear velocity of any point of a rotating body and its angular velocity. A point lying on a circle of radius R will travel a distance of 2πR in one revolution. Since the time of one revolution of the body is the period T, the module of the linear velocity of a point can be found as follows:
Since ω = 2πν, then
The modulus of centripetal acceleration of a point of a body moving uniformly around 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 examined the two simplest movements 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 motion, it is possible to describe the complex motion of an absolutely rigid body.
T, which the body spent on the way. Find the linear speed by dividing the path by the time it takes v=S/t.
To find the linear speed of a body that moves along a circular path, measure its radius R. After that, using a stopwatch, measure the time T spent by the body on one complete revolution. It is called the rotation period. To find the linear speed with which a body moves along a circular path, divide its length 2∙π∙R (circumference), π≈3.14, by the period of rotation v=2∙π∙R/T.
Determine the linear speed using its relation to the angular speed. To do this, use a stopwatch to find the time t during which the body describes an arc visible from the center at an angle φ. Measure this angle in and the radius of the circle R, which is the trajectory of the body. If the protractor measures in degrees, convert it to . To do this, multiply the number π by the readings of the protractor and divide by 180. For example, if the body described an arc of 30º, then this angle in radians is equal to π∙30/180=π/6. Considering that π≈3.14, then π/6≈0.523 radians. The central angle abutting the arc traversed by the body is called angular displacement, and the angular velocity is equal to the ratio of the angular displacement to , for which it is ω = φ/t. Find the linear speed by multiplying the angular speed by the radius of the trajectory v=ω∙R.
If there is a value of centripetal acceleration a that any body that moves in a circle has, find the linear speed. To do this, multiply the linear acceleration by the radius R of the circle representing the trajectory, and from the resulting number extract the square root v=√(a∙R).
They call it linear speed, with which the body moves along an arbitrary trajectory. Given the known length of the trajectory and the time it took to travel, find the linear speed in relation to length and time. Linear speed motion in a circle is equal to the product of the angular velocity and its radius. Also use other formulas to determine linear speed. It can be measured with a speedometer.
You will need
- stopwatch, protractor, tape measure or rangefinder, speedometer
Instructions
In the most general case, to determine the linear speed of a body at uniform , measure the length of the trajectory (the line along which the body moves) and divide by the length it took to cover this path v=S/t. If the movement is uneven, determine the linear speed using a speedometer or a special radar.
When a body moves in a circle, it has angular and linear velocities. To measure angular velocity, measure the central angle that describes the body in a circle over a certain period. For example, measure the time it takes a body to describe half a circle, in this case the central π radians (180º). Divide this angle by the time it took the body to travel half the circle, and you get the angular speed. If the angular speed body, then its linear speed, is equal to the product of the angular velocity and the radius of the circle along which the body is moving, which can be measured with a tape measure or range finder v=ω R.
Another way to determine the linear speed of a body moving in a circle. Using a stopwatch, measure the full body time around the circumference. This time is the period of rotation. Using a rangefinder or tape measure, measure the radius of the circular path along which the body moved. Calculate linear speed, dividing the product of the radius of the circle and 6.28 () by the time of its passage v = 6.28 R/t.
If the centripetal acceleration is known, which acts on every body moving in a circle with a constant speed yu, additionally measure its radius. In this case linear speed of a body moving in a circle is equal to the square root of the product of the centripetal acceleration and the radius of the circle.
Sources:
- linear speed in
To describe the movement of bodies along a complex trajectory, including a circle, kinematics uses the concepts of angular velocity, angular acceleration. Acceleration characterizes the change in the angular velocity of a body over time. In numerous kinematic problems, it is necessary to describe the motion of a body around moving and fixed points along a certain axis. At the same time, both speed and angular acceleration may change over time.
You will need
- - calculator.
Instructions
Remember that angular acceleration derivative with respect to , taken from the angular velocity (or ω). It's also that angular acceleration represents the second derivative taken with respect to time t from the angle of rotation. Angular acceleration can be written in the following form: →β= d →ω / dt. Thus, find the mean angular acceleration is possible from the increment of angular velocity to the increment of movement time: β avg. = Δω/Δt.
Find the angular velocity in order to calculate the angular acceleration. Let us assume that the rotation of a body around a fixed axis is described by the equation φ=f(t), and φ is the angle at a specific time t. Then, after a period of time Δt from moment t, the change in angle will be Δφ. Angular relationship between Δφ and Δt. Determine the angular velocity.
Find the angular mean acceleration according to the formula β avg. = Δω/Δt. That is, use a calculator to divide the change in angular velocity Δω by the known period of time during which the movement occurred. The quotient of division is the desired quantity. Write down the value found, expressing it in rad/s.
Please note that if the problem requires finding acceleration points of a rotating body. The speed of movement of any point of such a body is equal to the product of the angular velocity and the distance from the point to the axis of rotation. Wherein acceleration of a given point from two components: tangent and . The tangent is co-directed in a straight line with the speed with positive acceleration and in the opposite direction with negative acceleration. Let the distance from the point to the axis of rotation be designated R. And the angular velocity ω will be found by the formula: ω=Δv/Δt, where v is the linear speed of the body. To find the corner acceleration, divide the angular velocity by the distance between the point and the axis of rotation.
note
Determine precisely whether the axis around which the body is moving is mobile, since this is of fundamental importance for finding the angular acceleration. The rotation angle φ is a scalar quantity. In this case, the infinitesimal rotation, denoted by dφ, is a vector quantity. Its direction is determined by the right-hand rule (by the gimlet rule) and is directly related to the axis around which the body rotates.
Helpful advice
Remember that the angular acceleration vector is directed along the axis around which the body is moving. In this case, its direction coincides with the direction of movement during positive acceleration and is opposite to it during negative or slow motion.
The speed of the car is constantly changing while traveling. Determining what speed a car had at one point or another during the journey is very often done by both motorists themselves and the competent authorities. Moreover, there are a huge number of ways to find out the speed of a car.
Instructions
The easiest way to determine the speed of a car is familiar to everyone since school. To do this, you need to record the number of kilometers you have traveled and the time it took you to cover this distance. The speed of the car is calculated by: distance (km) divided by time (hours). This will give you the number you are looking for.
Option two is used when the car has stopped abruptly, but no one has taken basic measurements, such as time and distance. In this case, the speed of the car is calculated from its . There is even a special one for such calculations. But it can only be used if a mark is left on the road when braking.
So, the formula is as follows: the initial speed of the car is 0.5 x the braking rise time (m/s) x, the steady deceleration of the car during braking (m/s²) + the root of the braking distance (m) x, the steady deceleration of the car during braking (m/s²). The value called “steady-state deceleration of a car during braking” is fixed and depends only on what kind of asphalt was used. In the case of a dry road, substitute the number 6.8 into the formula - it is prescribed in GOST, used for calculations. For wet asphalt this value will be 5.
You can also determine the speed based on the braking distance using another formula. It looks like this: S = Ke x V x V / (254 x Fs). You need to substitute the following values into this formula: braking coefficient (Ke) - for this value is usually taken 1, speed at the beginning of braking (V), road adhesion coefficient (Fs) - for different weather conditions its value is determined: dry asphalt - 0 .7, wet - 0.4, compacted snow - 0.2, icy track - 0.1.
You can determine the speed of a car in a specific gear. To do this, you need the following values: number of crankshaft revolutions (Nc), dynamic wheel radius (R), gear ratio (in), main gear ratio (irn), initial vehicle speed (Va). Calculate the speed using the formula: Va = Nc x 60 x 2Pi x R / (1000 x in x irn).
When considering the movement of a body, we talk about its coordinates, speed, acceleration. Each of these parameters has its own formula depending on time, unless, of course, we are talking about chaotic movement.
Instructions
Let the body move straight and evenly. Then its speed is represented by a constant value, does not change with: v = const. Speed formula has the form v=v(const), where v(const) is a specific value.
Let the body move uniformly (uniformly accelerated or equally decelerated). As a rule, they only talk about uniformly accelerated motion, but in uniformly slow motion the acceleration is negative. Acceleration is usually a. Then the speed is expressed as a linear dependence on time: v=v0+a·t, where v0 is the initial speed, a is acceleration, t is time.
If you draw a graph of speed versus time, it will be a straight line. Acceleration is the tangent of the angle of inclination. With positive acceleration, the speed and straight line of speed rushes upward. With negative acceleration, the speed eventually reaches zero. Further, with the same value and direction of acceleration, the body can only move in the opposite direction.
Let the body move with a constant velocity. In this case, it has a centripetal acceleration a(c) directed towards the center of the circle. It is also called normal acceleration a(n). Linear speed and centripetal acceleration are related by the relation a=v?/R, where R is the direction along which the body moves.
The formula for the dependence of speed on time can have an arbitrary form. By definition, speed is the first derivative of a coordinate with respect to time: v=dx/dt. Therefore, if the dependence of the coordinate on time x=x(t) is given, the formula for the speed can be found by simple differentiation. For example, x(t)=5t?+2t-1. Then x"(t)=(5t?+2t-1)". That is, v(t)=5t+2.
If we further differentiate the speed formula, we can obtain acceleration, because acceleration is the first in time, and the second derivative of the coordinate: a=dv/dt=d?x/dx?. But the speed can also be obtained back from the acceleration by integration. You just need additional data. Typically, problems provide initial conditions.
When downloading a specific file from the Internet, it is interesting to know about the speed, as well as the time you have to wait until the entire operation is completed. This can be done using special software.
Since linear speed uniformly changes direction, the circular motion cannot be called uniform, it is uniformly accelerated.
Angular velocity
Let's choose a point on the circle 1 . Let's construct the radius. In a unit of time, the point will move to point 2 . In this case, the radius describes the angle. Angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T- this is the time during which the body makes one revolution.
Rotation frequency is the number of revolutions per second.
Frequency and period are interrelated by the relationship
Relationship with angular velocity
Linear speed
Each point on the circle moves at a certain speed. This speed is called linear. The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinding machine move, repeating the direction of instantaneous speed.
Consider a point on a circle that makes one revolution, the time spent is the period T. The path that a point travels is the circumference.
Centripetal acceleration
When moving in a circle, the acceleration vector is always perpendicular to the velocity vector, directed towards the center of the circle.
Using the previous formulas, we can derive the following relationships
Points lying on the same straight line emanating from the center of the circle (for example, these could be points that lie on the spokes of a wheel) will have the same angular velocities, period and frequency. That is, they will rotate the same way, but with different linear speeds. The further a point is from the center, the faster it will move.
The law of addition of speeds is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instantaneous velocities. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.
The Earth participates in two main rotational movements: diurnal (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, the cause of any acceleration is force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then the acting force is the elastic force.
If a body lying on a disk rotates with the disk around its axis, then such a force is the friction force. If the force stops its action, then the body will continue to move in a straight line
Consider the movement of a point on a circle from A to B. The linear speed is equal to v A And v B respectively. Acceleration is the change in speed per unit time. Let's find the difference between the vectors.
