Circular motion

A curvilinear motion

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This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

The particle takes a certain point as center of a circle radius Is the motion on the circumference of r, that is, when the particle moves, its trajectory is circumference The motion of is called "circular motion". It is one of the most common Curvilinear motion 。 E.g. motor rotor , wheels, pulleys, etc. The circular motion is divided into, Uniform circular motion And variable speed circular motion (such as rope/rod rotating ball in vertical plane Conical pendulum Sports). In circular motion, the most common and simplest is uniform circular motion (because the speed is vector So uniform circular motion actually means uniform rate Circular motion).

Chinese name: Circular motion
Foreign name: Circular Motion
Applicable fields: physical mechanics

Applied discipline: Physics
See publications: Physics Terms (Second Edition), Science Press
Time of publication: 1996^[2]

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Concept introduction

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In physics, Circular motion (circular motion) is to turn on a circle: a circular path or track. When considering the circular motion of an object, the volume of the object can be ignored and regarded as particle (except aerodynamics).

Examples of circular motion are: an artificial satellite follows its trajectory turn . Connect a stone with a rope and swing it in a circle. A racing car turns on the track. An electron vertically enters an average magnetic field , the rotation of a gear in the machine (any point on its surface or inside), the belt transmission device, the wheels of the train and the track at the turning.

Circumferential motion provides the acceleration required by moving objects with centripetal force. This centripetal force pulls the moving object to the center of the circular track. If there is no centripetal force, the object will follow Newton's first law inertia Ground Rectilinear motion 。 Even objects rate The speed and direction of the object are constantly changing. That is, in a uniform circular motion, the linear velocity changes (direction) while the angular velocity remains unchanged.

In life

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Trains crossing curves: actually doing circular movement^[1] , designed to be slightly higher than the inner rail, with Centripetal acceleration 。

Vehicle crossing arch bridge: it can also be regarded as circular movement. The supporting force of the bridge to the vehicle is centrifugal force, because the pressure of the vehicle to the bridge and the supporting force of the bridge to the vehicle are a pair of forces and Reaction force So the pressure is equal.

The car passes through a concave bridge: it can also be regarded as a circular motion. The supporting force of the bridge to the car is a centripetal force. Because the pressure of the car to the bridge and the supporting force of the bridge to the car are a pair of forces and reactions, the pressure is equal.

ferris wheel

Weightlessness in spacecraft: It is wrong to say that the reason why a spacecraft is weightless is that it is too far away from the earth, thus getting rid of the gravity of the earth. just because Gravity of the earth It is possible for the spacecraft, together with other crew members, to make a circular motion around the earth. The analysis here is only for circular orbits. In fact, the interior of any aircraft without resistance when the engine is turned off is a Complete weightlessness Environment. For example, all objects in a container thrown in any direction in the air Weightlessness 。

Ferris wheel of amusement park

Eccentric motion: objects in circular motion tend to fly away in the tangent direction due to inertia. But it didn't fly away, because the centripetal force was "pulling" it, keeping its distance from the center of the circle unchanged. Once the force suddenly disappears, the object will fly in the tangent direction. In addition to the sudden disappearance of the centripetal force, when the resultant force is not enough to provide the required centripetal force, the object will not fly along the tangent line, but will also gradually move away from the center of the circle, which is called centrifugal motion.

Features

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Uniform circular motion Features of, angular velocity ， cycle ， Linear velocity (Note: because the linear speed is a vector, the size of "linear speed" is constant, but the direction is always changing) and Centripetal acceleration The magnitude of is unchanged, and the direction of centripetal acceleration always points to the center of the circle.

Definition of linear velocity: the ratio of the arc length Δ L of the particle moving along the circumference to the time Δ t used is called linear velocity, or the product of angular velocity and radius.

Linear velocity Physical meaning : describes the speed of the particle moving along the circumference, which is a vector.

Definition of angular velocity: radian (radian system: 360 °=2 π) and the ratio of time t used. (Constant angular velocity in uniform circular motion)

Definition of period: the time it takes for an object in uniform circular motion to turn around for one week.

Definition of rotational speed: the number of revolutions per unit time of an object in uniform circular motion.

Main formula

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acceleration

Linear velocity

The linear speed can be derived from the above

To calculate the linear velocity, it can be used or derived

(Note: T is period, n represents speed, n and T can be converted each other, and the formula is T=1/n), π represents pi

Similarly, the angular velocity can be calculated by

among S Is the arc length, θ is the radian, r Is the radius, V Is the linear speed, a Is the acceleration, T Is the period, ω Is the angular velocity (unit: rad · s -1 ）。

Famous theory

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Any object needs a centripetal force when it moves in a circle, because it is constantly changing its speed. The speed of the object remains the same, but the direction changes all the time. Only a suitable centripetal force can maintain the motion of the object in a circular orbit. This acceleration (the velocity is a vector, which can be changed without changing the magnitude when changing the direction) is provided by the centripetal force. If this condition is not met, the object will leave the circular orbit. Note that the centripetal acceleration reflects the change speed of the linear velocity direction.

The direction of velocity of the object in circular motion is tangent to the circular path. The direction of the resultant force on a uniformly circular moving object always points to the center of the circle, that is, to change the direction of speed.

A centripetal force can keep an object in orbit. A good example is gravity 。 The earth's gravity gives the satellite the necessary force to move along its orbit.

In physics, the centripetal force is proportional to the square of the object's speed, its mass and the reciprocal of its radius:

(v is the linear velocity, ω is the angular velocity)

So if you know the force, mass and radius, you can calculate the rotation speed of the object. If we know the speed, mass and radius, we can calculate the force. The symbols are marked as follows:

Yes, external force=mass times acceleration, so:

Quality symbol removal - replaced by F and ma. Therefore, the acceleration can be calculated without knowing the mass of the object.

When a particle is in a plane When doing circular motion, the projection on another orthogonal plane is Simple harmonic motion The acceleration is constantly changing, just like the motion form of the spring vibrator.

If the object moves in a uniform circular motion along the circumference of radius R, and the time of one cycle of motion is T, the linear velocity is equal to the product of angular velocity and radius R.

When using this formula, it should be noted that the unit of angle must be radian. The above formula is valid only when the unit of angular velocity is radian/second.

Uniform motion

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Physical terms

1 Definition: particle along Circular motion , if it passes in any equal time arc With the same length, this kind of motion is called "uniform circular motion", also called "uniform rate circular motion", because the speed of the object in circular motion is constant, but the speed direction changes at any time.

2. Conditions for circular motion of objects: ① with Initial velocity ；② Subject to a constant size, direction and object movement speed The direction is always vertical, so it points center of a circle Force of（ centripetal force ）。 When an object moves in a uniform circular motion, although the speed remains the same, the direction of the speed changes all the time, so the uniform circular motion is Variable speed movement 。 And its centripetal direction acceleration The size of is unchanged, but the direction changes all the time, so the uniform circular motion is Variable acceleration motion 。 “ Uniform circular motion ”The word "uniform speed" in " rate Unchanged. The object in uniform circular motion still has acceleration, and the acceleration is constantly changing, because its acceleration direction is constantly changing, because its path of particle It is a circle, so uniform circular motion is a variable acceleration curve motion. The acceleration direction of uniform circular motion always points to the center of the circle. An object in circular motion with variable speed can always resolve an acceleration pointing to the center of the circle. The acceleration pointing to the center at the moment of direction is called centripetal acceleration.

Uniform velocity correlation formula

1、v( Linear velocity ):

(l represents arc length, t represents time, r represents radius, n represents frequency, ω represents angular velocity)

2、ω( angular velocity :)

(θ represents angle or radian)

3、T（ cycle ）:

4、f（ frequency ）:

5、F n （ centripetal force ）:

6、a n （ Centripetal acceleration ）:

7. The gravity acts as a centripetal force when the rope pulls the ball over the vertex, that is

Therefore, the minimum speed is

8、 J max ( Work Maximum):

When the rod pulls the ball, the minimum speed of v passing the vertex is 0

Derivation of the formula of centripetal force in uniform circular motion

Let the motion speed of a particle at A be

In a short time of exercise

After reaching point B, set the speed of

Due to the action of centripetal force, a point to the center of the circle is obtained

speed

, on

And

Under the joint action of

Speed of

Geometrically

And

Of included angle Equal to the included angle between OA and OB, when

Very hours

 v/v=s/r (Note: because the particle moves in a uniform circle

, s is the arc length, r is the radius)

therefore

⊿v=sv/r

Where △ v/△ t represents centripetal acceleration a, s/△ t represents linear velocity

therefore

Put the two-dimensional One-dimensional uniform circular motion

Establish a model: small ball with mass m and stiffness coefficient Is the spring of k (the original length is infinitely short) Rectangular coordinate system In x-y, make uniform circular motion with angular velocity ω and radius A.

here

knowable

There are

I.e

Similarly, there are

I.e

It can be known from this generalization that the projection of the ball on any line passing through the origin is in simple harmonic motion.

Variable speed movement

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Generally, the resultant force on the object moving in a circle is decomposed into radial component force (to keep the object moving in a circular orbit, that is, centripetal acceleration) and tangential component force (to change the speed of the object, that is, tangential acceleration).

The centripetal force is determined by the Instantaneous velocity decision.

In this case, the force on the object at the end of the rope can be divided into Radial component and Tangential component 。 The radial component can point to the center or outward.

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