What Is Circular Motion?

The motion of a particle on a circle with a certain point as the center of the circle and the radius r, that is, the motion of the particle whose circle is a circle is called "circle motion". It is one of the most common curvilinear movements. For example, motor rotors, wheels, pulleys, etc. all make circular motions. 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 motion in vertical plane). In circular motion, the most common and simplest is uniform velocity circular motion (because speed is a vector, uniform velocity circular motion actually refers to uniform velocity circular motion).

Chinese name: Circular motion

Applied discipline: physical

Overview of circular motion

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

Examples of circular motions are: a man-made satellite follows its trajectory, a stone is connected with a rope and swings in circles, a car turns on the track, an electron enters an average magnetic field vertically, and a gear rotates in the machine ( At any point on its surface and inside), belt drive, train wheels and turning tracks.

Circular motion provides centripetal force to the acceleration required by a moving object. This centripetal force pulls the moving object towards the center point of the circular trajectory. If there is no centripetal force, the object will inertially move in a straight line following Newton's first law. Even if the object velocity is constant, the velocity direction of the object is constantly changing. That is, in a uniform circular motion, the linear velocity changes (direction) while the angular velocity does not change.

Circular motion life

Train crossing curve: Actually do circular motion ^[1] , designed that the outer rail is slightly higher than the inner rail and has centripetal acceleration.

The relationship between uniform circular motion and simple harmonic motion (2 photos)

Car crossing arch bridge: It can also be seen as a circular motion. The support force of the bridge to the vehicle is, and because the pressure of the vehicle on the bridge and the support force of the bridge are a pair of acting and reaction forces, the pressure is the same. Also equal.

Car passing through a concave bridge: It can also be regarded as a circular motion. The support force of the bridge to the car is because the pressure of the car on the bridge and the support force of the bridge on the car are a pair of acting and reaction forces, so the pressure is also equal.

Weightlessness in spacecraft: Some people say that the weightlessness of a spacecraft is that it is too far away from the earth, and thus get rid of the gravity of the earth. This is wrong. It is because of the gravity of the earth that it is possible for the spacecraft and other occupants to make a circular motion around the earth. The analysis here is only for circular orbits. In fact, the interior of any aircraft that has its engine turned off and is not subject to resistance is a completely weightless environment

Ferris wheel . For example, in a container thrown in any direction in the air, all the objects in it are weightless.

Ferris wheel in playground

Centrifugal motion: Due to inertia, objects that make circular movements tend to fly away along the tangent direction. But it didn't fly away because the centripetal force was "pulling" it, keeping its distance from the center of the circle constant. Once the force suddenly disappears, the object flies away in a tangential direction. Except for the sudden disappearance of centripetal force, when the combined force is insufficient to provide the required centripetal force, although the object will not fly away along the tangent line, it will gradually move away from the center of the circle, which is called centrifugal motion.

Circular motion characteristics

The characteristics of uniform circular motion: the trajectory is circular, angular velocity, period, and the magnitude of linear velocity (Note: Because the linear velocity is a vector, the "linear velocity" is constant and the direction is constantly changing) and the magnitude of the centripetal acceleration is not the same Change, and the centripetal acceleration direction always points to the center of the circle.

Definition of linear velocity: The ratio of the arc length L passing by a particle moving in a circle to the time t used is called the linear velocity, or the product of angular velocity and radius.

The physical meaning of linear velocity: describes the speed of the particle moving along the circle, it is a vector.

Definition of angular velocity: the ratio of the radian (radian system: 360 ° = 2) that the radius turns to the time t used. (Angular velocity is constant in uniform circular motion)

Definition of period: the time it takes to make a circle moving at a constant speed.

Definition of rotation speed: the number of revolutions per unit time for an object that moves in a uniform circular motion.

The main formula of circular motion

Line speed

Linear velocity

The linear velocity can be deduced from the above

In addition to the linear velocity, v = 2r / T (Note: T is the period) = r = 2rn (Note: n represents the speed, n and T can be converted to each other, the formula is T = 1 / n) , represents the pi

Similarly, the angular velocity can be calculated using = radians / t = 2 / T = v / r = 2n

Where S is the arc length, r is the radius, V is the linear velocity, a is the acceleration, T is the period, and is the angular velocity (unit: rad / s).

Famous theory of circular motion

A centripetal force is required for any object in circular motion because it is constantly changing speed. The speed of the object is constant, but the direction is always changing. Only a proper amount of centripetal force can keep the object moving in a circular orbit. This acceleration (velocity is a vector and can be changed without changing its size) is provided by centripetal force. If this condition is not met, the object will leave the orbit. Note that centripetal acceleration reflects how quickly the direction of linear velocity changes.

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

Centripetal force keeps objects from falling out of orbit. A good example is gravity. Ground gravity gives artificial satellites the necessary force to make them orbit.

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

F = mv² / r, F = m²r (v is linear velocity, is angular velocity)

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

F = ma

Yes, combined external force = mass times acceleration, so:

a = v² / r = (2) ²r / T²

Mass Symbol RemovalReplace with F and ma. Therefore, the acceleration can be obtained without knowing the mass of the object.

When a particle moves in a circle in one plane, the projective motion in another orthogonal plane is a simple harmonic motion. Like the spring oscillator, the acceleration is constantly changing.

If the object moves at a constant velocity along a circle with a radius of R, and the time of one round of motion is T, then the magnitude of the linear velocity is equal to the product of the angular velocity and the radius R.

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

Uniform motion

Physical terms for circular motion

1 Definition: The mass moves along a circle. If the arcs passing through at any equal time are equal in length, this movement is called "constant-speed circular motion", also known as "uniform-speed circular motion". Change, but the speed direction changes at any time.

2 The conditions for the circular motion of the object: have the initial velocity; subject to a force (centripetal force) that has a constant size and the direction is always perpendicular to the direction of the object's speed of movement. When the object makes a constant-speed circular motion, although the magnitude of the speed does not change, the direction of the speed changes from time to time, so the constant-speed circular motion is a variable-speed motion. In addition, since the magnitude of the centripetal acceleration does not change during a uniform circular motion, but the direction changes at all times, the uniform circular motion is a variable acceleration motion. The term "uniform velocity" in the term "uniform velocity circular motion" merely means that the velocity does not change. An object that performs a uniform circular motion still has acceleration, and the acceleration is constantly changing, because its acceleration direction is constantly changing, and because its trajectory is circular, the 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 that performs variable-speed circular motion can always decompose an acceleration that points to the center of the circle. We refer to the acceleration that points to the center of the circle at all times as the centripetal acceleration.

Circular motion uniform speed related formula

1. v (linear velocity) = l / t = 2r / T = r = 2rf = 2nr (l represents arc length, t represents time, r represents radius, n is frequency, and is angular velocity)

2. (Angular Velocity) = / t = 2 / T = 2f ( means angle or radian)

3. T (period) = 2r / v = 2 /

4. f (frequency) = 1 / T

6. Fn (centripetal force) = mr² = mv² / r = mr4² / T² = mr4²f²

7. an (centripetal acceleration) = r² = v² / r = r4² / T² = r4²n²

8. Gravity acts as centripetal force when the rope pulls the ball over the apex, that is mg = mv² / r, so the minimum speed is v = (gr)

9, J max (the maximum value of work) = Fn × r

When the club pulls the ball, the minimum speed of v across the vertex is 0

Derivation of Centripetal Force Formula for Uniform Circular Motion

Let the velocity of a particle at A be Va. After a short period of time t, reach the point B. Let the velocity be Vb.

A centroid was obtained due to centripetal force

Velocity v, moves to point B under the combined action of v and Va, and reaches the speed of Vb

Then vector Va + vectorv = vector Vb, vectorv = vector Vb-vector Va

The angle between Va and Vb is equal to the angle between OA and OB by geometric method. When t is very small

v / v = s / r (explanation: because the mass points make a uniform circular motion, so Va = Vb = v, s represents the arc length, and r represents the radius)

So v = sv / r

v / t = s / t * v / r, where v / t represents centripetal acceleration a, and s / t represents linear velocity

So a = v² / r = r² = r4² / T² = r4²n²

F (centripetal force) = ma = mv² / r = mr² = m4² / T²r

One-dimensionalize the two-dimensional uniform circular motion in the plane

Establish a model: a small ball with mass m is connected to a spring with a stiffness coefficient of k (the original length is infinitely short), and a uniform velocity circular motion with an angular velocity and a radius A is performed in the plane rectangular coordinate system xy.

At this time, F (centripetal force) = kA = m (4 ^ 2 / T ^ 2) r, we know that T = 2k / m

On the x-axis, Vx = Vcos (t + ) Fx = kx = kAsin (t + ), that is x = kAsin (t + )

Similarly, there are Vy = Vsin (t + ) on the y axis, Fy = ky = kAsin (t + ), that is, y = kAcos (t + )

This generalization shows that the projection of the ball on any straight line passing through the origin is a simple harmonic motion.

Circular motion variable speed motion

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

The magnitude of the centripetal force is determined by the instantaneous speed of the moving object.

In this case, the force at the end of the rope can be divided into radial and tangential components. The radial component can be directed towards the center or outward.