The circular movement

The circular movement, also called, of curvilinear trajectory is much more abundant than the rectilinear movement. These are based on an axis of rotation and constant radius, whereby the trajectory is a circumference.

Uniform Circular Movement (UCM)

The uniform circular movement is present in many situations of daily life such as the plate of the microwave or the wheels of our vehicles among many others.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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In uniform circular motion the mobile describes a circular path with constant speed. That is, it runs through equal arcs in equal times, but at each moment it changes direction.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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The unit of measurement in the International System is radian. There is a simple mathematical relationship between the arcs described and the angles they support: “the angle is the relation between the arc and the radius with which it has been drawn”

   If we called ∆S the path arc and the angle swept by the radius, we have:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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The radian is the angle whose length of the arc is equal to the radius. Therefore, for a complete circumference it will be:

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Another unit for measuring angles are the sexagesimal degrees. But this unit is not used when measuring angular displacements.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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Linear speed and angular speed

Let’s imagine that we have a disk, like the one shown below, that rotates with a certain speed and in which we have marked two points, A and B. The two points describe a circular path movement, the two points describe the same angle but they do not travel the same distance since the radius of B is less than the radius of A.

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The linear velocity, “v”, is the speed with which a point moves along a circular path.

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If we imagine for a moment a record player that rotates with a certain speed and in which we have marked a point in one of its extremes.

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 The marked Point would describe an angle in the disk, with the angle swept, equation.pdf, in a time interval, ∆t. Therefore, the angular speed, “⍵”, en el UCM remains as: 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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The unit of angular speed in the International System is the radian per second (rad / s). The angular velocity is also expressed in revolutions per minute (rpm or rev / min), its equivalence being:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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When a disk rotates with some speed, the linear speed defined on the trajectory and the angular speed defined on the swept angle at a given time occur simultaneously. Therefore, it is possible to establish a relationship between the linear speed “v” and the angular one “⍵”.

 If the angular displacement and the angular speed are respectively:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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Clearing in the sencond equation and equaling we are finally:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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As:

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A clear example of this relationship can be shown with the wheels of bicycles, which thanks to them we advance (linear speed) due to the rotation made by the wheels (angular speed).

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | The circular movement

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If we set the value of the angular speed, ⍵, the greater of the radius of the wheel is, the greater the speed that the vehicle acquires.

Due to the Uniform Circular Motion is repeated periodically, it is also possible to study this movement according to periodic magnitudes, such as:

  • Period: It is the time that is used in a round or revolution. It is represented by “T” and is measured in seconds (s).
  • Frequency: It is the number of laps given by a mobile in 1 second. It is represented by  f”. It is the inverse of the period, therefore: 
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The frequency is measured in turns or cycles per second (c/s). Cycles per second are called hertz (Hz) although the most used unit of measure are the seconds minus 1 (s-1).

The angular speed of the body will be:

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Uniformly Accelerated Circular Movement (UACM)

Uniformly Accelerated Circular Movement occurs when a particle or solid body describes a circular path by steadily increasing or decreasing the speed in each unit of time. That is, the particle moves with a constant acceleration.

In the Uniform Circular Movement, linear speed, as being a vector tangent to the trajectory, varies its direction and direction along it. These changes in velocity induce an acceleration perpendicular to the trajectory, “an“, which is called centripetal acceleration, due to it is a vector which is always directed to the center of the circumference.

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Knowing that v = ⍵·R, we have the centripetal acceleration as:

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The centripetal acceleration of the Earth’s surface is responsible for well-visible phenomena such as, for example, the fact that the water in the basins is emptied with a combined movement of falling plus rotation, or the direction of rotation of the air masses atmospheric storm, in the northern hemisphere, rotates towards the center of the hemisphere in a counter-clockwise direction, while in the southern hemisphere, the rotation would be clockwise.

Parallelism between the Rectilinear Movement and the Circular Movement

Despite the obvious differences in their trajectory, there are certain similarities between the rectilinear and circular movement that we must mention and that highlight the similarities and equivalences in the concepts between both movements.

Given a rotación axis and the position of a point particle in circular or rotating movement, for a variación time ∆t or a given time dt, we have:

LINEAR MOVEMENT ANGULAR MOVEMENT
Position Arc
Speed Angular speed
Acceleration Angular acceleration
Mass Moment of inertia
Force Moment of force
Linear momentun Angular momentun

 

 

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