Earth motions

 

      Centrifugal force

 

   One of two bodies tied to a string around one centre, rotate a uniform motion. In the case of two bodies the rotation occurs around the centre of mass.

  When the string is cut they proceed along the tangent with a uniform rectilinear motion going away from the centre of rotation.

  The centrifugal force is a so-called pseudo force.

 

  The number of bodies can be varied in the tool bar.

  Button   keeps the trajectory visible.

  Buttons   and   allow to choose between a clockwise and anticlockwise rotation.

  The start is given by the button .

  To let the bodies free click on button .

 

 

 

 

 

      Force of Coriolis

 

  Three balls move, starting for the same point or going towards it, with a radial direction at 120° angles, leaving tracks at regular intervals of time.

  The trajectories of the three balls differ according to the coordinate system. When referring to a fixed coordinate system they coincide with the radii, whereas when referring to a rotating coordinate system they follow a curved shape. This is similar to what it is possible to what you can see on the Earth when there is movement without friction, like the winds.

  The force of Coriolis is a pseudo force.

 

  Buttons    and   allow the choice between a clockwise or anticlockwise rotation.

  With buttons   e   it is possible to choose the direction of the motion of the balls: from the centre towards the periphery (pole equator) or from the periphery towards the centre (equator - pole).

  With button  it is possible to have a view from upstairs or an axonometric view.

  For the choice of a fixed or rotating coordinate system use buttons   and  .

  Start by clicking the button .

 

 

 

 

 

      Foucault's pendulum

 

  The plane of oscillation of a pendulum is constant. This is true only in a Newtons coordinate system. The Earth rotates on its own axis and therefore the plane of oscillation of a pendulum, that is not at the equator, rotates in the opposite direction.

  In this simulation the pendulum oscillates on a rotating platform. The point of view are either that of an observer who is connected with the platform and sees the plane of oscillation of the pendulum rotate or that of an observer connected with a fixed coordinate system, who sees the platform rotate while the plane of oscillations stays still.

 

  For the choice of the fixed and/or rotating coordinate system use buttons   and  .

 

 

 

 

 

      Shifting aside the vertical

 

  A phenomenon due to the terrestrial rotation is that for which the bodies in free fall move away from the vertical.

  This is due to the fact that the tangential speed of the body caused by the terrestrial rotation at any height is greater than at level zero. Therefore the body which is dropped, keeps its higher horizontal speed compared to a body that is at the zero level and travels a longer way eastwards.

 

  The height of the body is chosen in the tool bar.

  To draw the trajectory click the button .

  Start by clicking the button .

  By clicking the button it is possible to start a new simulation with new features.

 

 

 

 

 

      Variation of the weight with the latitude

 

  The weight of a body varies according to the latitude. This is due to two concomitant causes::

  • The flatness of the Earth at the poles. In fact the force of attraction between two bodies follows the inverse square law of the distance between their centres. Therefore a body at the pole is attracted with more intensity than one at the equator.
  • The centrifugal force caused by the Earth's rotation, i.e. by the fact that the Earth is a rotating coordinate system. This force is proportional to the distance from the rotation axis; therefore it is maximum at the equator and nil at the poles. Besides it is opposed to the force of attraction.

  The screen shows an ellipse representing the Earth. Some inner concentric ellipses are used as scales to see the variation of the length of vectors. A circumference indicates the flatness of the poles.

  In this simulation vectors with arbitrary scales represent the force of attraction, the centrifugal force, the weight as a result of these two forces and the normal and tangential components of the centrifugal force to the Earth's surface.

 

  Button   shows the force of attraction.

  Button  shows the centrifugal force.

  Button  shows the weight and is active if the force of attraction and the centrifugal are visible.

  Button  shows the components of the centrifugal force and is active when the latter is visible.

  To vary the latitude use button .  Progress can be obtained with the key .

 

 

 

 

 

      Cyclone and anticyclone

 

  The simulation shows the formation of a cyclone and an anticyclone on the terrestrial surface due to the divergence and the convergence at a high altitude.

 

  In the tool bar it is possible to choose the point of view.

  By clicking the button it is possible to start a new simulation.

 

 

 

 

 

  Pseudo forces

 

The pseudo forces appear in a non-Newton coordinate system.

 

 

 

 

 

    Centre of mass

 

  Lets consider a system of bodies of mass along a line, at different distances from a point O.

  The centre of mass is the point:

  It is possible to broaden the definition to the space by considering also coordinate y and z.

 

 

 

 

 

  Trajectory

 

  The trajectory is the line followed by a moving body.

 

 

 

 

 

    Axonometry

 

  It is a graphic representation of spatial figures.

 

 

 

 

 

    Newton coordinate system

 

  We have a Newton coordinate system when it is motionless or it moves with a uniform rectilinear motion. The Earth is not a Newton coordinate system.

 

 

 

 

 

    Cartesian components of a vector.

 

  Given a vector and a curve or a surface, the components normal and tangential to the curve or the surface can be obtained by decomposing the vector along the tangential and normal (perpendicular) direction to the curve or the surface.

 

 

 

 

 

    Vector

 

  A vector is a quantity characterized by the fact that has a direction and a magnitude.

  It is represented by an oriented segment in which:

  • The direction is given by the line the segment is part and by the parallels.
  • The magnitude is given by the length of the arrow once a scale has been introduced.

 

 

 

 

 

    Force

 

  Force is that which makes the state of immobility or the uniform rectilinear motion of a free body vary (dynamic effects). It also produces deformation on the constraint bodies (static effects).

  Weight is the force with which bodies are attracted by the Earth. It is the resultant of the force of attraction and of the centrifugal force.

  The force is a vector quantity. The unit of measure is the Newton [kg m/sec²].

 

 

 

 

 

    Force of attracion

 

  This is the force which two masses M and m attract one another. It is proportional to the mass, and inversely proportional to the square of the distance between two centres.

.

  where G is the universal gravitational constant.

 

 

 

 

 

    Tangential speed

 

  It is the speed of a body that travel a curved trajectory (e.g. circolar motion).

 

 

 

 

 

    Circular uniform motion

 

  A body moves with a circular uniform motion when the trajectory is circular and the velocity is constant in magnitude and changes in direction. For this reason there is a centripetal acceleration.

 

 

 

 

 

    Rectilinear uniform motion

 

  A body moves with a rectilinear uniform motion when the trajectory is rectilinear and the velocity is constant.

 

 

 

 

 

    Proportionality

 

  • Between two quantities a and b there is a direct proportionality when the ratio a/b = k is constant.
  • Between two quantities a and b there is an inverse proportionality when the product a * b = k is constant.
  • Between two quantities a and b there is square proportionality when the ratio between one of them and the square of the other a /b² = k is constant.
  • Between two quantity a and b there is an inverse square proportionality when the product between one of them and the square of the other a * b² = k is constant.