PendulumA pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period.
Kepler's laws of planetary motionIn astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that: The orbit of a planet is an ellipse with the Sun at one of the two foci. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
Rotation around a fixed axisRotation around a fixed axis or axial rotation is a special case of rotational motion around a axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result.
Moving magnet and conductor problemThe moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant velocity, v, with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of reference of the conductor. The observable quantity in the experiment, the current, is the same in either case, in accordance with the basic principle of relativity, which states: "Only relative motion is observable; there is no absolute standard of rest".
Circular motionIn physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with a constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body. In circular motion, the distance between the body and a fixed point on the surface remains the same.
Gauss's law for gravityIn physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's law for gravity is often more convenient to work from than Newton's law. The form of Gauss's law for gravity is mathematically similar to Gauss's law for electrostatics, one of Maxwell's equations.
Pendulum (mechanics)A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated.
Rotation matrixIn linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae.
Double pendulumIn physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of coupled ordinary differential equations and is chaotic.
3D rotation groupIn mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation.
Inverted pendulumAn inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies.
WindWind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few hours, to global winds resulting from the difference in absorption of solar energy between the climate zones on Earth. The two main causes of large-scale atmospheric circulation are the differential heating between the equator and the poles, and the rotation of the planet (Coriolis effect).
