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Phasor notation (also known as angle notation) is a mathematical notation used in electronics engineering and electrical engineering.A vector whose polar coordinates are magnitude and angle is written . [13] can represent either the vector (, ) or the complex number + =, according to Euler's formula with =, both of which have magnitudes of 1.
Kepler's laws apply only in the limited case of the two-body problem. Voltaire and Émilie du Châtelet were the first to call them "Kepler's laws". Nearly a century later, Isaac Newton had formulated his three laws of motion. In particular, Newton's second law states that a force F applied to a mass m produces an acceleration a given by the ...
Euler's second law states that the rate of change of angular momentum L about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force acting on that body M about that point: [1] [4] [5]
The mass of an object as measured in its own frame of reference is called its rest mass or invariant mass and is sometimes written . If an object moves with velocity v {\displaystyle \mathbf {v} } in some other reference frame, the quantity m = γ ( v ) m 0 {\displaystyle m=\gamma (\mathbf {v} )m_{0}} is often called the object's "relativistic ...
where I s is the moment of inertia, ω s is the angular velocity of spin about the spin axis, m is the mass, g is the acceleration due to gravity, θ is the angle between the spin axis and the axis of precession and r is the distance between the center of mass and the pivot. The torque vector originates at the center of mass.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
where μ is the reduced mass and r is the relative position r 2 − r 1 (with these written taking the center of mass as the origin, and thus both parallel to r) the rate of change of the angular momentum L equals the net torque N = = ˙ ˙ + ¨ , and using the property of the vector cross product that v × w = 0 for any vectors v and w ...
the mass–energy equivalence formula which gives the energy in terms of the momentum and the rest mass of a particle. The equation for the mass shell is also often written in terms of the four-momentum ; in Einstein notation with metric signature (+,−,−,−) and units where the speed of light c = 1 {\displaystyle c=1} , as p μ p μ ≡ p ...