Search results
Results from the WOW.Com Content Network
The momentum of the object at time t is therefore p(t) = m(t)v(t). One might then try to invoke Newton's second law of motion by saying that the external force F on the object is related to its momentum p(t) by F = dp / dt , but this is incorrect, as is the related expression found by applying the product rule to d(mv) / dt : [17]
This shows that , and the polhode, stay on a closed loop, in the object's moving frame of reference. The orientation of the body in space thus has two degrees of freedom. Firstly, some point on the "taco edge" has to align with L , {\displaystyle \mathbf {L} ,} which is a constant vector in absolute space.
The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a, the vector ax is in C. [2]
The moment of force, or torque, is a first moment: =, or, more generally, .; Similarly, angular momentum is the 1st moment of momentum: =.Momentum itself is not a moment.; The electric dipole moment is also a 1st moment: = for two opposite point charges or () for a distributed charge with charge density ().
Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each.
Examples of integrals of motion are the angular momentum vector, =, or a Hamiltonian without time dependence, such as (,) = + (). An example of a function that is a constant of motion but not an integral of motion would be the function C ( x , v , t ) = x − v t {\displaystyle C(x,v,t)=x-vt} for an object moving at a constant speed in one ...
Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of [mass][length][time] −1. Mathematically, the duality between position and momentum is an example of Pontryagin duality .
The trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.