Search results
Results from the WOW.Com Content Network
These are known as the Navier–Stokes equations. [35] The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction i and force in direction j, there is a stress component σ ij. The nine components make up the Cauchy stress tensor σ, which includes both pressure and shear.
F is the resultant force applied, t 1 and t 2 are times when the impulse begins and ends, respectively, m is the mass of the object, v 2 is the final velocity of the object at the end of the time interval, and; v 1 is the initial velocity of the object when the time interval begins. Impulse has the same units and dimensions (MLT −1) as momentum.
Newton's second law, in modern form, states that the time derivative of the momentum is the force: =. If the mass m {\displaystyle m} does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of the velocity, which is the acceleration: [ 21 ] F = m d v d t ...
The force in the equation is not the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as = since m is a constant in Newtonian mechanics. Newton's second law applies to point-like particles, and to all points in a rigid body.
As written in the Cauchy momentum equation, the stress terms p and τ are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the flow motion. [12]
where p i = momentum of particle i, F ij = force on particle i by particle j, and F E = resultant external force (due to any agent not part of system). Particle i does not exert a force on itself. Torque. Torque τ is also called moment of a force, because it is the rotational analogue to force: [8]
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 ().
Internal forces between the particles that make up a body do not contribute to changing the momentum of the body as there is an equal and opposite force resulting in no net effect. [3] The linear momentum of a rigid body is the product of the mass of the body and the velocity of its center of mass v cm. [1] [4] [5]