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The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the magnitude of the vector, for example, = + +. Each vector equation represents three scalar equations.
A differential equation of motion, usually identified as some physical law (for example, F = ma) and applying definitions of physical quantities, is used to set up an equation for the problem. [ clarification needed ] Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a ...
The equation for universal gravitation thus takes the form: F = G m 1 m 2 r 2 , {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},} where F is the gravitational force acting between two objects, m 1 and m 2 are the masses of the objects, r is the distance between the centers of their masses , and G is the gravitational constant .
This is simplest to express for the case of a single point mass, in which is a function (,), and the point mass moves in the direction along which changes most steeply. In other words, the momentum of the point mass is the gradient of S {\displaystyle S} : v = 1 m ∇ S . {\displaystyle \mathbf {v} ={\frac {1}{m}}\mathbf {\nabla } S.}
Newton's law of motion for a particle of mass m written in vector form is: = , where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration (that is, acceleration in an inertial frame) of the particle, given by: = , where r is the position vector of the particle (not to be confused with radius, as ...
The solution of the equations is a flow velocity.It is a vector field—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time.
The total mass is the zeroth moment of mass. The center of mass is the 1st moment of mass normalized by total mass: R = 1 M ∑ i r i m i {\textstyle \mathbf {R} ={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}} for a collection of point masses, or 1 M ∫ r ρ ( r ) d 3 r {\textstyle {\frac {1}{M}}\int \mathbf {r} \rho (\mathbf {r} )\,d^{3}r} for ...
A mass quickly rotating about its axis in a ball-shaped device defines an angular momentum. When the person exercising tilts the ball, a force results which even increases the rotational speed when reacted to specifically by the user. Examples of using conservation of angular momentum for practical advantage are abundant.