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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.
By Newton's second law, the cause of acceleration is a net force acting on the object, which is proportional to its mass m and its acceleration. The force, usually referred to as a centripetal force , has a magnitude [ 7 ] F c = m a c = m v 2 r {\displaystyle F_{c}=ma_{c}=m{\frac {v^{2}}{r}}} and is, like centripetal acceleration, directed ...
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law): = =, where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration.
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: [ 22 ] F = m d v d t ...
Since the definition of acceleration is a = dv/dt, the second law can be written in the simplified and more familiar form: F = m a . {\displaystyle \mathbf {F} =m\mathbf {a} \,.} So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle.
The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law or their derivative form, Lagrangian mechanics. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the ...
To find an acceleration, consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of >) derive a system of equations for the acceleration (a). As a sign convention, assume that a is positive when downward for and upward for .
Consider Newton's second law, = where F is the sum of forces on the object, m is mass, and a is the acceleration. Newton's equation can be applied to the tangential axis only. This is because only changes in speed are of concern and the bob is forced to stay in a circular path.
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