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So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force from this equation is the force needed to hold the load motionless on the inclined plane, or push it up at a constant velocity. If the input force is greater than this, the load will accelerate up the plane.
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.
[4] [5] [6] A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined.
In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. [1] More generally, the term motion is a synonym for surjective isometry in metric geometry, [2] including elliptic geometry and hyperbolic ...
The kinematics equations of serial and parallel robots can be viewed as relating parameters, such as joint angles, that are under the control of actuators to the position and orientation [T] of the end-effector. From this point of view the kinematics equations can be used in two different ways.
= (Equation II: angle of projectile launch) Note that the sine function is such that there are two solutions for θ {\displaystyle \theta } for a given range d h {\displaystyle d_{h}} . The angle θ {\displaystyle \theta } giving the maximum range can be found by considering the derivative or R {\displaystyle R} with respect to θ ...
Eliminating the angular velocity dθ/dt from this radial equation, [47] ¨ = +. which is the equation of motion for a one-dimensional problem in which a particle of mass μ is subjected to the inward central force −dV/dr and a second outward force, called in this context the (Lagrangian) centrifugal force (see centrifugal force#Other uses of ...
In the inertial frame, the differential equation is not always helpful in solving for the motion of a general rotating rigid body, as both I in and ω can change during the motion. One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant.
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