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A string or rope is often idealized as one dimension, having fixed length but being massless with zero cross section. If there are no bends in the string, as occur with vibrations or pulleys , then tension is a constant along the string, equal to the magnitude of the forces applied by the ends of the string.
An equation for the acceleration can be derived by analyzing forces. Assuming a massless, inextensible string and an ideal massless pulley, the only forces to consider are: tension force (T), and the weight of the two masses (W 1 and W 2). To find an acceleration, consider the forces affecting each individual mass.
where is the applied tension on the line, is the resulting force exerted at the other side of the capstan, is the coefficient of friction between the rope and capstan materials, and is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle =).
In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g. the partial derivative of L with respect to the z velocity component of particle 2, defined by v z,2 = dz 2 /dt, is ...
The rod or cord is massless, inextensible and always remains under tension. The bob is a point mass. The motion occurs in two dimensions. The motion does not lose energy to external friction or air resistance. The gravitational field is uniform. The support is immobile. The differential equation which governs the motion of a simple pendulum is
In climbing, a Tyrolean traverse is a technique that enables climbers to cross a void between two fixed points, such as between a headland and a detached rock pillar (e.g. a sea stack), or between two points that enable the climbers to cross over an obstacle such as chasm or ravine, or over a fast moving river. [1]
Also, A μ is the electromagnetic 4-vector potential. There exists an equivalent formulation of the relativistic Lagrangian, which has two advantages: it allows for a generalization to massless particles and tachyons; it is based on an energy functional instead of a length functional, such that it does not contain a square root.
The four-velocity at any point of world line () is defined as: = where is the four-position and is the proper time. [ 1 ] The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined for tachyonic world lines, where the ...