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For example, consider a system consisting of an object that is being lowered vertically by a string with tension, T, at a constant velocity. The system has a constant velocity and is therefore in equilibrium because the tension in the string, which is pulling up on the object, is equal to the weight force , mg ("m" is mass, "g" is the ...
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
The angular deficit δ is linearly related to the string tension (= mass per unit length), i.e. the larger the tension, the steeper the cone. Therefore, δ reaches 2π for a certain critical value of the tension, and the cone degenerates to a cylinder. (In visualizing this setup one has to think of a string with a finite thickness.)
Geodesics for massless particles are called "null geodesics", since they lie in a "light cone" or "null cone" of spacetime (the null comes about because their inner product via the metric is equal to 0), massive particles follow "timelike geodesics", and hypothetical particles that travel faster than light known as tachyons follow "spacelike ...
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]