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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 physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
Consider a pendulum of mass m and length ℓ, which is attached to a support with mass M, which can move along a line in the -direction. Let x {\displaystyle x} be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle θ {\displaystyle \theta } from the vertical.
If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher. [1] A string, tied at A , is kept in tension by W , a suspended weight, and two bridges, B and the movable bridge C , while D is a freely moving wheel; all allowing one to demonstrate Mersenne's laws regarding tension and length [ 1 ]
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This type of very "slack" slackline provides a wide array of opportunities for both swinging and static maneuvers. A rodeo line has no tension in it, while both traditional slacklines and tightropes are tensioned. This slackness in the rope or webbing allows it to swing at large amplitudes and adds a different dynamic.