The type of rubber band used inside some baseballs and golf...

The type of rubber band used inside some baseballs and golf balls obeys Hooke's law over a wide range of elonga- tion of the band. A segment of this material has an unstretched length $\ell$ and a mass $m .$ When a force $F$ is applicd, the band stretches an additional length $\Delta \ell$ . (a) What is the speed (in terms of $m, \Delta \ell,$ and the spring constant $k$ ) of transverse waves on this stretched rubber band? (b) Using your answer to (a), show that the time reguired for a transerse pulse to travel the length of the rubber band is proportional to 1$/ \sqrt{\Delta \ell}$ i and is constant if $\Delta \ell \gg \ell$

The type of rubber band used inside some baseballs and golf balls obeys Hooke's law over a wide range of elonga-
tion of the band. A segment of this material has an unstretched
length $\ell$ and a mass $m .$ When a force $F$ is applicd, the band
stretches an additional length $\Delta \ell$ . (a) What is the speed (in terms of $m, \Delta \ell,$ and the spring constant $k$ ) of transverse waves
on this stretched rubber band? (b) Using your answer to (a),
show that the time reguired for a transerse pulse to travel the length of the rubber band is proportional to 1$/ \sqrt{\Delta \ell}$ i
and is constant if $\Delta \ell \gg \ell$

Key Concepts

Scaling of Wave Propagation Time

The time required for a wave pulse to traverse the length of a stretched band is determined by the ratio of the band’s length to the wave speed (t = L/v). When the tension is related to the band’s extension via Hooke’s law, the dependence of the travel time on the extension can be explicitly derived. This scaling behavior, particularly the inverse square root dependence on the extension (i.e., t ? 1/??l), illustrates how increased stretching can lead to shorter transit times, which is a vital concept in the study of dynamic systems and elastic materials.

Hooke's Law

Hooke’s law describes the relationship between the force applied to an elastic material and the resulting extension. In the context of rubber bands and other elastic materials, it states that the force F is proportional to the extension ?l (i.e., F = k?l), where k is the spring constant. This law is fundamental when considering how forces affect the material's dimensions and energy storage, especially under small to moderate deformations.

Transverse Wave Speed on a Stretched String

The speed of transverse waves on a stretched string or rubber band is determined by the square root of the ratio of the tension in the string (T) to its linear mass density (?), given by v = ?(T/?). This relation is crucial for understanding how waves propagate along a stretched elastic medium and is derived from balancing forces and applying Newton’s laws to a small segment of the medium.

Linear Mass Density

Linear mass density, denoted by ?, refers to the mass per unit length of a stretched string or band, which is calculated as ? = m / L, where m is the total mass and L is the total stretched length of the string. This concept is important in wave dynamics, as the mass distribution along the medium affects how quickly waves can move through it.

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