A rope of total mass m and length L is suspended vertically....

A rope of total mass $m$ and length $L$ is suspended vertically. Show that a transverse pulse travels the length of the rope in a time interval $\Delta t=2 \sqrt{L} / g .$ (Suggestion: First find an expression for the wave speed at any point a distance $x$ from the lower end by considering the tension in the rope as resulting from the weight of the segment below that point.)

Key Concepts

Transverse Waves

Transverse waves are disturbances that propagate perpendicular to the direction of the medium. In the context of a rope, a transverse pulse is such a disturbance that travels along the length of the rope, and its speed depends on the properties of the rope such as tension and mass density.

Variable Tension in a Hanging Rope

In a vertically hanging rope, the tension is not constant but increases with distance from the bottom due to the weight of the rope below any given point. This variable tension is crucial since it influences the local wave speed along the rope, with regions higher up experiencing greater tension due to supporting more rope weight.

Wave Speed in a Variable Medium

The wave speed in a medium like a rope is given by the square root of the tension divided by the mass per unit length. When the tension varies along the rope, the wave speed also becomes a function of position. This relationship is foundational in determining how the pulse propagates through different parts of the rope.

Integration of Differential Time Intervals

To determine the total time taken by a pulse to traverse the rope, one must consider the pulse's differential travel time over infinitesimally small segments of the rope. By expressing the differential time as the ratio of an infinitesimal segment length to the locally varying wave speed, and then integrating along the entire rope, the overall propagation time can be computed.

Transcript

00:01 In this problem it is given that we have suppose here a rope of mass m and length l like this.

00:10 Okay and which is suspended vertically as shown in this particular figure then we have to show that a transverse wave a transverse wave travels along this wire from its bottom to top in time.

00:31 Interval of 2 root l upon g where l is the length of this particular rope.

00:40 We know that the velocity of transverse wave traveling along a rope or string we can find by the relation v is equal to root of t upon mu where t that is tension in that wire or rope and mu we have that is mass per unit length.

01:01 Now here as this rope is uniform, so it's mass per unit length that is constant for this given wire, but as this wire is suspended vertically and if i consider the various points of over here of this wire, then they will observe this tension which is variable.

01:25 So how this tension varies with respect to this position of point? over to this wire to find out that tension as tension in this wire is because of the weight of this wire so if i consider a wire of length wire then whatever be its mass we have so that will exert the tension over to this remaining part of wire so we have to find out the mass of the wire of length wire so if i consider y as a length or for y length of wire i can say for y length of wire it's mass we can find as mu into y as mu is you know that mass per unit length now the tension of tension that is present in this wire we can find by the weight of this why that much length of wire so here here if i put this value of mass of length of wire y as mu y.

02:41 So the tension over to this wire, i can see this mu y g.

02:46 And putting this value in this relation, i will get velocity is equal to root of mu y g upon so this mu mu mu get cancelled...

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