Two identical spring, each of stiffness k are welded to each other at point P. The other two ends are fixed to the edge of a smooth horizontal tube as shown. A particle of mass m is welded at P. The entire system is horizontal. The period of oscillation of the particle in the direction of x is a)b)c)d)Correct answer is option 'D'. Can you explain this answer?

Question

Two identical spring, each of stiffness k are welded to each other at point P. The other two ends are fixed to the edge of a smooth horizontal tube as shown. A particle of mass m is welded at P. The entire system is horizontal. The period of oscillation of the particle in the direction of x is  a)b)c)d)Correct answer is option 'D'. Can you explain this answer?

Answer

Let us assume a force dF is applied at P in positive x - direction. This will stretch each spring by dl inducing a spring force dFs in each spring.

Let the static deformation of the system is dx (along the x-direction). The particle is in equilibrium. So,

Using Pythagoras theorem,

Here y is constant.

Answer

Answer will be d but sin( theta) is also coming in square root.please check

Answer

If x is the extension then x sin theta will be the extension in the spring therefore two springs are at an angle of 180 - 2 theta so will have to take the component in the direction opposite to the displacement therefore the net restoring force will be 2K x sin square theta and you will get your answer

Answer

Sometime around 500 million years ago, green algae floating in shallow coastal waters adapted to take advantage of the nearby dry land. Two major obstacles faced the early pioneers: the need to support their bodies and the threat of drying out. While some terrestrial plants (mosses, liverworts and hornworts) adapted to survive periods of dessication but remained dependent on water, others overcame these challenges by developing roots to anchor them, a waxy layer to prevent water loss and vascular tissues to transport water and provide structural support. Fossils like Cooksoniaand Aglaophyton show how these features were developed and elaborated over tens of millions of years, adjusting the shape and structure of plants for a terrestrial life. The plants of the early Devonian(420-390 million years ago) were relatively small; most were only a few centimeters tall, though Psilophyton could reach tens of centimeters in height. By the mid-Devonian (around 390 million years ago), tree-like ('arborescent') plants appear in the fossil record, rising to dominance through the Carboniferous (360-300 million years ago). The Carboniferous is named for these early forests, which became a rich supply of the coal that fuels our precarious civilization (carbo='coal' + ferre='to carry'), though coal is also found in beds from other geological eras. Carboniferous forests were composed of lycopsids (club mosses), sphenopsids (horse tails), and ferns; in the mid- to late-Carboniferous, progymnosperms and early seeds plants appeared at tropical latitudes as ferns overtook lycopsids and sphenopsids to become the dominant plants in the forests. The earliest trees known in the fossil record are Wattieza and Archaeopteris (which shouldn't be confused with the similarly named Archaeopteryx). Around 10m tall, Archaeopteris was a progymnosperm, a group of trees that looked like modern gymnosperms but reproduced via spores instead of seeds.Archaeopteris is considered a likely ancestor of extant gymnosperms. Capped with large fronds, it had a woody stem with annual growth rings recording how it thickened over the course of its life; like gymnosperms, it sported an extensive root system. Unlike gymnosperms, however, Archaeopteris also reproduced via spores borne on special leaves, in the same manner as ferns do today.

Answer

Let us assume a force dF is applied at P in positive x - direction. This will stretch each spring by dl inducing a spring force dFs in each spring.

Let the static deformation of the system is dx (along the x-direction). The particle is in equilibrium. So,

Using Pythagoras theorem,

Here y is constant.

Answer

Let us assume a force dF is applied at P in positive x - direction. This will stretch each spring by dl inducing a spring force dFs in each spring.

Let the static deformation of the system is dx (along the x-direction). The particle is in equilibrium. So,

Using Pythagoras theorem,

Here y is constant.

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