6. Class 11th Physics | Thermal Expansion | Tension in a Clamped Wire | by Ashish Arora


now, in this section we’ll discuss about,
tension developed in a clamped wire. say we’re having, a metal wire which is clamped between
2 rigid supports. say, these are rigid supports. and the wire is just taut. almost zero tension
is considered. say if we increase the temperature, then obviously the wire will expand, and there’ll
be a sag in the wire. so we can say that a sag is developed if temperature is increased,
it expands. and the tension will still remain zero. but say if we decrease the temperature,
and if it is just taut, say its temperature is decreased by delta t. in this situation
the wire will have a tendency to contract due to, thermal contraction phenomena. its
just the reverse of, thermal expansion. in this situation say, we open the wire from
1 end, then what happens is, we can say, the wire will contract upto, this much of length,
say which is delta l. earlier it was l not, it’ll be contracted by a length delta l
and we can write, thermal contraction. in length, will be delta l, which can be written
as, alpha l not delta t, length is l not. here alpha is the coefficient of linear expansion
of this wire. in this situation, if it remains clamped at, the rigid support, in that case,
whichever length is contracted due to thermal contraction, will be the same compensated
by elastic expansion. or if you can assume, say, it is contracted, and if i hold this
end and pull it and tie it up again at the rigid support, we can say that a tension is
developed in it. so the tension developed will be f, and the wire is, elastically expanded,
and, again it is, clamped at the rigid support. so in this situation, this tension, we can
call it f, which is developed within the length of this wire. so we can find out very easily
the tension developed in the wire by using, the young modulus of this metal wire. as we
know, young’s modulus. of material, can very easily be related by strain and stress
developed in the material. like, young modu-lus can be written as, stress by strain. so in
this situation, stress can be written as, f upon, s. where s is the cross sectional
area of the wire, which is equal to s. so the stress developed in the wire due to, elastic
elongation is, f by s, because f is the tension developed due to elastic elongation. and a
strain can directly be written as, delta l by l not. or you must consider the initial
length to be l not minus delta l, which can be approximately taken as l not. so in this
situation, this tension in, wire, can be written as, y s, delta l by, l not. and delta l by
l not from here we can write as alpha delta t, so it can be written as, y s, alpha delta
t. so, due to fall in temperature, if temperature fall is, delta t. and due to fall in temperature
by delta t, we can write, tension in wire, is equal to, y s, alpha delta t. so just keep
this thing in your mind. this is also very important result, that is quite useful in
different kinds of numerical problems.

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