Question
A wire gets stretched by $4mm$ due to a certain load. If the same load is applied to a wire of same material with half the length and double the diameter of the first wire. What will be the change in its length?
✓
Answer
Given. $l_1=4 mm =4 \times 10^{-3} m$
$L _2=\frac{ L _1}{2}, D _2=2 D _{,} r _2=2 r _1$
To find: Change in length $\left(I_2\right)$
Formula: $Y =\frac{ FL }{ Al }=\frac{ FL }{\pi r ^2 l}$
Calculation: From formula,
$ Y _1=\frac{ F _1 L _1}{\pi r _1{ }^2 l_1}$
$Y _2=\frac{ F _2 L _2}{\pi r _2{ }^2 l_2} $
Dividing equation (ii) by equation (i),
$\frac{ Y _2}{ Y _1}=\frac{\frac{ F _2 L _2}{\pi r _2^2 l_2}}{\frac{ F _1 L _1}{\pi r _1^2 l_1}}$
Since same load is applied on same wire, $Y_1=Y_2$ and $F_1=F_2$
$\therefore \quad \frac{ L _1}{ r _1^2 l_1}=\frac{ L _2}{ r _2^2 l_2}$
....[From (iii)]
$ l_2 =\frac{ L _2 \times r _1^2 \times l_1}{ r _2^2 \times L _1}$
$ =\frac{ L _2 \times r _1^2 \times l_1}{4 r _1^2 \times 2 \times L _2}$
$ =\frac{l_1}{8}$
$ =\frac{4 \times 10^{-3}}{8}$
$= 0.5 \times 10^{-3} m$
$= 0.5 mm$
The new change in length of the wire is 0.5 mm.
$L _2=\frac{ L _1}{2}, D _2=2 D _{,} r _2=2 r _1$
To find: Change in length $\left(I_2\right)$
Formula: $Y =\frac{ FL }{ Al }=\frac{ FL }{\pi r ^2 l}$
Calculation: From formula,
$ Y _1=\frac{ F _1 L _1}{\pi r _1{ }^2 l_1}$
$Y _2=\frac{ F _2 L _2}{\pi r _2{ }^2 l_2} $
Dividing equation (ii) by equation (i),
$\frac{ Y _2}{ Y _1}=\frac{\frac{ F _2 L _2}{\pi r _2^2 l_2}}{\frac{ F _1 L _1}{\pi r _1^2 l_1}}$
Since same load is applied on same wire, $Y_1=Y_2$ and $F_1=F_2$
$\therefore \quad \frac{ L _1}{ r _1^2 l_1}=\frac{ L _2}{ r _2^2 l_2}$
....[From (iii)]
$ l_2 =\frac{ L _2 \times r _1^2 \times l_1}{ r _2^2 \times L _1}$
$ =\frac{ L _2 \times r _1^2 \times l_1}{4 r _1^2 \times 2 \times L _2}$
$ =\frac{l_1}{8}$
$ =\frac{4 \times 10^{-3}}{8}$
$= 0.5 \times 10^{-3} m$
$= 0.5 mm$
The new change in length of the wire is 0.5 mm.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Draw neat labelled diagram for a bar magnet
What are the two different modes of communication?
A train is moving eastward at 10 m/sec. A waiter is walking eastward at 1.2m/sec; and a fly is flying toward the north across the waiter’s tray at 2 m/s. What is the velocity of the fly relative to Earth.
Derive the expressions for the magnifying power and the length of a compound microscope using two convex lenses.
A hiker stands on the edge of a cliff $490 m$ above the ground and throws a stone horizontally with an initial speed of $15 \ ms^{-1}.$ Neglecting air resistance, find the time taken by the stone to reach the ground, and the speed with which it hits the ground. Take $= 9.8 \ m/s^2.$
→A TV tower has a height of $200\ m$ . How much population is covered by TV transmission if the average population density around the tower is $1000 / km ^2$ ? (Radius of the Earth $=6.4 \times 10^6 m$ )
→A factory owner kept five identical spheres between two wooden blocks on a ruler as shown in figure. He called all his workers and told them to take reading, to check their efficiency and knowledge.
$i.$ What is the area of central sphere?
$ii.$ What is the absolute error in reading of diameter of second sphere?
→$i.$ What is the area of central sphere?
$ii.$ What is the absolute error in reading of diameter of second sphere?
A constant retarding force of $50 N$ is applied to a body of mass $20 \ kg$ moving initially with a speed of $15 \ ms^{-1}.$ How long does the body take to stop?
→How a thermometer is calibrated?
Derive the relation between radian and degree. Also find out 1” and 1’ in terms of their respective values in radian. (Take π = 3.1416)