An elastic wire is found to be elongated by 1cm when a mass is gently hung from it. It extends by 3cm more, …..

Q: An elastic wire is found to be elongated by 1 cm when a mass is gently hung from it. It extends by 3 cm more, when the attached mass is made to move uniformly in a horizontal circle performing 2 rev s-1. The length of the unstretched wire (in cm) is (Given, g = 9.81 ms-2)

(a) $\displaystyle \frac{24-\sqrt{15}}{\sqrt{15}}\times 4$

(b) $\displaystyle \frac{24}{\sqrt{15}}\times 4$

(c) $\displaystyle \frac{4}{\sqrt{5}}$

(d) $\displaystyle \frac{4-\sqrt{15}}{2}$

Ans: (a)

Sol:

Let the length of unstretched wire is L0

here , ω = 2πν = 4 π

T cosθ = m g …(i)

T sinθ = m ω2 r

T sinθ = m (4 π)2 r …(ii)

On dividing ‘

tanθ = 16 π2 r/g

As wire is elastic. So, it obeys Hooke’s law.

Initially , mg = k(1)

when mass rotates ,

T = k(4)

T = 4 mg

From (i)

4 mg cosθ = mg

cosθ = 1/4

hence , tanθ = √(15)

√(15) = 16 π2 r/g

$\displaystyle r = \frac{g \sqrt{15}}{16 \pi^2}$

$\displaystyle r = \frac{981 \sqrt{15}}{16 (3.14)^2}$

r = 24 cm

From figure ,

$\displaystyle sin\theta = \frac{24}{L_0 + 4}$

$\displaystyle \frac{\sqrt{15}}{4} = \frac{24}{L_0 + 4}$

$\displaystyle L_0 = \frac{24-\sqrt{15}}{\sqrt{15}}\times 4$

The pressure that has to be applied to the end of a steel wire of length 10cm to keep its length constant when…..

Q: The pressure that has to be applied to the end of a steel wire of length 10cm to keep its length constant when its temperature is raised by 100°C is (for steel, Young’s modulus is 2 × 1011 Nm-2 and coefficient of thermal expansion is 11 × 10-5 K-1)

(a) 2.2 × 108 Pa

(b) 2.2 × 109 Pa

(c) 2.2 × 107 Pa

(d) 2.2 × 106 Pa

Ans: (a)

Two wires are made of the same material and have the same volume. However, wire 1 has cross-sectional area A…..

Q: Two wires are made of the same material and have the same volume. However, wire 1 has cross-sectional area A and wire 2 has cross-sectional area 3A. If the length of wire 1 increases by Δx on applying force F, then how much force is needed to stretch wire 2 by the same amount?

(a) F

(b) 4F

(c) 6F

(d) 9F

Ans: (d)

One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal…..

Q: One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal thin copper wire of length L are radius R. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is

(a) 0.25

(b) 0.50

(c) 2.00

(d) 4.00

Ans: (c)