SATHEE: Properties of Matter 1 Question 21

Properties of Matter 1 Question 21

21. A thin rod of negligible mass and area of cross-section $4 \times 10^{- 6} m^{2}$ , suspended vertically from one end, has a length of $0.5 m$ at $100^{\circ} C$ . The rod is cooled to $0^{\circ} C$ , but prevented from contracting by attaching a mass at the lower end. Find

(a) this mass and

(b) the energy stored in the rod, given for the rod. Young’s modulus $= 10^{11} N / m^{2}$ , coefficient of linear expansion $= 10^{- 5} K^{- 1}$ and $g = 10 m / s^{2}$ .

(1997 C, 5M)

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Solution:

(a) The change in length due to decrease in temperature,

$\begin{aligned} Δ l_{1} & = L α Δ θ = (0.5) (10^{- 5}) (0 - 100) \\ Δ l_{1} & = - 0.5 \times 10^{- 3} m \end{aligned}$

Negative sign implies that length is decreasing. Now, let $M$ be the mass attached to the lower end. Then, change in length due to suspension of load is

$\begin{aligned} Δ l_{2} = \frac{(M g) L}{A Y} & = \frac{(M) (10) (0.5)}{(4 \times 10^{- 6}) (10^{11})} \\ Δ l_{2} & = (1.25 \times 10^{- 5}) M \\ Δ l_{1} + Δ l_{2} & = 0 \\ or (1.25 \times 10^{- 5}) M & = (0.5 \times 10^{- 3}) \\ ∴ & M & = (\frac{0.5 \times 10^{- 3}}{1.25 \times 10^{- 5}}) k g \\ or & M & = 40 k g \end{aligned}$

(b) Energy stored,

At $0^{\circ} C$ the natural length of the wire is less than its actual length; but since a mass is attached at its lower end, an elastic potential energy is stored in it. This is given by

$U = \frac{1}{2} (\frac{A Y}{L}) (Δ l)^{2}$

Here, $Δ l = | Δ l_{1} | = Δ l_{2} = 0.5 \times 10^{- 3} m$

Substituting the values,

$U = \frac{1}{2} (\frac{4 \times 10^{- 6} \times 10^{11}}{0.5}) {(0.5 \times 10^{- 3})}^{2} = 0.1 J$

NOTE

Comparing the equation

$Y = \frac{F / A}{Δ / / L} or F = (\frac{A Y}{L}) Δ l$

with the spring equation $F = K \cdot Δ x$ , we find that equivalent spring constant of a wire is $k = (\frac{A Y}{L})$

Therefore, potential energy stored in it should be

$U = \frac{1}{2} k (Δ I)^{2} = \frac{1}{2} (\frac{A Y}{L}) (Δ I)^{2}$

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Properties of Matter 1 Question 21

21. A thin rod of negligible mass and area of cross-section $4 \times 10^{- 6} m^{2}$ , suspended vertically from one end, has a length of $0.5 m$ at $100^{\circ} C$ . The rod is cooled to $0^{\circ} C$ , but prevented from contracting by attaching a mass at the lower end. Find

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Properties of Matter 1 Question 21

21. A thin rod of negligible mass and area of cross-section 4×10−6m2, suspended vertically from one end, has a length of 0.5m at 100∘C. The rod is cooled to 0∘C, but prevented from contracting by attaching a mass at the lower end. Find

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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21. A thin rod of negligible mass and area of cross-section $4 \times 10^{- 6} m^{2}$ , suspended vertically from one end, has a length of $0.5 m$ at $100^{\circ} C$ . The rod is cooled to $0^{\circ} C$ , but prevented from contracting by attaching a mass at the lower end. Find