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PHXI11:THERMAL PROPERTIES OF MATTER

366751 Statement A :
Copper expands more than glass for same rise in temperature.
Statement B :
Copper is far better conductor of heat than glass.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI11:THERMAL PROPERTIES OF MATTER

366752 If two rods of length $L$ and $2 L$ having coefficient of linear expansion $α$ and $2 α$ respectively are connected so that total length becomes $3 L$ , the average coefficient of linear expansion of the composition rod equals:

1

\frac{3}{2} α

2

\frac{5}{2} α

3

\frac{5}{3} α

4 None of these

Explanation:

Effective linear coefficient
$\begin{matrix} Δ L = Δ L_{1} + Δ L_{2} \\ L α Δ T = α_{1} L_{1} Δ T + α_{2} L_{2} Δ T \\ α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}} = \frac{L (α) + 2 L (2 α)}{3 L} \end{matrix}$
$= \frac{5 α}{3}$

PHXI11:THERMAL PROPERTIES OF MATTER

366753 A copper and a tungsten plate having a thickness $d$ each are riveted together so that at $0^{\circ} C$ they form a flat bimetallic plate. Find the average radius of the curvature of this plate at temperature T. The coefficient of linear expansion for Copper and tungsten are $α_{C u}$ and $α_{t}$ .

1

δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

2

δ \frac{[1 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

3

δ \frac{[2 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

4

δ \frac{[1 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

Explanation:

The length of copper plate at a temperature $T$ is $l_{C} = l_{0} (1 + α_{C} T)$ , where $l_{0}$ is the length of copper plate at $0^{\circ} C$ . The length of the tungsten plate is $l_{t} = l_{0} (1 + α_{t} T)$
supporting img

Length of copper plate $l_{C} = ϕ (R + δ / 2)$
Consequently,
$ϕ (R + δ / 2) = l_{0} (1 + α_{C} T) (1)$
Similarly for tungsten plate
$ϕ (R - δ / 2) = l_{0} (1 + α_{t} T) (2)$
To eliminate the unknown quantities, and $l_{0}$ we divide the equation (1) by (2)
$\begin{aligned} \Rightarrow \frac{(R + δ / 2)}{(R - δ / 2)} = \frac{(1 + α_{C} T)}{(1 + α_{t} T)} \\ \Rightarrow R = δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]} \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366754 As a result of a temperature rise of $32^{\circ} C$ , a bar with a crack at its centre buckles upward as shown in the figure. If the fixed distance is $L_{0} = 3.77 m$ and the coefficient of linear expansion is $25 \times 10^{- 6} /^{\circ} C$ , find $x$ , the distance to which the centre rises.
supporting img

1

6.0 c m

2

3.5 c m

3

4 c m

4

7.5 c m

Explanation:

Let $\frac{L}{2}$ is the length of each part after rising temperature. From the right angle triangle
${(\frac{L}{2})}^{2} = x^{2} + {(\frac{L_{0}}{2})}^{2}$
$\frac{L^{2}}{4} = x^{2} + \frac{L_{0}^{2}}{4}$
$x^{2} = \frac{1}{4} (L^{2} - L_{0}^{2})$
$x = \frac{1}{2} \sqrt{L^{2} - L_{0}^{2}} = \frac{L_{0}}{2} \sqrt{{(1 + α Δ T)}^{2} - 1}$
$x = \frac{L_{0}}{2} \sqrt{1 + 2 α Δ T - 1} = \frac{L_{0}}{2} \sqrt{2 α Δ T}$
$x = \frac{3.77}{2} \times \sqrt{2 \times 25 \times 10^{- 6} \times 32}$
$x = 7.5 c m$

PHXI11:THERMAL PROPERTIES OF MATTER

366751 Statement A :
Copper expands more than glass for same rise in temperature.
Statement B :
Copper is far better conductor of heat than glass.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI11:THERMAL PROPERTIES OF MATTER

366752 If two rods of length $L$ and $2 L$ having coefficient of linear expansion $α$ and $2 α$ respectively are connected so that total length becomes $3 L$ , the average coefficient of linear expansion of the composition rod equals:

1

\frac{3}{2} α

2

\frac{5}{2} α

3

\frac{5}{3} α

4 None of these

Explanation:

Effective linear coefficient
$\begin{matrix} Δ L = Δ L_{1} + Δ L_{2} \\ L α Δ T = α_{1} L_{1} Δ T + α_{2} L_{2} Δ T \\ α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}} = \frac{L (α) + 2 L (2 α)}{3 L} \end{matrix}$
$= \frac{5 α}{3}$

PHXI11:THERMAL PROPERTIES OF MATTER

366753 A copper and a tungsten plate having a thickness $d$ each are riveted together so that at $0^{\circ} C$ they form a flat bimetallic plate. Find the average radius of the curvature of this plate at temperature T. The coefficient of linear expansion for Copper and tungsten are $α_{C u}$ and $α_{t}$ .

1

δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

2

δ \frac{[1 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

3

δ \frac{[2 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

4

δ \frac{[1 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

Explanation:

The length of copper plate at a temperature $T$ is $l_{C} = l_{0} (1 + α_{C} T)$ , where $l_{0}$ is the length of copper plate at $0^{\circ} C$ . The length of the tungsten plate is $l_{t} = l_{0} (1 + α_{t} T)$
supporting img

Length of copper plate $l_{C} = ϕ (R + δ / 2)$
Consequently,
$ϕ (R + δ / 2) = l_{0} (1 + α_{C} T) (1)$
Similarly for tungsten plate
$ϕ (R - δ / 2) = l_{0} (1 + α_{t} T) (2)$
To eliminate the unknown quantities, and $l_{0}$ we divide the equation (1) by (2)
$\begin{aligned} \Rightarrow \frac{(R + δ / 2)}{(R - δ / 2)} = \frac{(1 + α_{C} T)}{(1 + α_{t} T)} \\ \Rightarrow R = δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]} \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366754 As a result of a temperature rise of $32^{\circ} C$ , a bar with a crack at its centre buckles upward as shown in the figure. If the fixed distance is $L_{0} = 3.77 m$ and the coefficient of linear expansion is $25 \times 10^{- 6} /^{\circ} C$ , find $x$ , the distance to which the centre rises.
supporting img

1

6.0 c m

2

3.5 c m

3

4 c m

4

7.5 c m

Explanation:

Let $\frac{L}{2}$ is the length of each part after rising temperature. From the right angle triangle
${(\frac{L}{2})}^{2} = x^{2} + {(\frac{L_{0}}{2})}^{2}$
$\frac{L^{2}}{4} = x^{2} + \frac{L_{0}^{2}}{4}$
$x^{2} = \frac{1}{4} (L^{2} - L_{0}^{2})$
$x = \frac{1}{2} \sqrt{L^{2} - L_{0}^{2}} = \frac{L_{0}}{2} \sqrt{{(1 + α Δ T)}^{2} - 1}$
$x = \frac{L_{0}}{2} \sqrt{1 + 2 α Δ T - 1} = \frac{L_{0}}{2} \sqrt{2 α Δ T}$
$x = \frac{3.77}{2} \times \sqrt{2 \times 25 \times 10^{- 6} \times 32}$
$x = 7.5 c m$

PHXI11:THERMAL PROPERTIES OF MATTER

366751 Statement A :
Copper expands more than glass for same rise in temperature.
Statement B :
Copper is far better conductor of heat than glass.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI11:THERMAL PROPERTIES OF MATTER

366752 If two rods of length $L$ and $2 L$ having coefficient of linear expansion $α$ and $2 α$ respectively are connected so that total length becomes $3 L$ , the average coefficient of linear expansion of the composition rod equals:

1

\frac{3}{2} α

2

\frac{5}{2} α

3

\frac{5}{3} α

4 None of these

Explanation:

Effective linear coefficient
$\begin{matrix} Δ L = Δ L_{1} + Δ L_{2} \\ L α Δ T = α_{1} L_{1} Δ T + α_{2} L_{2} Δ T \\ α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}} = \frac{L (α) + 2 L (2 α)}{3 L} \end{matrix}$
$= \frac{5 α}{3}$

PHXI11:THERMAL PROPERTIES OF MATTER

366753 A copper and a tungsten plate having a thickness $d$ each are riveted together so that at $0^{\circ} C$ they form a flat bimetallic plate. Find the average radius of the curvature of this plate at temperature T. The coefficient of linear expansion for Copper and tungsten are $α_{C u}$ and $α_{t}$ .

1

δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

2

δ \frac{[1 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

3

δ \frac{[2 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

4

δ \frac{[1 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

Explanation:

The length of copper plate at a temperature $T$ is $l_{C} = l_{0} (1 + α_{C} T)$ , where $l_{0}$ is the length of copper plate at $0^{\circ} C$ . The length of the tungsten plate is $l_{t} = l_{0} (1 + α_{t} T)$
supporting img

Length of copper plate $l_{C} = ϕ (R + δ / 2)$
Consequently,
$ϕ (R + δ / 2) = l_{0} (1 + α_{C} T) (1)$
Similarly for tungsten plate
$ϕ (R - δ / 2) = l_{0} (1 + α_{t} T) (2)$
To eliminate the unknown quantities, and $l_{0}$ we divide the equation (1) by (2)
$\begin{aligned} \Rightarrow \frac{(R + δ / 2)}{(R - δ / 2)} = \frac{(1 + α_{C} T)}{(1 + α_{t} T)} \\ \Rightarrow R = δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]} \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366754 As a result of a temperature rise of $32^{\circ} C$ , a bar with a crack at its centre buckles upward as shown in the figure. If the fixed distance is $L_{0} = 3.77 m$ and the coefficient of linear expansion is $25 \times 10^{- 6} /^{\circ} C$ , find $x$ , the distance to which the centre rises.
supporting img

1

6.0 c m

2

3.5 c m

3

4 c m

4

7.5 c m

Explanation:

Let $\frac{L}{2}$ is the length of each part after rising temperature. From the right angle triangle
${(\frac{L}{2})}^{2} = x^{2} + {(\frac{L_{0}}{2})}^{2}$
$\frac{L^{2}}{4} = x^{2} + \frac{L_{0}^{2}}{4}$
$x^{2} = \frac{1}{4} (L^{2} - L_{0}^{2})$
$x = \frac{1}{2} \sqrt{L^{2} - L_{0}^{2}} = \frac{L_{0}}{2} \sqrt{{(1 + α Δ T)}^{2} - 1}$
$x = \frac{L_{0}}{2} \sqrt{1 + 2 α Δ T - 1} = \frac{L_{0}}{2} \sqrt{2 α Δ T}$
$x = \frac{3.77}{2} \times \sqrt{2 \times 25 \times 10^{- 6} \times 32}$
$x = 7.5 c m$

PHXI11:THERMAL PROPERTIES OF MATTER

366751 Statement A :
Copper expands more than glass for same rise in temperature.
Statement B :
Copper is far better conductor of heat than glass.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI11:THERMAL PROPERTIES OF MATTER

366752 If two rods of length $L$ and $2 L$ having coefficient of linear expansion $α$ and $2 α$ respectively are connected so that total length becomes $3 L$ , the average coefficient of linear expansion of the composition rod equals:

1

\frac{3}{2} α

2

\frac{5}{2} α

3

\frac{5}{3} α

4 None of these

Explanation:

Effective linear coefficient
$\begin{matrix} Δ L = Δ L_{1} + Δ L_{2} \\ L α Δ T = α_{1} L_{1} Δ T + α_{2} L_{2} Δ T \\ α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}} = \frac{L (α) + 2 L (2 α)}{3 L} \end{matrix}$
$= \frac{5 α}{3}$

PHXI11:THERMAL PROPERTIES OF MATTER

366753 A copper and a tungsten plate having a thickness $d$ each are riveted together so that at $0^{\circ} C$ they form a flat bimetallic plate. Find the average radius of the curvature of this plate at temperature T. The coefficient of linear expansion for Copper and tungsten are $α_{C u}$ and $α_{t}$ .

1

δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

2

δ \frac{[1 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

3

δ \frac{[2 + (α_{c} + α_{t}) T]}{[(α_{c} - α_{t}) T]}

4

δ \frac{[1 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]}

Explanation:

The length of copper plate at a temperature $T$ is $l_{C} = l_{0} (1 + α_{C} T)$ , where $l_{0}$ is the length of copper plate at $0^{\circ} C$ . The length of the tungsten plate is $l_{t} = l_{0} (1 + α_{t} T)$
supporting img

Length of copper plate $l_{C} = ϕ (R + δ / 2)$
Consequently,
$ϕ (R + δ / 2) = l_{0} (1 + α_{C} T) (1)$
Similarly for tungsten plate
$ϕ (R - δ / 2) = l_{0} (1 + α_{t} T) (2)$
To eliminate the unknown quantities, and $l_{0}$ we divide the equation (1) by (2)
$\begin{aligned} \Rightarrow \frac{(R + δ / 2)}{(R - δ / 2)} = \frac{(1 + α_{C} T)}{(1 + α_{t} T)} \\ \Rightarrow R = δ \frac{[2 + (α_{c} + α_{t}) T]}{[2 (α_{c} - α_{t}) T]} \end{aligned}$

PHXI11:THERMAL PROPERTIES OF MATTER

366754 As a result of a temperature rise of $32^{\circ} C$ , a bar with a crack at its centre buckles upward as shown in the figure. If the fixed distance is $L_{0} = 3.77 m$ and the coefficient of linear expansion is $25 \times 10^{- 6} /^{\circ} C$ , find $x$ , the distance to which the centre rises.
supporting img

1

6.0 c m

2

3.5 c m

3

4 c m

4

7.5 c m

Explanation:

Let $\frac{L}{2}$ is the length of each part after rising temperature. From the right angle triangle
${(\frac{L}{2})}^{2} = x^{2} + {(\frac{L_{0}}{2})}^{2}$
$\frac{L^{2}}{4} = x^{2} + \frac{L_{0}^{2}}{4}$
$x^{2} = \frac{1}{4} (L^{2} - L_{0}^{2})$
$x = \frac{1}{2} \sqrt{L^{2} - L_{0}^{2}} = \frac{L_{0}}{2} \sqrt{{(1 + α Δ T)}^{2} - 1}$
$x = \frac{L_{0}}{2} \sqrt{1 + 2 α Δ T - 1} = \frac{L_{0}}{2} \sqrt{2 α Δ T}$
$x = \frac{3.77}{2} \times \sqrt{2 \times 25 \times 10^{- 6} \times 32}$
$x = 7.5 c m$