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PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369763 Assertion :
The stretching of a coil is determined by its shear modulus.
Reason :
Shear modulus changes only shape of a coil keeping length unchanged.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369764 Assertion :
A hollow shaft is found to be stronger than a solid shaft made of same size and material.
Reason :
The torque required to produce a given twist in hollow cylinder is greater than that required to twist a solid cylinder of same size and material.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369765 Consider a solid cylinder having length $L$ and radius $a$ . The upper end of the cylinder is fixed to the ceiling. The lower end of the cylinder is twisted by an angle $θ$ then find the restoring torque produced in the rod. Given that $η$ is the shear modulus.
supporting img

1

\frac{π η θ a^{4}}{2 L}

2

\frac{η θ a^{3}}{L}

3

\frac{η θ a^{4}}{L}

4

\frac{2 π η θ a^{4}}{L}

Explanation:

Consider two points $P$ and $q$ as shown in the figure. After rotating the rod the point $q$ gets shifted to $q^{'}$ . Let $ϕ$ is the angle of shear.
supporting img

From the figure $q q^{'} = a θ = L ϕ$
Consider a circular slice of radius $r$ and width dr as shown in the figure.

The small shear force on the slice is $η = \frac{d F / d A}{\tan ϕ}$ $d F = d A η \tan ϕ$
$\begin{aligned} d F = (2 π r d r) η (\frac{r θ}{L}) \\ d τ = d F r = \frac{2 π η θ r^{3} d r}{L} \\ τ = \frac{2 π η θ}{L} \int_{0}^{a} r^{3} d r \\ τ = \frac{2 π η θ}{L} [\frac{a^{4}}{4}] = \frac{π η θ a^{4}}{2 L} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369766 The edge of an aluminium cube is $10 c m$ long. One face of the cube is firmly fixed to a vertical wall. A mass of $100 k g$ is then attached to the opposite face of the cube. The shear modulus of aluminium is $25 G P a$ Then the vertical deflection of this face

1

3 \times 10^{- 7} m

2

4 \times 10^{- 7} m

3

8 \times 10^{- 7} m

4

2 \times 10^{- 7} m

Explanation:

$Δ ℓ = \frac{F \times ℓ}{A \times η} = \frac{100 \times 10 \times 10^{- 1}}{100 \times 10^{- 4} \times 25 \times 10^{9}} = 4 \times 10^{- 7}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369763 Assertion :
The stretching of a coil is determined by its shear modulus.
Reason :
Shear modulus changes only shape of a coil keeping length unchanged.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369764 Assertion :
A hollow shaft is found to be stronger than a solid shaft made of same size and material.
Reason :
The torque required to produce a given twist in hollow cylinder is greater than that required to twist a solid cylinder of same size and material.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369765 Consider a solid cylinder having length $L$ and radius $a$ . The upper end of the cylinder is fixed to the ceiling. The lower end of the cylinder is twisted by an angle $θ$ then find the restoring torque produced in the rod. Given that $η$ is the shear modulus.
supporting img

1

\frac{π η θ a^{4}}{2 L}

2

\frac{η θ a^{3}}{L}

3

\frac{η θ a^{4}}{L}

4

\frac{2 π η θ a^{4}}{L}

Explanation:

Consider two points $P$ and $q$ as shown in the figure. After rotating the rod the point $q$ gets shifted to $q^{'}$ . Let $ϕ$ is the angle of shear.
supporting img

From the figure $q q^{'} = a θ = L ϕ$
Consider a circular slice of radius $r$ and width dr as shown in the figure.

The small shear force on the slice is $η = \frac{d F / d A}{\tan ϕ}$ $d F = d A η \tan ϕ$
$\begin{aligned} d F = (2 π r d r) η (\frac{r θ}{L}) \\ d τ = d F r = \frac{2 π η θ r^{3} d r}{L} \\ τ = \frac{2 π η θ}{L} \int_{0}^{a} r^{3} d r \\ τ = \frac{2 π η θ}{L} [\frac{a^{4}}{4}] = \frac{π η θ a^{4}}{2 L} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369766 The edge of an aluminium cube is $10 c m$ long. One face of the cube is firmly fixed to a vertical wall. A mass of $100 k g$ is then attached to the opposite face of the cube. The shear modulus of aluminium is $25 G P a$ Then the vertical deflection of this face

1

3 \times 10^{- 7} m

2

4 \times 10^{- 7} m

3

8 \times 10^{- 7} m

4

2 \times 10^{- 7} m

Explanation:

$Δ ℓ = \frac{F \times ℓ}{A \times η} = \frac{100 \times 10 \times 10^{- 1}}{100 \times 10^{- 4} \times 25 \times 10^{9}} = 4 \times 10^{- 7}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369763 Assertion :
The stretching of a coil is determined by its shear modulus.
Reason :
Shear modulus changes only shape of a coil keeping length unchanged.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369764 Assertion :
A hollow shaft is found to be stronger than a solid shaft made of same size and material.
Reason :
The torque required to produce a given twist in hollow cylinder is greater than that required to twist a solid cylinder of same size and material.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369765 Consider a solid cylinder having length $L$ and radius $a$ . The upper end of the cylinder is fixed to the ceiling. The lower end of the cylinder is twisted by an angle $θ$ then find the restoring torque produced in the rod. Given that $η$ is the shear modulus.
supporting img

1

\frac{π η θ a^{4}}{2 L}

2

\frac{η θ a^{3}}{L}

3

\frac{η θ a^{4}}{L}

4

\frac{2 π η θ a^{4}}{L}

Explanation:

Consider two points $P$ and $q$ as shown in the figure. After rotating the rod the point $q$ gets shifted to $q^{'}$ . Let $ϕ$ is the angle of shear.
supporting img

From the figure $q q^{'} = a θ = L ϕ$
Consider a circular slice of radius $r$ and width dr as shown in the figure.

The small shear force on the slice is $η = \frac{d F / d A}{\tan ϕ}$ $d F = d A η \tan ϕ$
$\begin{aligned} d F = (2 π r d r) η (\frac{r θ}{L}) \\ d τ = d F r = \frac{2 π η θ r^{3} d r}{L} \\ τ = \frac{2 π η θ}{L} \int_{0}^{a} r^{3} d r \\ τ = \frac{2 π η θ}{L} [\frac{a^{4}}{4}] = \frac{π η θ a^{4}}{2 L} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369766 The edge of an aluminium cube is $10 c m$ long. One face of the cube is firmly fixed to a vertical wall. A mass of $100 k g$ is then attached to the opposite face of the cube. The shear modulus of aluminium is $25 G P a$ Then the vertical deflection of this face

1

3 \times 10^{- 7} m

2

4 \times 10^{- 7} m

3

8 \times 10^{- 7} m

4

2 \times 10^{- 7} m

Explanation:

$Δ ℓ = \frac{F \times ℓ}{A \times η} = \frac{100 \times 10 \times 10^{- 1}}{100 \times 10^{- 4} \times 25 \times 10^{9}} = 4 \times 10^{- 7}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369763 Assertion :
The stretching of a coil is determined by its shear modulus.
Reason :
Shear modulus changes only shape of a coil keeping length unchanged.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369764 Assertion :
A hollow shaft is found to be stronger than a solid shaft made of same size and material.
Reason :
The torque required to produce a given twist in hollow cylinder is greater than that required to twist a solid cylinder of same size and material.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369765 Consider a solid cylinder having length $L$ and radius $a$ . The upper end of the cylinder is fixed to the ceiling. The lower end of the cylinder is twisted by an angle $θ$ then find the restoring torque produced in the rod. Given that $η$ is the shear modulus.
supporting img

1

\frac{π η θ a^{4}}{2 L}

2

\frac{η θ a^{3}}{L}

3

\frac{η θ a^{4}}{L}

4

\frac{2 π η θ a^{4}}{L}

Explanation:

Consider two points $P$ and $q$ as shown in the figure. After rotating the rod the point $q$ gets shifted to $q^{'}$ . Let $ϕ$ is the angle of shear.
supporting img

From the figure $q q^{'} = a θ = L ϕ$
Consider a circular slice of radius $r$ and width dr as shown in the figure.

The small shear force on the slice is $η = \frac{d F / d A}{\tan ϕ}$ $d F = d A η \tan ϕ$
$\begin{aligned} d F = (2 π r d r) η (\frac{r θ}{L}) \\ d τ = d F r = \frac{2 π η θ r^{3} d r}{L} \\ τ = \frac{2 π η θ}{L} \int_{0}^{a} r^{3} d r \\ τ = \frac{2 π η θ}{L} [\frac{a^{4}}{4}] = \frac{π η θ a^{4}}{2 L} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369766 The edge of an aluminium cube is $10 c m$ long. One face of the cube is firmly fixed to a vertical wall. A mass of $100 k g$ is then attached to the opposite face of the cube. The shear modulus of aluminium is $25 G P a$ Then the vertical deflection of this face

1

3 \times 10^{- 7} m

2

4 \times 10^{- 7} m

3

8 \times 10^{- 7} m

4

2 \times 10^{- 7} m

Explanation:

$Δ ℓ = \frac{F \times ℓ}{A \times η} = \frac{100 \times 10 \times 10^{- 1}}{100 \times 10^{- 4} \times 25 \times 10^{9}} = 4 \times 10^{- 7}$

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