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PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369933 Assertion :
Addition of carbon to iron increases the elastic property.
Reason :
Carbon and iron belong to same group of elements.

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

369934 A uniform ring of mass $M$ of outside radius $r_{2}$ is fitted tightly with a shaft of radius $r_{1}$ . If the shaft is rotated with a constant angular acceleration $α$ about it's axis, the moment of the elastic force in the ring about the axis of rotation is
supporting img

1

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

2

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

3

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

4

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

Explanation:

Consider a circular slice having radius $x$ and width $d x$ . The mass of the slice is $d m$ and its areal density is
$σ = \frac{M}{π (r_{2}^{2} - r_{1}^{2})}$
supporting img

The toque required to rotate the slice with angular acceleration $α$ is
$\begin{aligned} d τ = (d m a_{t}) x = (d m α x) x \\ d τ = d m α x^{2} = μ (2 π x d x) α x^{2} \\ \int d τ = 2 π μ x α \int x^{3} d x \\ τ = \frac{2 π α M}{π r_{2}^{2} - r_{1}^{2}} \int_{1_{1}}^{r_{2}} x^{3} d x = \frac{2 α M}{r_{2}^{2} - r_{1}^{2}} [\frac{r_{2}^{4} - r_{1}^{4}}{4}] \\ τ = \frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369935 Assertion :
The size of a hydrogen balloon increases as it rises in air.
Reason :
The material of the balloon can be easily stretched.

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

369936 Assertion :
Railway tracks are laid on small sized wooden sleepers.
Reason :
Large sized wooden sleepers are preferred to decrease pressure on ground.

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

369937 Proof resilience is related to

1 potential energy stored in the elastic body

2 stiffness of beam

3 elastic fatigue

4 elastic relaxation

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369933 Assertion :
Addition of carbon to iron increases the elastic property.
Reason :
Carbon and iron belong to same group of elements.

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

369934 A uniform ring of mass $M$ of outside radius $r_{2}$ is fitted tightly with a shaft of radius $r_{1}$ . If the shaft is rotated with a constant angular acceleration $α$ about it's axis, the moment of the elastic force in the ring about the axis of rotation is
supporting img

1

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

2

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

3

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

4

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

Explanation:

Consider a circular slice having radius $x$ and width $d x$ . The mass of the slice is $d m$ and its areal density is
$σ = \frac{M}{π (r_{2}^{2} - r_{1}^{2})}$
supporting img

The toque required to rotate the slice with angular acceleration $α$ is
$\begin{aligned} d τ = (d m a_{t}) x = (d m α x) x \\ d τ = d m α x^{2} = μ (2 π x d x) α x^{2} \\ \int d τ = 2 π μ x α \int x^{3} d x \\ τ = \frac{2 π α M}{π r_{2}^{2} - r_{1}^{2}} \int_{1_{1}}^{r_{2}} x^{3} d x = \frac{2 α M}{r_{2}^{2} - r_{1}^{2}} [\frac{r_{2}^{4} - r_{1}^{4}}{4}] \\ τ = \frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369935 Assertion :
The size of a hydrogen balloon increases as it rises in air.
Reason :
The material of the balloon can be easily stretched.

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

369936 Assertion :
Railway tracks are laid on small sized wooden sleepers.
Reason :
Large sized wooden sleepers are preferred to decrease pressure on ground.

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

369937 Proof resilience is related to

1 potential energy stored in the elastic body

2 stiffness of beam

3 elastic fatigue

4 elastic relaxation

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369933 Assertion :
Addition of carbon to iron increases the elastic property.
Reason :
Carbon and iron belong to same group of elements.

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

369934 A uniform ring of mass $M$ of outside radius $r_{2}$ is fitted tightly with a shaft of radius $r_{1}$ . If the shaft is rotated with a constant angular acceleration $α$ about it's axis, the moment of the elastic force in the ring about the axis of rotation is
supporting img

1

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

2

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

3

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

4

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

Explanation:

Consider a circular slice having radius $x$ and width $d x$ . The mass of the slice is $d m$ and its areal density is
$σ = \frac{M}{π (r_{2}^{2} - r_{1}^{2})}$
supporting img

The toque required to rotate the slice with angular acceleration $α$ is
$\begin{aligned} d τ = (d m a_{t}) x = (d m α x) x \\ d τ = d m α x^{2} = μ (2 π x d x) α x^{2} \\ \int d τ = 2 π μ x α \int x^{3} d x \\ τ = \frac{2 π α M}{π r_{2}^{2} - r_{1}^{2}} \int_{1_{1}}^{r_{2}} x^{3} d x = \frac{2 α M}{r_{2}^{2} - r_{1}^{2}} [\frac{r_{2}^{4} - r_{1}^{4}}{4}] \\ τ = \frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369935 Assertion :
The size of a hydrogen balloon increases as it rises in air.
Reason :
The material of the balloon can be easily stretched.

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

369936 Assertion :
Railway tracks are laid on small sized wooden sleepers.
Reason :
Large sized wooden sleepers are preferred to decrease pressure on ground.

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

369937 Proof resilience is related to

1 potential energy stored in the elastic body

2 stiffness of beam

3 elastic fatigue

4 elastic relaxation

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369933 Assertion :
Addition of carbon to iron increases the elastic property.
Reason :
Carbon and iron belong to same group of elements.

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

369934 A uniform ring of mass $M$ of outside radius $r_{2}$ is fitted tightly with a shaft of radius $r_{1}$ . If the shaft is rotated with a constant angular acceleration $α$ about it's axis, the moment of the elastic force in the ring about the axis of rotation is
supporting img

1

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

2

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

3

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

4

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

Explanation:

Consider a circular slice having radius $x$ and width $d x$ . The mass of the slice is $d m$ and its areal density is
$σ = \frac{M}{π (r_{2}^{2} - r_{1}^{2})}$
supporting img

The toque required to rotate the slice with angular acceleration $α$ is
$\begin{aligned} d τ = (d m a_{t}) x = (d m α x) x \\ d τ = d m α x^{2} = μ (2 π x d x) α x^{2} \\ \int d τ = 2 π μ x α \int x^{3} d x \\ τ = \frac{2 π α M}{π r_{2}^{2} - r_{1}^{2}} \int_{1_{1}}^{r_{2}} x^{3} d x = \frac{2 α M}{r_{2}^{2} - r_{1}^{2}} [\frac{r_{2}^{4} - r_{1}^{4}}{4}] \\ τ = \frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369935 Assertion :
The size of a hydrogen balloon increases as it rises in air.
Reason :
The material of the balloon can be easily stretched.

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

369936 Assertion :
Railway tracks are laid on small sized wooden sleepers.
Reason :
Large sized wooden sleepers are preferred to decrease pressure on ground.

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

369937 Proof resilience is related to

1 potential energy stored in the elastic body

2 stiffness of beam

3 elastic fatigue

4 elastic relaxation

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369933 Assertion :
Addition of carbon to iron increases the elastic property.
Reason :
Carbon and iron belong to same group of elements.

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

369934 A uniform ring of mass $M$ of outside radius $r_{2}$ is fitted tightly with a shaft of radius $r_{1}$ . If the shaft is rotated with a constant angular acceleration $α$ about it's axis, the moment of the elastic force in the ring about the axis of rotation is
supporting img

1

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

2

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

3

\frac{M (r_{2}^{4} + r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})}

4

\frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} + r_{1}^{2})}

Explanation:

Consider a circular slice having radius $x$ and width $d x$ . The mass of the slice is $d m$ and its areal density is
$σ = \frac{M}{π (r_{2}^{2} - r_{1}^{2})}$
supporting img

The toque required to rotate the slice with angular acceleration $α$ is
$\begin{aligned} d τ = (d m a_{t}) x = (d m α x) x \\ d τ = d m α x^{2} = μ (2 π x d x) α x^{2} \\ \int d τ = 2 π μ x α \int x^{3} d x \\ τ = \frac{2 π α M}{π r_{2}^{2} - r_{1}^{2}} \int_{1_{1}}^{r_{2}} x^{3} d x = \frac{2 α M}{r_{2}^{2} - r_{1}^{2}} [\frac{r_{2}^{4} - r_{1}^{4}}{4}] \\ τ = \frac{M (r_{2}^{4} - r_{1}^{4}) α}{2 (r_{2}^{2} - r_{1}^{2})} \end{aligned}$

PHXI09:MECHANICAL PROPERTIES OF SOLIDS

369935 Assertion :
The size of a hydrogen balloon increases as it rises in air.
Reason :
The material of the balloon can be easily stretched.

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

369936 Assertion :
Railway tracks are laid on small sized wooden sleepers.
Reason :
Large sized wooden sleepers are preferred to decrease pressure on ground.

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

369937 Proof resilience is related to

1 potential energy stored in the elastic body

2 stiffness of beam

3 elastic fatigue

4 elastic relaxation