Show that strain energy per unit volume of a strained wire is 1 2 × s t r e s s × s t r a i n .

The work done by the external applied force during stretching is stored as potential energy in the wire and is called as strain energy in the wire. If we assume a wire of length L , area of cross-section A is fixed at one end and stretched by suspending a load M from the other end, due to this load extension of wire is x Hence, Potential energy = work done by external applied force = 1 2 × l o a d × e x t e n s i o n Strain energy = 1 2 × l o a d × e x t e n s i o n Now, strain energy per unit volume = 1 2 × l o a d × e x t e n s i o n v o l u m e = 1 2 × l o a d × e x t e n s i o n a r e a o f c r o s s e c t i o n × l e n g t h = 1 2 × l o a d a r e a o f c r o s s e c t i o n × e x t e n s i o n l e n g t h = 1 2 × s t r e s s × s t r a i n

Show that strain energy per unit volume of a strained wire is 1 2 × s t r e s s × s t r a i n .

The work done by the external applied force during stretching is stored as potential energy in the wire and is called as strain energy in the wire. If we assume a wire of length L , area of cross-section A is fixed at one end and stretched by suspending a load M from the other end, due to this load extension of wire is x Hence, Potential energy = work done by external applied force = 1 2 × l o a d × e x t e n s i o n Strain energy = 1 2 × l o a d × e x t e n s i o n Now, strain energy per unit volume = 1 2 × l o a d × e x t e n s i o n v o l u m e = 1 2 × l o a d × e x t e n s i o n a r e a o f c r o s s e c t i o n × l e n g t h = 1 2 × l o a d a r e a o f c r o s s e c t i o n × e x t e n s i o n l e n g t h = 1 2 × s t r e s s × s t r a i n

MEDIUM

Earn 100

Show that strain energy per unit volume of a strained wire is $\frac{1}{2} \times s t r e s s \times s t r a i n$ .

Important Questions on Mechanical Properties of Solids

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

The elastic behaviour of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of $5 \times 10^{- 4}$ is _____ $kJ m^{- 3}$ . Assume that material is elastic upto the linear strain of $5 \times 10^{- 4}$ .

Question Image

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A wire of Young's modulus

Y

is subjected to a stress

S

. The elastic potential energy per unit volume of the wire is given by

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

If $S$ is stress and

Y

is Young's modulus of material of a wire, the energy stored in the wire per unit volume is

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

The Young's modulus of steel is twice that of brass. Two wires of same length and of same area of cross-section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of:

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A wire of length

L

and area of cross section

A

is made of material of Young’s modulus

Y

. It is stretched by an amount

x

. The work done in stretching the wire is

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Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A load of mass

M kg

is suspended from a steel wire of length

2 m

and radius

1.0 mm

in Searle's apparatus experiment. The increase in length produced in the wire is

4.0 mm .

Now the load is fully immersed in a liquid of relative density

2 .

The relative density of the material of load is

8 .

The new value of increase in length of the steel wire is:

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

What is the elastic potential energy stored in a stretched steel wire of length

4 m .

The wire is stretched through

4 mm

and consists of cross-sectional area of

8 {mm}^{2} .

(Young's modulus of steel

= 2.0 \times 10^{11} N m^{- 2}

)

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Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

The bulk modulus of a spherical object is

B

. If it is subjected to uniform pressure

p

, the fractional decrease in radius is

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is

1 : 4,

the ratio of their diameters is:

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

Consider a metallic wire of length

10 m

. An external force applied results in an elongation of

5 mm

. What is the potential energy stored per unit volume
[Young's modulus of wire

Y = 16 \times 10^{10} N m^{- 2}

]

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

An external pressure

P

is applied on a cube at

0 ° C

so that it is equally compressed from all sides.

K

is the bulk modulus of the material of the cube and

α

is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

When a block of mass

M

is suspended by a long wire of length

L

, the length of the wire becomes

(L + l) .

The elastic potential energy stored in the extended wire is

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Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A metal rod of length

L

and cross-sectional area

A

is heated through

T ° C

. What is the force required to prevent the expansion of the rod lengthwise?

[Y

= Young's modulus of the material of rod,

α =

coefficient of linear expansion

]

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

An aluminium rod with Young’s modulus

Y = 7.0 \times 10^{10} N m^{- 2}

undergoes elastic strain of

0.04 %

. The energy per unit volume stored in the rod in

SI

unit

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A steel rail of length

5 m

and area of cross section

40 {cm}^{2}

is prevented from expanding along its length while the temperature rises by

10 ° C

. If coefficient of linear expansion and Young's modulus of steel are

1.2 \times 10^{- 5} K^{- 1}

and

2 \times 10^{11} N m^{- 2}

respectively, the force developed in the rail is approximately:

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area

a

floats on the surface of the liquid, covering entire cross-section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere

(\frac{d r}{r})

, is:

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

In materials like aluminium and copper, the correct order of magnitude of various elastic modulii is :

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A stone of mass

20 g

is projected from a rubber catapult of length

0.1 m

and area of cross section

10^{- 6} m^{2}

stretched by an amount

0.04 m .

The velocity of the projected stone is

m s^{- 1} .

(Young's modulus of rubber

= 0.5 \times 10^{9} N m^{- 2}

)

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

A boy’s catapult is made of rubber cord which is

42 c m

long, with

6 m m

diameter of cross-section and of negligible mass. The boy keeps a stone weighing

0.02 k g

on it and stretches the cord by

20 c m

by applying a constant force. When released, the stone flies off with a velocity of

20 m s^{- 1}

. Neglect the change in the area of cross-section of the cord while stretched. The Young’s modulus of rubber is closest to:

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Solids>Elastic Potential Energy in a Stretched Wire

One end of a slack wire (Young's modulus,

Y

, length,

L

and cross-sectional area,

A

) is clamped to a rigid wall and the other end to a block (mass

m

) which rests on a smooth horizontal plane. The block is set in motion with a speed,

v

. What is the maximum distance the block will travel after the wire becomes taut?

Show that strain energy per unit volume of a strained wire is 12×stress×strain.

Important Questions on Mechanical Properties of Solids

Learn and Prepare for any exam you want !

Show that strain energy per unit volume of a strained wire is $\frac{1}{2} \times s t r e s s \times s t r a i n$ .