Hooke’s Law - Practice Questions & MCQ

Hooke’s law states that if the deformation is small, the stress in a body is proportional to the corresponding strain, i.e.,

$
\begin{gathered}
\text { Stress } \propto \text { Strain } \\
\Rightarrow \text { Stress }=E(\text { Strain }) \\
\Rightarrow E=\frac{\text { Stress }}{\text { Strain }}
\end{gathered}
$

Where E is called as Modulus of elasticity and it depends on the nature of the material and temperature of the body and is independent of the dimensions of the body.

Unit of Modulus of elasticity $=\mathrm{N} / \mathrm{m}^2$
- Modulus of elasticity is of three types:-
a. Young's Modulus(Y):- It is defined as the ratio of longitudinal stress to longitudinal strain.

$
Y=\frac{\text { longitudinal stress }}{\text { longitudinal strain }}=\frac{F / A}{\Delta L / L}=\frac{F l}{A \Delta L}
$

b. Shear Modulus or Modulus of rigidity(G):- It is defined as the ratio of shearing stress to the shearing strain.

$
G=\frac{\text { shearing stress }}{\text { shearing strain }}=\frac{F / A}{x / L}=\frac{F l}{A x}=\frac{F}{A \phi}
$

c. Bulk Modulus(B):- It is defined as the ratio of volume stress to the volume strain.

$
\begin{gathered}
\text { Volumestress }=\frac{F}{A}=\text { Pressure } \\
B=-\frac{P}{\Delta V / V}
\end{gathered}
$
 

where $\mathrm{P}=$ increase in pressure, $\mathrm{V}=$ original volume, $\Delta V=$ change in volume
The negative sign indicates that with the increase in pressure, volume decreases by $\Delta V$.
- Compressibility(C):- The reciprocal of bulk modulus is called as compressibility.

$
C=\frac{1}{B}=-\frac{\Delta V / V}{P}
$
 

• Unit of compressibility:- N^-1m²
• Poisson's ratio:- The ratio of lateral strain to longitudinal strain is called Poisson’s ratio ().

1. Lateral strain : The ratio of change in radius or diameter to the original radius or diameter is called lateral strain.
2. Longitudinal strain : The ratio of change in length to the original length is called longitudinal strain.

$\sigma=-\frac{\Delta d / d}{\Delta l / l}$

where negative sign is due to decrease in the transverse length