– Relation between tensile (compreasive) strain $t$ and strese o:
\[
t=a / E \text {. }
\]
where A is Young’s modulus.
– Relation between lateral compresive (tensile) strsin $e^{\prime}$ and longitudinal iensile (compresive) strain e:
\[
\boldsymbol{\epsilon}^{\prime}=-\boldsymbol{\mu} \text {, }
\]
where $\mu$ is Poisson’s ratie.
– Relation between shear strain $\gamma$ and tancential stress $\mathrm{z}:$
\[
\mathbf{\gamma}=\tau / G \text {, }
\]
where $G$ is shear modulus.
– Compresibility:
\[
B=-\frac{1}{T} \frac{d}{d p} .
\]
– Volume density of elastic strain energy:
\[
=-\boldsymbol{E} e^{\mathrm{N} / 2}, \|-\sigma \boldsymbol{\gamma}^{/ / 2}
\]
1.290. What pressure has to be applied to the ends of a steel cylinder to keep its length constant on raising its temperature by $100^{\circ} \mathrm{C}$ ?
1.291. What internal pressure (in the absence of an external pressure) can be sustained
(a) by a glass tube; (b) by a glass spherical nask, if in both cases the wall thickness is equal to $\Delta r=1.0 \mathrm{~mm}$ and the radius of the tube and the flask equals $r=25 \mathrm{~mm}$ ?
1.292. A horizontally oriented copper rod of length $t=1.0 \mathrm{~m}$ is rotated about a vertical axis passing through its middle. What is the number of $\mathrm{rps}$ at which this rod ruptures?
1.293. A ring of radius $r=25 \mathrm{~cm}$ made of lead wire is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring. What is the number of rps at which the ring ruptures?
1.294. A steel wire of diameter $d=1.0 \mathrm{~mm}$ is stretched horizontally between two clamps located at the distance $l=2.0 \mathrm{~m}$ from each other. A weight of mass $m=0.25 \mathrm{~kg}$ is suspended from the midpoint $O$ of the wire. What will the resulting descent of the point $O$ be in centimetres?
1.295. A uniform elastic plank moves over a smooth horizontal plane due to a constant force $F$, distributed uniformly over the end hace. The surface of the end face is equal to $S$, and Young’s modulus of the material to $\boldsymbol{E}$. Find the compressive strain of the plank in the direction of the acting force.
1.296. A thin uniform copper rod of length $l$ and mass $m$ rotates uniformly with an angular velocity $\omega$ in a horizontal plane about a vertical axis passing through one of its ends. Determine the tension in the rod as a function of the distance $r$ from the rotation axis. Find the elongation of the rod.
1.297. A solid copper eylinder of length $t=65 \mathrm{~cm}$ is placed on a horizontal surface and subjected to a vertical compressive force $F=1000 \mathrm{~N}$ directed downward and distributed uniformly over the end face. What will be the resulting change of the volume of the cylinder in cuhic millimetres?
1.298. A copper rod of length $t$ is suspended from the ceiling by one of its ends. Find:
(a) the elongation $\Delta l$ of the rod due to its own weight;
(b) the relative increment of its volume $\Delta V / V$.
1.299. A bar made of material whose Young’s modulus is equal to $E$ and Poisson’s ratio to $\mu$ is subjected to the hydrostatic pressure p. Find:
(a) the fractional decrement of its volume:
(b) the relationship between the compresibility $\beta$ and the elastic constants $E$ and $\mu$.
Show that Poisson’s ratio $\mu$ cannot exceed $1 / 2$.
1.300. One end of a steel rectangular girder is embedded into a wall (Fig. 1.74). Due to gravity it sags slightly. Find the radius of curvature of the neutral layer (see the dotied line in the figure) in the vicinity of the point $O$ if the length of the protruding section of
Fig. 1.74.
the girder is equal to $l=6.0 \mathrm{~m}$ and the thickness of the girder equals $h=10 \mathrm{~cm}$.
1.301. The bending of an elastic rod is described by the elastic curve passing through centres of gravity of rod’s cross-sections. At small bendings the equation of this curve takes the form
\[
N(x)=E I \frac{d^{2} y}{d x^{2}},
\]
where $N(x)$ is the bending moment of the elastic forces in the crosssection corresponding to the $x$ coordinate, $E$ is Young’s modulus, $I$ is the moment of inertia of the cross-section relative to the axis passing through the neutral layer ( $I=\int z^{2} d S$, Fig. 1.75).

Suppose one end of a steel rod of a square cross-section with side $a$ is embedded into a wall, the protruding section being of length $l$
Fig. 1.75.
Fig. 1.76.
(Fig. 1.76). Assuming the mass of the rod to be negligible, find the shape of the elastic curve and the deflection of the rod $\lambda$, if its end $A$ experiences
(a) the bending moment of the couple $N_{0}$;
(b) a force $F$ oriented along the $y$ axis.
1.302. A steel girder of length $l$ rests freely on two supports (Fig. 1.77). The moment of inertia of its cross-section is equal to $I$ (see the foregoing problem). Neglecting the mass of the girder and assuming the sagging to be slight, find the deflection $\lambda$ due to the force $F$ applied to the middle of the girder.
1.303. The thickness of a rectangular steel girder equals $h$. Using the equation of Problem 1.301, find the deflection $\lambda$ caused by the weight of the girder in two cases:
(a) one end of the girder is embedded into a wall with the length of the protruding section being equal to $l$ (Fig. 1.78a);
(b) the girder of length $2 l$ rests freely on two supports (Fig. 1.78b).
1.304. A steel plate of thickness $h$ has the shape of a square whose side equals $l$, with $h \ll l$. The plate is rigidly fixed to a vertical axle
Fig. 1.77.
$O O$ which is rotated with a constant angular acceleration $\beta$ (Fig. 1.79). Find the deflection $\lambda$, assuming the sagging to be small.
1.305. Determine the relationship between the torque $N$ and the torsion angle $\varphi$ for
(a) the tube whose wall thickness $\Delta r$ is considerably less than the tube radius;
(b) for the solid rod of circular cross-section. Their length $l$, radius $r$, and shear modulus $G$ are supposed to be known.
Fig. 1.78.
Fig. 1.79.
1.306. Calculate the torque $N$ twisting a steel tube of length $l=$ $=3.0 \mathrm{~m}$ through an angle $\varphi=2.0^{\circ}$ about its axis, if the inside and outside diameters of the tube are equal to $d_{1}=30 \mathrm{~mm}$ and $d_{2}=$ $=50 \mathrm{~mm}$.
1.307. Find the maximum power which can be transmitted by means of a steel shaft rotating about its axis with an angular velocity $\omega=120 \mathrm{rad} / \mathrm{s}$, if its length $l=200 \mathrm{~cm}$, radius $r=1.50 \mathrm{~cm}$, and the permissible torsion angle $\varphi=2.5^{\circ}$.
1.308. A uniform ring of mass $m$, with the outside radius $r_{2}$, is fitted tightly on a shaft of radius $r_{1}$. The shaft is rotated about its axis with a constant angular acceleration $\beta$. Find the moment of elastic forces in the ring as a function of the distance $r$ from the rotation axis.
1.309. Find the elastic deformation energy of a steel rod of mass $m=3.1 \mathrm{~kg}$ stretched to a tensile strain $\varepsilon=1.0 \cdot 10^{-3}$.
1.310. A steel cylindrical rod of length $l$ and radius $r$ is suspended by its end from the ceiling.
(a) Find the elastic deformation energy $U$ of the rod.
(b) Define $U$ in terms of tensile strain $\Delta l / l$ of the rod.
1.311. What work has to be performed to make a hoop out of a steel band of length $l=2.0 \mathrm{~m}$, width $h=6.0 \mathrm{~cm}$, and thickness $8=2.0 \mathrm{~mm}$ ? The process is assumed to proceed within the elasticity range of the material.
1.312. Find the elastic deformation energy of a steel rod whose ene end is fixed and the other is twisted through an angle $5=6.0^{\circ}$. The length of the rod is equal to $l=1.0 \mathrm{~m}$, and the radius to $r=$ $=10 \mathrm{~mm}$.
1.313. Find how the volume density of the elastic deformation energy is distributed in a steel rod depending on the distance $r$ from its axis. The length of the rod is equal to $l$, the torsion angle to $\%$.
1.314. Find the volume density of the elastie deformation energy in fresh water at the depth of $h=1000 \mathrm{~m}$.