IMO Practice Test — Mechanical Properties of Solids
12 Questions • 15 min • Olympiad level
15:00
Question 1 of 12
Two wires of the same material and length have radii in the ratio 1:2. Under the same load, the ratio of their extensions is:
$1:2$
$2:1$
$4:1$
$1:4$
Explanation: $\Delta L=\frac{FL}{AY}\propto\frac{1}{r^2}$; with radii $1:2$, extensions are $\frac{1}{1}:\frac{1}{4}=4:1$.
Question 2 of 12
A wire of length $L$ and area $A$ has Young's modulus $Y$. If it is cut into two equal halves, the Young's modulus of each half is:
$\frac{Y}{2}$
$Y$
$2Y$
$4Y$
Explanation: Young's modulus is a material property; it does not change when the wire is cut.
Question 3 of 12
A wire stretches by $\Delta L$ under a load. If both its length and radius are doubled with the same load, the new extension is:
$\frac{\Delta L}{2}$
$\Delta L$
$2\Delta L$
$\frac{\Delta L}{4}$
Explanation: $\Delta L\propto\frac{L}{r^2}$; new $=\frac{2L}{(2r)^2}=\frac{2L}{4r^2}=\frac{1}{2}\cdot\frac{L}{r^2}$, so it halves.
Question 4 of 12
The work done in stretching a wire by an amount $x$ (extension proportional to force) is proportional to:
$x$
$x^2$
$\sqrt{x}$
$\frac{1}{x}$
Explanation: $U=\frac{1}{2}kx^2$ since the restoring force is proportional to $x$, so $U\propto x^2$.
Question 5 of 12
If the stress on a wire is doubled (still elastic), the elastic energy stored per unit volume becomes:
double
half
four times
unchanged
Explanation: $u=\frac{1}{2}\frac{(\text{stress})^2}{Y}\propto(\text{stress})^2$, so doubling stress quadruples $u$.
Question 6 of 12
A material has bulk modulus $B$. The fractional change in volume produced by a pressure $P$ is:
$\frac{P}{B}$
$\frac{B}{P}$
$PB$
$\frac{1}{PB}$
Explanation: $B=\frac{P}{\Delta V/V}$, so $\frac{\Delta V}{V}=\frac{P}{B}$.
Question 7 of 12
For a perfectly incompressible material, the bulk modulus and Poisson's ratio are respectively:
zero and $0$
infinite and $0.5$
infinite and $0$
zero and $0.5$
Explanation: Incompressible means $\Delta V=0$, so $B\to\infty$ and Poisson's ratio takes its limiting value $0.5$.
Question 8 of 12
Two rods of the same material and length, with cross-sectional areas $A$ and $2A$, support the same load. The ratio of elastic energy stored (rod 1 : rod 2) is:
$1:2$
$2:1$
$1:1$
$1:4$
Explanation: $U=\frac{F^2 L}{2AY}\propto\frac{1}{A}$; areas $A:2A$ give energies $\frac{1}{A}:\frac{1}{2A}=2:1$.
Question 9 of 12
The depression of a beam loaded at its centre varies with its length $L$ as:
$L$
$L^2$
$L^3$
$\frac{1}{L^3}$
Explanation: $\delta=\frac{WL^3}{4bd^3Y}\propto L^3$.
Question 10 of 12
A load is suspended from a steel and a copper wire of equal length and area in series. The stress in each wire is:
greater in steel
greater in copper
the same in both
zero in copper
Explanation: The same force acts on the same area in each wire, so the stress is identical; only the strains differ.
Question 11 of 12
If a wire obeying Hooke's law is stretched to twice its natural length (hypothetically elastic throughout), the energy stored compared with a stretch to 1.5 times its natural length increases by a factor of about:
$2$
$4$
$\frac{(1.0)^2}{(0.5)^2}$
$1.5$
Explanation: Energy $\propto(\text{extension})^2$; extensions are $L$ and $0.5L$, so the ratio is $\frac{(1.0)^2}{(0.5)^2}=4$.
Question 12 of 12
A spherical ball of bulk modulus $B$ is taken to a depth where the pressure is $P$. The fractional decrease in its radius is approximately:
$\frac{P}{B}$
$\frac{P}{3B}$
$\frac{3P}{B}$
$\frac{P}{2B}$
Explanation: $\frac{\Delta V}{V}=\frac{P}{B}$ and $\frac{\Delta V}{V}=3\frac{\Delta r}{r}$, so $\frac{\Delta r}{r}=\frac{P}{3B}$.