Two rods $A$ and $B$ of identical dimensions are at temperature $30\,^{\circ}C$. If $A$ is heated up to $180\,^{\circ}C$ and $B$ up to $T\,^{\circ}C$,then the new lengths are the same. If the ratio of the coefficients of linear expansion of $A$ and $B$ is $4:3$,then the value of $T$ is........$^{\circ}C$.

$A$ pendulum clock loses $12\;s$ a day if the temperature is $40^{\circ}C$ and gains $4\;s$ a day if the temperature is $20^{\circ}C$. The temperature at which the clock will show correct time,and the coefficient of linear expansion $(\alpha)$ of the metal of the pendulum shaft are respectively:

An aluminium and steel rod having same lengths and cross-sections are joined to make a total length of $120 \ cm$ at $30^{\circ} C$. The coefficients of linear expansion of aluminium and steel are $24 \times 10^{-6} /{ }^{\circ} C$ and $1.2 \times 10^{-5} /{ }^{\circ} C$, respectively. The length of this composite rod when its temperature is raised to $100^{\circ} C$, is . . . . . . $cm$. (in $cm$)

$A$ rod of length $20 \, cm$ is made of metal $A$. It expands by $0.075 \, cm$ when its temperature is raised from $0^{\circ}C$ to $100^{\circ}C$. Another rod of a different metal $B$ having the same length expands by $0.045 \, cm$ for the same change in temperature. $A$ third rod of the same length is composed of two parts,one of metal $A$ and the other of metal $B$. This rod expands by $0.06 \, cm$ for the same change in temperature. The portion made of metal $A$ has the length ............. $cm$.

When the temperature of a body increases from $T$ to $T+\Delta T$, its moment of inertia increases from $I$ to $I+\Delta I$. If $\alpha$ is the coefficient of linear expansion of the material of the body, then $\frac{\Delta I}{I}$ is (neglect higher orders of $\alpha$ ).

Three uniform thin aluminium rods each of length $2 \,m$ form an equilateral triangle $PQR$ as shown in the figure. The midpoint of the rod $PQ$ is at the origin of the coordinate system. If the temperature of the system of rods increases by $50^{\circ} C$, the increase in $y$-coordinate of the centre of mass of the system of the rods is ............ $mm$. (Coefficient of volume expansion of aluminium $= 12 \sqrt{3} \times 10^{-6} \,K^{-1}$)