A thin uniform rod of length $$l$$ and mass $$m$$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $$\omega $$. Its center of mass rises to a maximum height of:

The height at which the acceleration due to gravity becomes $${g \over 9}$$ (where $$g=$$ the acceleration due to gravity on the surface of the earth) in terms of $$R,$$ the radius of the earth, is:

Two wires are made of the same material and have the same volume. However wire $$1$$ has cross-sectional area $$A$$ and wire $$2$$ has cross-sectional area $$3A.$$ If the length of wire $$1$$ increases by $$\Delta x$$ on applying force $$F,$$ how much force is needed to stretch wire $$2$$ by the same amount?

Statement - 1: The temperature dependence of resistance is usually given as $$R = {R_0}\left( {1 + \alpha \,\Delta t} \right).$$ The resistance of wire changes from $$100\Omega $$ to $$150\Omega $$ when its temperature is increased from $${27^ \circ }C$$ to $${227^ \circ }C$$. This implies that $$\alpha = 2.5 \times {10^{ - 3}}/C.$$

Statement - 2: $$R = {R_0}\left( {1 + \alpha \,\Delta t} \right)$$ is valid only when the change in the temperature $$\Delta T$$ is small and $$\Delta T = \left( {R - {R_0}} \right) < < {R_0}.$$