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Units & measurements test

  1. Write the dimensional formula of the following:

    1. Work
    2. Angular Velocity
    3. Frequency
    4. Gravitational Constant
    5. Planks Constant

  2. Find whether the following formula is dimensionally consistent:

    \[T=\frac{k}{R}\sqrt{\frac{r^3}{g}}\]

    where, \(T\) = time period, \(k\) = constant, \(R\), \(r\) = radii and \(g\) = acceleration due to gravity.

  3. If the following formula is dimensionally consistent, then find the dimensions of \(a\) and \(b\):

    \[ E = a g + \frac{1}{2} bv^2 \]

    where, \(g\) = acceleration due to gravity, \(E\) = Total energy, \(v\) = velocity.

  4. If the acceleration due to gravity of a planet depends on the gravitational constant \(G\), mass of planet \(M\) and radius of planet \(R\), then derive an equation of acceleration due to gravity \(g\).

  5. The value of plank's constant in SI units is \(h=6.626\times 10^{-34} \, Js\) [Joule second]. Find the value of planks constant in a new system of units which has gram [\(g\)], nanometer [\(nm\)] and half minute [\(30 s\)] as the base units of mass, length and time.

    \(1nm = 10^{-9} m\)