Question

The dimension of planck's constant is the same as that of the
A. Linear impulse
B. Work
C. Linear momentum
D. Angular

Answer 1

Hint: In this we first, start with a simple formula that contains the planck's constant that is the energy of a wave \[E = h\gamma \] then we substitute the dimension of the E and $\gamma $ to get the dimension of the planck's constant as \[\left[ h \right] = \left[ {M{L^2}{T^{ - 1}}} \right]\]. Now we find the dimensions of the options that is the dimension of Linear impulse is \[\left[ {ML{T^{ - 1}}} \right]\], Angular Momentum is \[\left[ {M{L^2}{T^{ - 1}}} \right]\], energy is $\left[ {M{L^2}{T^{ - 2}}} \right]$, the pressure is \[\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]\] we can see that the dimensions of angular momentum match planck's constant.

Complete Step-by-Step solution:
We first start with the most basic and easy formula that contains planck's constant in its expression that is the formula for the energy of a wave which is proportional to the frequency or inversely proportional to the wavelength that is
\[E \propto \gamma \]
\[ \Rightarrow E = h\gamma \]
Where E is the energy of the wave having a unit of joule (J) and dimension $\left[ {M{L^2}{T^{ - 2}}} \right]$.
             $\gamma $ is the frequency having a unit of Hertz (Hz) and dimension $\left[ {{T^{ - 1}}} \right]$.
And H is the planck's constant
\[ \Rightarrow h = \dfrac{E}{\gamma }\]
To find the dimensions of planck's constant h
\[ \Rightarrow \left[ h \right] = \dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {{T^{ - 1}}} \right]}}\]
\[ \Rightarrow \left[ h \right] = \left[ {M{L^2}{T^{ - 1}}} \right]\]
Now we find the dimension of the option given in the question that is
The dimension of Linear impulse is \[\left[ {ML{T^{ - 1}}} \right]\].
The dimension of the Angular Momentum \[\left[ {M{L^2}{T^{ - 1}}} \right]\].
The dimension of energy is already mentioned above that is $\left[ {M{L^2}{T^{ - 2}}} \right]$.
The dimension of pressure is \[\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]\].
From these, we can see that the dimensions of angular momentum are matching with the dimensions of the planck's constant that is \[\left[ {M{L^2}{T^{ - 1}}} \right]\].

Note: For these types of questions, we need to remember some basic dimensions of values like energy, force, impulse, momentum, pressure. Then we need to be well versed with the basic equations of the above mentions. After that using the known dimensions we can find the dimensions of the unknown values of the dimensions of some famous constant like planck's constant.