Units and Measu...

In the relation, $$\displaystyle P=\frac{\alpha }{\beta }e^{{\alpha z}/{k\theta }}$$ $$P$$ is pressure, $$Z$$ is distance, $$K$$ is Boltzmann constant and $$\displaystyle \theta $$ is the temperature. The dimensions of $$\displaystyle \beta $$ will be 

If $$E, M, J$$ and $$G$$ respectively denote energy, mass, angular momentum and universal gravitational constant, the quantity, which has the same dimensions as the dimensions of $$\frac{EJ^2}{M^5G^2}$$ is:

Match the following physical quantities with their respective dimensional formula:

Out of the following pairs, choose the pair in which the physical quantities do not have identical dimension?

Amol has three objects, X, Y and Z. He puts X into a measuring cylinder with $$30$$ mL of water. Then he puts Y and Z one by one into the same measuring cylinder as shown.
Which of the objects has/have a volume less than $$30$$ mL?

Assertion: The dimensional formula for relative velocity is same as that of the change in velocity.
Reason: Relative velocity of P w.r.t. Q is the ratio of velocity of P and that of Q.

The Van der Waal's equation of $$'n'$$ moles of a real gas is
$$\displaystyle \left( P+\frac { a }{ { V }^{ 2 } }  \right) \left( V-b \right) =nRT$$
Where $$P$$ is pressure, $$V$$ is volume, $$T$$ is absolute temperature, $$R$$ is molar gas constant and $$a, b, c$$ are Van der Waal constants. The dimensional formula for $$ab$$ is:

A beam balance of unequal arm length is used by an unscruplus trader. When an object is weighed on left pan. The weight $$i$$ found to be $${W}_{1}$$. When the object is weighed on the right plan, the weight is found to be $${W}_{2}$$. If $${W}_{1}\neq {W}_{2}$$, the correct weight of the object is

The dimension of $$\left( \frac { 1 }{ 2 }  \right) { \varepsilon  }_{ 0 }{ E }^{ 2 }$$ is ($${ \varepsilon  }_{ 0 }$$: permittivity of free space, Electric field)