Monday, October 3, 2016

Units and Dimensions Problems and Solutions One

Units are used to measure physical quantities. Dimensional formula represent the fundamental quantities and its powers in a given physical quantity. Any equation written correctly as per physics shall satisfy principle of homogeneity. According to this concept, dimensional formula of  a equation shall be having the same dimensions. This tells that we can add or subtract only similar physical quantities.

Problem One

If J is angular momentum and E is the kinetic energy of a body, we need to know a mathematical form represents what ?

Solution

To solve this problem, we need to write the dimensional formula of both the physical quantities. We know that angular momentum is defined as moment of momentum. By writing its formula, we can identify its dimensional formula. We also know that kinetic energy is another format of energy and it has dimensions similar to wok done.
Thus by writing them we can solve the problem as shown in the diagram below.

So it is nothing but moment of inertia.

Problem two

This problem deals with Rydberg constant, planks constant and velocity of the light. We get this in Bohr's atomic model and it has units that are reciprocal of length.

Planks constant is based on quantum concept and it has dimensions similar to energy per time.

Solution

By substituting the dimensional formula, we can solve the problem as shown below. The answer tells you that the product gives dimensions of energy as shown below.

Problem three

A complex mathematical equation is given that is made with certain physical quantities. We need to find the representation of the physical quantity that leads to ?

Solution

Here e is the charge of an electron. We know that current is defined as rate of flow of charge that is charge per time. Thus the dimensions of the charge are the product of current and time.

Permittivity is the ability of a medium or free space that is passing electric field through it. We can find its dimensions basing on Columb's inverse square law.

G is universal gravitational constant and we can measure it using inverse square law between the two masses.

Thus by substituting all of them we can get the answer as shown in the diagram below.

Problem Four

The equation given in this problem is called Vandarwalls Equation.
This is used to explain the behavior of real gases and the relation between gas parameters.

Solution

We can add or subtract physical quantities of same type. Thus what ever we have added to pressure shall have the dimensions of pressure. What ever we have subtracted with volume also shall have the same dimensions of volume. Taking this principle of homogeneity into consideration, we can solve the problem as shown below.

Problem five

This problem is also a mathematical equation and we need to find the dimensional formula of one physical quantity of it.

Solution

The left hand side of the equation is velocity that is rate of change of displacement and the right hand side shall be the same. We also know that the exponential functions cannot have any dimensional formula. Thus the power of exponential function shall have zero dimensions. Taking that into consideration we can find the dimensions of B and hence dimensions of A as shown in the diagram below.

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