## Thursday, October 6, 2016

### Units and Dimensions Problems and Solutions Three

Let us solve a problem where we are going to convert a physical quantity from from units of measurement to other. The physical quantity that we are trying to convert is is acceleration to gravity. It is nothing but acceleration and is defined as the rate of change of velocity.

Problem one

Solution

We need to solve this problem basing on principle of homogeneity. Writing the dimensional analysis  on both the sides and then substituting the appropriate units on both the sides of the equation, we can solve the problem as shown in the diagram below.

Problem Two

The second problem deals with expressing energy not in normal terms of mass, length and time as fundamental quantities but representing it in terms of force, acceleration and time. The problem is as shown in the diagram below.

Solution

We need to express in terms of the above mentioned physical quantities. Time is a common term in both the formats. We need to express length and mass in terms of force and acceleration so that the problem can be solved as shown below.

Problem Three

In this problem we need to solve the dimension of a unknown physical quantity used in the given equation. The left hand side of the equation is the number of particles crossing per unit area and per unit time. Thus shall have dimensions of per area and per time as number won't have any dimensions.

Solution

The other side of the equation has unknown physical quantity D and we need to find it. Numbers are the number of particles crossing the given space per unit volume on the right hand side of the equation and X is displacement itself.

By equating the dimensions on both the sides of the equation, we can solve the problem as shown in the diagram below.

Problem Four

The problem is regarding measuring the mean deviation of the given parameters. Length of the cylinder is measured in multiple times and each time different reading is obtained as shown in the diagram below.

Solution

To measure the mean deviation, first we need to measure the average of the given values. Average is simply sum of the readings by the number of the readings.

We can now measure how much each reading is different from the other readings. That is called deviation and we need to measure the deviation of each reading from the average. We can then measure the average of these deviations and it is called mean deviation. Thus we are able to certify that the given value is the sum of mean value and sum or difference of the mean deviation. Thus though we are not able to say exactly what is the value is, we can at least say that the reading vary between the given range.

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