Scientific Study

Controlled variables are kept constant during an experiment so that only the independent variable affects the dependent variable. This makes the result fair, accurate and reliable.

The dependent variable is the solubility of the salt (the amount of salt that dissolves). It changes as the temperature (the independent variable) is changed.

(d) Newton. Newton (the unit of force) is a derived unit because it is obtained from fundamental units: $1\,\text{N} = 1\,\text{kg m s}^{-2}$. Kilogram, second and kelvin are fundamental (base) units.

Given equation: $v^2 = u^2 + 2as$

Here $v$, $u$ are velocities, $a$ is acceleration, $s$ is displacement.

Unit (dimension) of each term:

• $v^2 = (\text{m s}^{-1})^2 = \text{m}^2\text{s}^{-2}$
• $u^2 = (\text{m s}^{-1})^2 = \text{m}^2\text{s}^{-2}$
• $2as = (\text{m s}^{-2})(\text{m}) = \text{m}^2\text{s}^{-2}$ (the number 2 has no unit)

All terms have the same unit $\text{m}^2\text{s}^{-2}$, so the equation is valid (homogeneous).

A controlled variable is a variable that is kept constant (unchanged) throughout an experiment so that it does not affect the result. Example: in a plant growth test with different fertilizer amounts, the type of soil and amount of water are controlled.

• Independent variable: the amount (availability) of sunlight given to the plant.
• Controlled variable: any one of the amount of water, type of soil, type/species of plant, or temperature, kept the same for all plants.

(The dependent variable is the growth of the plant.)

(d) Asiatic Pennywort (Ghodtapre/Mandukparni). It is used traditionally to improve memory and intelligence.

Yarsagumba (Cordyceps sinensis). Other examples: Panchaunle (Dactylorhiza hatagirea), Kutki.

(i) Force applied to stretch a rubber band

• Independent variable: applied force (load).
• Dependent variable: extension (increase in length) of the rubber band.
• Reason: When the applied force is changed, the extension of the rubber band changes. So force is the manipulated cause and extension is the measured effect. Other factors such as initial length and temperature are kept constant.

(ii) Distance travelled by a stone thrown

• Independent variable: initial velocity (speed) of projection and angle of projection (launch angle). If only one is being studied, the other is kept constant.
• Dependent variable: horizontal distance (range) travelled by the stone.
• Reason: The distance travelled depends on how fast and at what angle the stone is thrown. Changing the initial speed or launch angle changes the distance; hence these are independent variables and distance is the dependent variable. Other factors such as mass of the stone and air resistance are kept constant.

(i) More growth of plant in sunlight

• Independent variable: sunlight (its amount).
• Dependent variable: growth of the plant.

(ii) Stretch of a catapult and distance covered by the stone

• Independent variable: the stretch of the catapult.
• Dependent variable: the distance covered by the stone.

A fundamental unit is a unit that is independent of other units and is not derived from any other unit. Example: metre (unit of length), kilogram, second.

A fundamental unit is independent and cannot be expressed in terms of other units (e.g. kg), whereas a derived unit is obtained by combining fundamental units (e.g. newton $= \text{kg m s}^{-2}$).

• Independent variable: light (lighted room or dark room, i.e. the presence of light).
• Dependent variable: the growth rate of the plants.

The dependent variable is the amount of solute dissolved (the concentration of the solution formed), which depends on factors like the amount of solvent or temperature.

Given equation: $v^2 = u^2 + 2as^2$

• $v^2 = \text{m}^2\text{s}^{-2}$
• $u^2 = \text{m}^2\text{s}^{-2}$
• $2as^2 = (\text{m s}^{-2})(\text{m}^2) = \text{m}^3\text{s}^{-2}$

The term $2as^2$ has the unit $\text{m}^3\text{s}^{-2}$, which is different from $\text{m}^2\text{s}^{-2}$ of the other terms. The units are not the same on both sides, so the equation is incorrect. (The correct equation is $v^2 = u^2 + 2as$.)

Variables help us to find the cause-and-effect relationship between factors in an experiment, so that the study is systematic, fair, and gives reliable results.

The other two types are independent and dependent variables.

• Independent variable: e.g. the amount of fertilizer applied to plants (changed by the experimenter).
• Dependent variable: e.g. the height of the plants (it changes as the fertilizer amount changes).

(b) Amount of fertilizer. It is the variable deliberately changed by the experimenter; the height of the plant is the dependent variable, while water and soil type are kept constant.

Given equation: $s = ut + \frac{1}{2}a^2t$

• $s = \text{m}$
• $ut = (\text{m s}^{-1})(\text{s}) = \text{m}$
• $\frac{1}{2}a^2t = (\text{m s}^{-2})^2(\text{s}) = \text{m}^2\text{s}^{-4}\cdot\text{s} = \text{m}^2\text{s}^{-3}$

The last term has unit $\text{m}^2\text{s}^{-3}$, not m. The terms have different units, so the equation is not valid. (The correct form is $s = ut + \frac{1}{2}at^2$.)

(c) controlled variable. A variable that the researcher keeps constant during the study so that it does not influence the result is called a controlled variable.

• Controlled variable: any one of age, sex, body weight, diet, or health condition of the persons studied (kept same for all).
• Dependent variable: blood pressure (it changes with the amount of alcohol consumed, the independent variable).

An independent variable can be changed deliberately by the experimenter, so its effect on the dependent variable can be studied and cause-and-effect relations can be found.

A dependent variable is the variable whose value depends on, and changes with, the independent variable; it is what we measure in an experiment.

Example: when the temperature is changed, the solubility of a salt changes. Solubility is the dependent variable.

The density of water is the dependent variable, because its value changes with temperature (the independent variable).

To show: $P = IV$ is homogeneous.

Fundamental units: power $P$ = work/time $= \dfrac{\text{kg m}^2\text{s}^{-2}}{\text{s}} = \text{kg m}^2\text{s}^{-3}$.

Voltage $V = \dfrac{\text{work}}{\text{charge}} = \dfrac{\text{kg m}^2\text{s}^{-2}}{\text{A s}} = \text{kg m}^2\text{s}^{-3}\text{A}^{-1}$.

Current $I = \text{A}$.

Right side: $IV = \text{A} \times \text{kg m}^2\text{s}^{-3}\text{A}^{-1} = \text{kg m}^2\text{s}^{-3}$.

Left side $= \text{kg m}^2\text{s}^{-3}$ = right side. The units on both sides are equal, so the equation $P = IV$ is homogeneous.