Unit 1: Scientific Study
Important Points
Science: The organized knowledge learned through observation, study, experimentation, and reasoning is called Science.
Scientific Study: The systematic process of collecting information, analyzing it, and deriving conclusions about a topic or phenomenon is called Scientific Study.
Variables in Scientific Research: Any quantity, factor, or condition that can be controlled, changed, or measured in an experiment is called a variable.
Scientific Experiment
Types of Variables
Variables are classified into three types:
- Independent Variable
- Dependent Variable
- Controlled Variable
Independent Variable: A variable that can be changed by the researcher throughout the experiment is called an Independent Variable.
Example: In the effect of light on plant growth, sunlight is the independent variable.
Dependent Variable: A variable that cannot be changed by the researcher throughout the experiment is called a Dependent Variable.
Example: In an experiment on the effect of sunlight on plant growth, the growth of the plant is the dependent variable.
Controlled Variable: A variable that is kept constant throughout the experiment by the researcher is called a Controlled Variable.
Example: Initial conditions for plant growth such as water, fertilizer, and air supply.
Physical Quantity & Units
Physical Quantity: Quantities that can be measured and expressed mathematically are called Physical Quantities.
Examples: Mass, length, time, volume, force, etc.
Unit: A specific standard basis used to measure physical quantities is called a Unit.
Examples: meter (m), kilogram (kg), second (s), kelvin (K), etc.
✨ Remember: Units help us compare measurements accurately!
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Units & Unit Analysis
Types of Units
Fundamental Units: The independent units that cannot be derived from other units are called Fundamental Units.
Examples: Meter, kilogram, second.
Derived Units: The units formed by combining two or more fundamental units are called Derived Units.
Examples: Newton (kg·m/s²), Joule (kg·m²/s²), etc.
Fundamental Units in SI System
| S.N. | Fundamental Quantity | SI Unit | Symbol |
|---|---|---|---|
| 1 | Length | meter | m |
| 2 | Mass | kilogram | kg |
| 3 | Time | second | s |
| 4 | Temperature | kelvin | K |
| 5 | Electric Current | ampere | A |
| 6 | Luminous Intensity | candela | cd |
| 7 | Amount of Substance | mole | mol |
m
kg
s
SI Units
Examples of Derived Units
| S.N. | Physical Quantity | Formula | SI Unit | Symbol |
|---|---|---|---|---|
| 1 | Area | A = l × b | square meter | m² |
| 2 | Volume | V = l × b × h | cubic meter | m³ |
| 3 | Density | d = m/V | kg per cubic meter | kg/m³ |
| 4 | Velocity | v = s/t | meter per second | m/s |
| 5 | Acceleration | a = (v-u)/t | m/s² | m/s² |
| 6 | Force | F = m × a | Newton | N (kg·m/s²) |
| 7 | Work/Energy | W = F × d | Joule | J (kg·m²/s²) |
| 8 | Power | P = W/t | Watt | W (kg·m²/s³) |
| 9 | Pressure | P = F/A | Pascal | Pa (kg/m·s²) |
| 10 | Frequency | f = v/λ | Hertz | Hz (s⁻¹) |
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Unit Analysis & Questions
Unit Analysis of Equations
Unit Analysis: The process of verifying the consistency and validity of physical equations through the analysis of units is called Unit Analysis.
Rule: Both sides of the equation must have consistent fundamental units for it to be valid.
v² = u² + 2as
Verification:
LHS: Unit of v² = (m/s)² = m²/s² → (i)
RHS: Unit of u² = (m/s)² = m²/s²
Unit of 2as = (m/s²) × m = m²/s²
∴ RHS = m²/s² + m²/s² = m²/s² → (ii)
From (i) and (ii), units of LHS = units of RHS.
Hence, the equation is valid.
LHS: Unit of v² = (m/s)² = m²/s² → (i)
RHS: Unit of u² = (m/s)² = m²/s²
Unit of 2as = (m/s²) × m = m²/s²
∴ RHS = m²/s² + m²/s² = m²/s² → (ii)
From (i) and (ii), units of LHS = units of RHS.
Hence, the equation is valid.
Very Short Answer Questions
1. Prove that the unit of gravitational constant is a derived unit.
We know: G = F·d²/(m₁m₂)
Unit of F = N = kg·m/s²
Unit of d² = m²
Unit of m₁m₂ = kg²
Thus, Unit of G = (kg·m/s²)·m²/kg² = m³/(kg·s²) = m³·kg⁻¹·s⁻²
The unit depends on the fundamental units m, kg, s.
Hence, it is a derived unit.
Unit of F = N = kg·m/s²
Unit of d² = m²
Unit of m₁m₂ = kg²
Thus, Unit of G = (kg·m/s²)·m²/kg² = m³/(kg·s²) = m³·kg⁻¹·s⁻²
The unit depends on the fundamental units m, kg, s.
Hence, it is a derived unit.
2. A student has to test the change in effort applied and the number of pulleys used. Identify a controlled variable and a dependent variable from: number of pulleys, load of object, effort applied, friction produced in the pulleys, and type of pulley.
Controlled Variables: Friction produced by the pulley, type of pulley, load of the object.
Dependent Variable: Effort applied.
Dependent Variable: Effort applied.
3. Write the fundamental units involved in the unit of pressure.
We know: P = F/A = N/m²
In fundamental units: Pa = (kg·m/s²)/m² = kg·m⁻¹·s⁻²
Fundamental units: kg, m, s
In fundamental units: Pa = (kg·m/s²)/m² = kg·m⁻¹·s⁻²
Fundamental units: kg, m, s
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4. Rita wanted to test the eating habits of her dog. She decided to study how the amount of food and the time of giving food affects the speed at which the dog eats. What is wrong with the design of Rita’s experiment?
Mistake: Rita used two independent variables (amount of food and feeding time) in one experiment.
Correction: Conduct two separate experiments:
• Study the relationship between the amount of food and eating speed.
• Study the relationship between feeding time and eating speed.
Keep other factors (food type, bowl, environment) constant.
Correction: Conduct two separate experiments:
• Study the relationship between the amount of food and eating speed.
• Study the relationship between feeding time and eating speed.
Keep other factors (food type, bowl, environment) constant.
5. Perform unit analysis of the equation: s = ut + ½at²
LHS: Unit of s = m → (i)
RHS: Unit of ut = (m/s) × s = m
Unit of ½at² = (m/s²) × s² = m
∴ RHS = m + m = m → (ii)
From (i) and (ii), LHS unit = RHS unit.
The equation is valid.
RHS: Unit of ut = (m/s) × s = m
Unit of ½at² = (m/s²) × s² = m
∴ RHS = m + m = m → (ii)
From (i) and (ii), LHS unit = RHS unit.
The equation is valid.
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Reasons, Differences & Long Answers
Give Reasons
1. The unit of mass is called a fundamental unit.
The unit of mass is kg, which does not depend on other units.
2. The unit of volume is a derived unit.
The unit of volume is m³, which is obtained from the product of three fundamental units of length. So, it is a derived unit.
3. Why is it necessary to control variables during an experiment?
While studying the effect of one variable, uncontrolled variables can make the results unreliable.
4. The equation v² = ut is not valid.
LHS: Unit of v² = (m/s)² = m²/s²
RHS: Unit of ut = (m/s) × s = m
LHS unit ≠ RHS unit.
Hence, the equation is not homogeneous and is invalid.
RHS: Unit of ut = (m/s) × s = m
LHS unit ≠ RHS unit.
Hence, the equation is not homogeneous and is invalid.
5. The independent variable is called the right variable, and the dependent variable is called the left variable.
While expressing the relation between two variables in an equation, the dependent variable is written on the left side, and the independent variable is written on the right side.
6. The dependent variable is called the vertical or y-variable, and the independent variable is called the horizontal or x-variable.
While plotting a graph, the dependent variable is always plotted on the Y-axis and the independent variable on the X-axis.
X-axis (Independent)
Y-axis (Dependent)
Differences Between
1. Fundamental Units and Derived Units
| Fundamental Units | Derived Units |
|---|---|
| Independent of other units. | Formed by combining fundamental units. |
| Fixed in number (7 in SI). | Large in number. |
| Used to find derived units. | Not generally used to find other units. |
| Examples: m, kg, s | Examples: N, J, Pa |
2. Independent Variable and Controlled Variable
| Independent Variable | Controlled Variable |
|---|---|
| Can be changed by the researcher. | Kept constant by the researcher. |
| It is the cause of the experiment. | It is kept the same. |
3. Dependent Variable and Controlled Variable
| Dependent Variable | Controlled Variable |
|---|---|
| Cannot be changed by the researcher. | Kept constant by the researcher. |
| It is observed and measured. | It is kept the same. |
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Long Answer Questions
1. Write the advantages and limitations of unit analysis of equations.
Advantages:
- To prove the accuracy of physical equations.
- To relate different physical quantities.
- To find the dimension of a new physical quantity.
- To convert the unit of one system into another.
- Constants in physical equations cannot be found.
- An equation may be correct in unit analysis but not scientifically.
- Useful in multiplication and division, but not in addition and subtraction.
- Does not give information about dimensional constants.
2. Differentiate between dependent and independent variables. Identify them in: (i) The force applied to stretch a rubber band. (ii) Distance travelled by a thrown stone.
Differences:
Identification:
(i) Independent Variable: Force applied to stretch a rubber band.
(ii) Dependent Variable: Distance travelled by a thrown stone.
| Dependent Variable | Independent Variable |
|---|---|
| Cannot be changed by the researcher. | Can be changed by the researcher. |
| It is the effect of the experiment. | It is the cause of the experiment. |
(i) Independent Variable: Force applied to stretch a rubber band.
(ii) Dependent Variable: Distance travelled by a thrown stone.
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Bright Idea!
3. Prove that ‘Watt’ is a derived unit. Also, check the homogeneity of: s = ut² + ½at²
Proof that Watt is derived unit:
We know: P = W/t = (F·d)/t = (m·a·d)/t
In units: Watt = (kg × m/s² × m)/s = kg·m²/s³ = kg·m²·s⁻³
It depends on kg, m, s. Hence, it is a derived unit.
Homogeneity check for s = ut² + ½at²:
LHS: Unit of s = m
RHS: Unit of ut² = (m/s) × s² = m·s (not consistent)
Unit of ½at² = (m/s²) × s² = m
Since terms on RHS do not have the same unit, the equation is not homogeneous and is invalid.
We know: P = W/t = (F·d)/t = (m·a·d)/t
In units: Watt = (kg × m/s² × m)/s = kg·m²/s³ = kg·m²·s⁻³
It depends on kg, m, s. Hence, it is a derived unit.
Homogeneity check for s = ut² + ½at²:
LHS: Unit of s = m
RHS: Unit of ut² = (m/s) × s² = m·s (not consistent)
Unit of ½at² = (m/s²) × s² = m
Since terms on RHS do not have the same unit, the equation is not homogeneous and is invalid.
4. How is unit consistency verified in an equation? Explain with an example.
Unit consistency is verified by checking whether the physical quantities on the LHS and RHS have the same fundamental units. If they match, the equation is dimensionally consistent and valid.
Example: Verify v = u + at
LHS: v → m/s
RHS: u + at → m/s + (m/s²) × s = m/s + m/s = m/s
Units match → equation is valid.
Example: Verify v = u + at
LHS: v → m/s
RHS: u + at → m/s + (m/s²) × s = m/s + m/s = m/s
Units match → equation is valid.
5. Niva claimed the formulas for power and pressure are P = mV/A and P = mv² respectively. Check their validity using unit analysis.
(a) For P = mV/A:
Unit of m = kg
Unit of V = m/s
Unit of A = m²
RHS unit = (kg × m/s)/m² = kg/(m·s)
LHS unit of power P = kg·m²/s³
LHS ≠ RHS → invalid
(b) For P = mv²:
RHS unit = kg × (m/s)² = kg·m²/s²
LHS unit of pressure P = kg/(m·s²)
LHS ≠ RHS → invalid
Unit of m = kg
Unit of V = m/s
Unit of A = m²
RHS unit = (kg × m/s)/m² = kg/(m·s)
LHS unit of power P = kg·m²/s³
LHS ≠ RHS → invalid
(b) For P = mv²:
RHS unit = kg × (m/s)² = kg·m²/s²
LHS unit of pressure P = kg/(m·s²)
LHS ≠ RHS → invalid
Study Tips
✨ Memorize the 7 fundamental SI units
✨ Practice unit analysis with different equations
✨ Remember: Independent = Cause, Dependent = Effect
✨ Always check units in physics problems
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