1.2 Derived Units in Terms of Base Units
Derived units are physical quantities obtained by multiplying or dividing SI base units. They can be expressed as products or quotients of these base units.
Examples of Unit Derivation
- Force: Calculated as mass×acceleration. Since acceleration is velocity/time, and velocity is displacement/time, the SI unit (newton, N) is expressed as kg m s −2 .
- Work: Calculated as Force×displacement. The SI unit (joule, J) is expressed as kg m 2 s −2 .
Common Derived Units Table
| Derived Quantity | SI Unit | Symbol | In Terms of Base Units |
|---|---|---|---|
| Force | newton | N | kg m s −2 |
| Work | joule | J | kg m 2 s −2 |
| Power | watt | W | kg m 2 s −3 |
| Pressure | pascal | Pa | kg m −1 s −2 |
| Electric Charge | coulomb | C | A s |
1.3 Dimensions of Physical Quantities
Dimension refers to the qualitative nature of a physical quantity. Quantities of the same nature (e.g., length, height, distance) share the same dimensions and are measured in the same units.
Representation
- Dimensions are represented by a capital letter enclosed in square brackets [ ].
- Dimensions of derived quantities are found by multiplying or dividing the dimensions of the base quantities.
Dimensions of Base Quantities
- Mass: [M]
- Length: [L]
- Time: [T]
- Electric Current: [I]
- Temperature: [θ]
- Intensity of Light: [J]
- Amount of Substance: [N]
Essential Terms in Dimensional Analysis
- Dimensional Variables: Quantities that have dimensions and variable magnitudes (e.g., velocity, force, energy).
- Dimensional Constants: Quantities that have dimensions but a constant magnitude (e.g., Planck’s constant h, Gravitational constant G, speed of light c).
- Dimensionless Variables: Quantities with no dimensions but variable magnitudes (e.g., plane angle, strain, coefficient of friction).
- Dimensionless Constants: Quantities with no dimensions and constant magnitudes (e.g., pure numbers, π, exponential constant e).
1.3.1 Advantages of Dimensions
Dimensional analysis allows us to verify equations, derive formulas, and determine units by treating dimensions as algebraic quantities. The fundamental rule is that quantities can only be added or subtracted if they share the same dimensions.
i. The Homogeneity of an Equation
To be physically correct, both sides of an equation must have identical dimensions. This is known as the principle of homogeneity of dimensions.
- Example: Checking the equation $v_f = v_i + at$.
- Dimensions of L.H.S (velocity): $[LT^{-1}]$
- Dimensions of R.H.S: $[LT^{-1}] + [LT^{-2}][T] = [LT^{-1}] + [LT^{-1}]$
- Since both sides result in $[LT^{-1}]$, the equation is dimensionally correct.
ii. To Derive a Possible Formula
You can derive a relationship between physical quantities by identifying the factors a quantity depends on and using dimensional consistency to find their powers.
Process for Derivation (Example: Wavelength of Matter Waves):
- Identify dependencies: Wavelength ($\lambda$) depends on Planck’s constant ($h$), velocity ($v$), and mass ($m$).
- Write the proportionality: $\lambda \propto h^a m^b v^c$.
- Substitute dimensions for all variables: $[L] = [ML^2T^{-1}]^a [M]^b [LT^{-1}]^c$.
- Equate the powers of $M, L,$ and $T$ from both sides to create algebraic equations.
- Solve for $a, b,$ and $c$ (e.g., $a=1, b=-1, c=-1$).
- Substitute these values back into the original relation: $\lambda = (\text{constant}) \frac{h}{mv}$.
1.3.2 Limitations of Dimensional Analysis
- It cannot distinguish between different physical quantities that share the same dimensions (e.g., work, energy, and torque all have dimensions $[ML^2T^{-2}]$).
- It cannot be used to derive formulas involving trigonometric, exponential, or logarithmic functions.
- It cannot determine the value of dimensionless constants in a formula.
- A dimensionally correct equation is not guaranteed to be physically correct, though a dimensionally wrong equation is always physically wrong.
Worked Example: Time Period of a Simple Pendulum
Goal: Derive a formula for Time Period ($T$) based on mass ($m$), length ($l$), angle ($\theta$), and gravity ($g$).
- Initial relation: $T = (\text{constant}) m^a l^b \theta^c g^d$.
- Apply dimensions: $[M^0L^0T] = [M]^a [L]^b [LL^{-1}]^c [LT^{-2}]^d$.
- Equate powers:
- For $M$: $a = 0$
- For $L$: $b + d = 0$
- For $T$: $-2d = 1 \rightarrow d = -1/2$
- Solve for $b$: $b - 1/2 = 0 \rightarrow b = 1/2$.
- Final Formula: $T = (\text{constant}) \sqrt{\frac{l}{g}}$. (Note: Experimental constant is $2\pi$).
Assignment 1.2: Dimensional Consistency
Which relationship is dimensionally consistent with acceleration ($[LT^{-2}]$)? (Where $x$ = distance, $t$ = time, $v$ = velocity)
- (a) $v/t^2$
- (b) $v/x^2$
- (c) $v^2/t$
- (d) $v^2/x$ (Correct: $[LT^{-1}]^2 / [L] = [L^2T^{-2}] / [L] = [LT^{-2}]$)
Study Guide: Precision, Accuracy, and Uncertainties
1. Precision vs. Accuracy
Science relies on observations and measurements. Understanding the distinction between these two terms is fundamental:
- Precision: How closely multiple measurements of the same quantity agree with each other. It is associated with the least count of the measuring instrument.
- Smaller least count = Greater precision.
- Indicated by absolute uncertainty.
- Accuracy: How closely a measurement agrees with the standard or true value.
- Indicated by fractional or percentage uncertainty.
- Smaller fractional uncertainty = Greater accuracy.
2. Errors in Measurement
An error is defined as the difference between the true value and the observed value:
Error = True Value – Observed Value
Common Causes of Error:
- Negligence or inexperience of a person.
- Using faulty apparatus.
- Inappropriate methods or techniques.
Types of Errors:
- Personal Error
- Systematic Error
- Random Error
3. Understanding Uncertainty
Uncertainty is the range of possible values within which the true value of a measurement lies. It quantifies the variability in data.
- Absolute Uncertainty: Equal to the least count of the measuring instrument.
- Example: A meter rod with a least count of 0.1 cm has an absolute uncertainty of $\pm 0.1$ cm.
- Fractional Uncertainty: The ratio of absolute uncertainty to the measured value.
- $\text{Fractional Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}}$
- Percentage Uncertainty: The fractional uncertainty expressed as a percentage.
- $\text{Percentage Uncertainty} = \text{Fractional Uncertainty} \times 100\%$
4. Summary of Key Relationships
| Term | Related To... | Improved By... |
|---|---|---|
| Precision | Absolute Uncertainty / Least Count | Using an instrument with a smaller least count. |
| Accuracy | Fractional / Percentage Uncertainty | Reducing the magnitude of fractional error. |
1.5.1 Rules for Calculating Uncertainties in Final Result
When performing mathematical operations on measured quantities (x and y) with uncertainties (Δx and Δy), the propagated uncertainty (Δz) in the final result (z) is determined by specific rules.
a) Rule for Addition and Subtraction
For addition ($z = x + y$) or subtraction ($z = x - y$), absolute uncertainties are added to find the total uncertainty.
- Formula: $\Delta z = \pm(\Delta x + \Delta y)$
- Example: $(24.0 \pm 0.1) \text{ cm} + (30.0 \pm 0.1) \text{ cm} \rightarrow \Delta z = \pm 0.2 \text{ cm}$
b) Rule for Multiplication and Division
For multiplication ($z = xy$) or division ($z = x/y$), percentage uncertainties are added.
- Rule: $\%$ \text{ uncertainty in } z = \% \text{ uncertainty in } x + \% \text{ uncertainty in } y$
c) Rule for Power of a Quantity
The total uncertainty in a value raised to a power is the percentage uncertainty multiplied by that power.
- Formula for $z = x^n$: $\%$ \text{ uncertainty in } z = n \times (\% \text{ uncertainty in } x)$
d) Uncertainties in Average Values
The uncertainty in the average value of multiple measurements is found through these steps:
- Calculate the average of the measured values.
- Find the deviation of each value from the average.
- Calculate the mean deviation (average of the deviations), which acts as the uncertainty.
e) Uncertainty in Timing Experiments
To find the uncertainty in the time period (T) of a vibrating body:
- Calculate Time Period: $T = \frac{\text{Time of multiple vibrations}}{\text{No. of vibrations}}$
- Calculate Uncertainty ($\Delta T$): $\Delta T = \frac{\text{Least Count (L.C) of device}}{\text{No. of vibrations}}$
Worked Examples
Example: Resistance (Multiplication/Division Rule)
Given: $V \pm \Delta V = (7.3 \pm 0.1) \text{ volts}$ and $I \pm \Delta I = (2.73 \pm 0.051) \text{ ampere}$
- Calculate R: $R = V/I = 7.3/2.73 = 2.7 \ \Omega$
- Find % uncertainties:
- V: $(0.1 / 7.3) \times 100\% = 1\%$
- I: $(0.05 / 2.73) \times 100\% = 2\%$
- Total % uncertainty in R: $1\% + 2\% = 3\%$
- Final Result: $(2.7 \pm 0.08) \ \Omega$
Example: Surface Area of a Disc (Power Rule)
Given: $r = 2.25 \text{ cm}$, $\Delta r = \pm 0.01 \text{ cm}$
- Calculate Area ($A = \pi r^2$): $3.14 \times 2.25^2 = 15.90 \text{ cm}^2$
- Find % uncertainty in r: $(0.01 / 2.25) \times 100\% = 0.4\%$
- Find % uncertainty in A: $2 \times 0.4\% = 0.8\%$
- Convert to absolute uncertainty: $0.8\% \times 15.90 \text{ cm}^2 = 0.13 \text{ cm}^2$
- Final Result: $(15.90 \pm 0.13) \text{ cm}^2$
| Q. No. | Question / Working | Correct Option |
|---|---|---|
| 1 | $\text{Percentage Uncertainty} = \left( \frac{\text{Uncertainty}}{\text{Measured Value}} \right) \times 100$ $= \left( \frac{0.02}{0.50} \right) \times 100 = 4\%$ | B. 4% |
| 2 | $\text{Percent Error} = \left( \frac{|\text{Expected} - \text{Actual}|}{\text{Expected}} \right) \times 100$ $= \left( \frac{|171.9 - 154.8|}{171.9} \right) \times 100 \approx 9.9\%$ | D. 9.9% |
| 3 | The readings are consistent ($7.32\text{ g}$), meaning they are precise. However, they are far from the standard ($2.00\text{ g}$), meaning they are not accurate. | D. precise but not accurate |
| 4 | From $F = \frac{G m_1 m_2}{r^2} \Rightarrow G = \frac{F r^2}{m_1 m_2}$ $[G] = \frac{[MLT^{-2}][L^2]}{[M^2]} = [M^{-1}L^3T^{-2}]$ | C. $[M^{-1}L^3T^{-2}]$ |
| 5 | Precision refers to the closeness of two or more measurements to each other. | C. precise |
| 6 | $\text{Average} = \frac{3.0+3.2+3.4+2.8+3.1}{5} = 3.1\text{ s}$ $\text{Max Deviation} = |3.4 - 3.1| = 0.3\text{ s}$ | A. $\pm 0.3\text{ s}$ |
| 7 | Force, Weight, and Tension all have dimensions of Force $[MLT^{-2}]$. Modulus of Elasticity is $\frac{\text{Stress}}{\text{Strain}}$, which is $[ML^{-1}T^{-2}]$. | C. modulus of elasticity |
| 8 | Pressure = $\frac{\text{Force}}{\text{Area}} = \frac{MLT^{-2}}{L^2} = M^1 L^{-1} T^{-2}$ Matches $a = -1, b = 1, c = -2$. | B. pressure, if $a = -1, b = 1, c = -2$ |
| 9 | $10^6 + 10^3 = 1,000,000 + 1,000 = 1,001,000$. In scientific notation: $1.001 \times 10^6$. The order of magnitude is $10^6$. | C. $10^6$ |
| 10 | Checking dimensions for $v = \sqrt{\lambda g}$: $[L T^{-1}] = \sqrt{[L][L T^{-2}]} = \sqrt{L^2 T^{-2}} = [L T^{-1}]$. The dimensions match. | A. $v = \sqrt{\lambda g}$ |
1.1 Draw a table to show a reasonable estimate of some physical quantities.
| Physical Quantity | Reasonable Estimate |
|---|---|
| Mass of an adult | $70\text{ kg}$ |
| Height of a room | $3\text{ m}$ |
| Time between heartbeats | $0.8\text{ s}$ |
| Mass of an apple | $100\text{ g}$ |
1.2 Express the units of the following derived quantities in terms of base units.
- (a) Force: $F = ma \Rightarrow \text{kg} \cdot \text{m} \cdot \text{s}^{-2}$
- (b) Work: $W = Fd \Rightarrow (\text{kg} \cdot \text{m} \cdot \text{s}^{-2}) \cdot \text{m} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}$
- (c) Power: $P = \frac{W}{t} \Rightarrow \frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{s}} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}$
- (d) Pressure: $P = \frac{F}{A} \Rightarrow \frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}^2} = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}$
- (e) Electric Charge: $Q = It \Rightarrow \text{A} \cdot \text{s}$
1.3 Why is it important to use an instrument with the smallest resolution?
Using an instrument with a smaller resolution (smaller least count) reduces the absolute uncertainty in a measurement. It allows for more precise readings and ensures that the recorded data is closer to the actual value, increasing the overall reliability of the experiment.
1.4 What is the importance of increasing the number of readings in an experiment?
Increasing the number of readings helps to minimize random errors. By taking the average (mean) of multiple readings, the effect of unpredictable fluctuations is reduced, making the final result more accurate and representative of the true value.
1.5 What is the difference between precision and accuracy?
| Precision | Accuracy |
|---|---|
| Describes the closeness of two or more measurements to each other. | Describes the closeness of a measured value to a standard or known true value. |
| Depends on the resolution of the instrument. | Depends on the reduction of systematic errors. |
1.6 What is the principle of homogeneity of dimensions?
The principle states that the dimensions of all terms on both sides of a physical equation must be the same. For an equation to be physically correct, you can only add or subtract physical quantities if they have the same dimensions.
1.7 Ball drop measurement calculation.
(i) What is the uncertainty of the results?
Readings: $6.2, 6.0, 6.4, 6.1, 5.8$ (all in seconds). $\text{Average Value} = \frac{6.2 + 6.0 + 6.4 + 6.1 + 5.8}{5} = 6.1\text{ s}$ $\text{Maximum Deviation} = |6.4 - 6.1| = 0.3\text{ s}$ (or $|5.8 - 6.1| = 0.3\text{ s}$). $\text{Uncertainty} = \pm 0.3\text{ s}$
(ii) How should the resulting time be expressed?
The time should be expressed as: $t = (6.1 \pm 0.3)\text{ s}$
1.8 Find the dimensions of Planck’s constant $h$.
Given: $E = hf$ $h = \frac{E}{f}$ Dimensions of Energy ($E$) = $[ML^2T^{-2}]$ Dimensions of Frequency ($f$) = $[T^{-1}]$ $[h] = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$
1.9 Justify why all measurements contain some uncertainty.
All measurements have uncertainty due to three main factors:
- Instrument limitations: Every tool has a limited resolution or least count.
- Environmental factors: Changes in temperature, pressure, or humidity can affect the measurement.
- Human factors: Limitations in human senses or reaction time (e.g., parallax error or timing errors).
Numerical Solutions
1.1 Heartbeat Estimation
Calculation: $60 \text{ years} \times 365 \text{ days} \times 24 \text{ hours} \times 60 \text{ minutes} \times 72 \text{ bpm} \approx 2.2 \times 10^9$.
Ans: $10^9$
1.2 Dimensions
- a) $v^2 / ax$: $\frac{[LT^{-1}]^2}{[LT^{-2}][L]} = \frac{L^2T^{-2}}{L^2T^{-2}} = \text{Dimensionless}$
- b) $at^2 / 2$: $[LT^{-2}][T^2] = [L]$
Ans: (a) No, (b) [L]
1.3 Percentage Uncertainty in A
Formula: $\%A = 2(\%X) + 2(\%Y) + \%Z$
Calculation: $2(1\%) + 2(1\%) + 2\% = 6\%$.
Ans: 6%
1.4 SI Base Unit of Constant c
Formula: $c = F / (rv)$
Dimensions: $\frac{[MLT^{-2}]}{[L][LT^{-1}]} = [ML^{-1}T^{-1}]$
Ans: kg m⁻¹s⁻¹
1.5 Homogeneity of P = ρgh
- LHS (Pressure): $[ML^{-1}T^{-2}]$
- RHS (ρgh): $[ML^{-3}][LT^{-2}][L] = [ML^{-1}T^{-2}]$
- Conclusion: LHS = RHS; the equation is homogeneous.
1.6 Protons in a Bacterium
Calculation: $10^{-15} \text{ kg} / 10^{-27} \text{ kg} = 10^{12}$.
Ans: $10^{12}$ protons
1.7 Hydrogen Atoms across Sun
Calculation: $10^5 \text{ km} / 10^{-14} \text{ km} = 10^{19}$.
Ans: $10^{19}$ atoms
1.8 Power and Uncertainty
- a) Power: $P = I^2R = 3^2 \times 13 = 117 \text{ W}$.
- b) % Uncertainty in I: $(0.1 / 3) \times 100 \approx 3\%$.
- c) % Uncertainty in R: $(0.5 / 13) \times 100 \approx 3.84\%$.
- d) Absolute Uncertainty in P: Total % Uncertainty $= 2(3.33\%) + 3.84\% = 10.5\%$. Absolute $= 117 \times 0.105 \approx 11.7 \text{ W}$.
Ans: 117 W, 3%, 3.84%, 11.7 W