Question 1

1. A hypothetical physical quantity X is defined by the relation $X = \frac{a^2 b}{c \sqrt{d}}$. If the percentage errors in the measurement of a, b, c and d are 1%, 2%, 3% and 4% respectively, what is the maximum possible percentage error in the calculation of X?

A) 6%B) 8%C) 9%D) 12%

Answer

✓ Correct Answer: Option C

Explanation: Solution: The relative error propagation formula for X is given by: $$ \frac{\Delta X}{X} = 2\frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c} + \frac{1}{2}\frac{\Delta d}{d} $$ Substituting the given percentage parameters: $$Percentage Error = (2 \times 1%) + (1 \times 2%) + (1 \times 3%) + \left(\frac{1}{2} \times 4%\right) = 2 + 2 + 3 + 2 = 9%.$$

Question 2

2. In a new system of units, the unit of Mass is 10 kg, the unit of Length is 100 m, and the unit of Time is 1 minute. What is the magnitude of 1 Newton in this new system?

0.36 units

Question 3

3. The Van der Waals equation for a real gas is given by $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$. What are the dimensional components of the constant a?

A) $[ML^5 T^{-2}]$B) $[ML^2 T^{-2}]$C) $[ML^{-1} T^{-2}]$D) $[L^6]$

Answer

✓ Correct Answer: Option A

Explanation: Solution: By the dimensional principle of homogeneity, quantities with identical dimensions can only be added or subtracted together. Thus, the dimensions of P must equal the dimensions of $\frac{a}{V^2}$. $$ [a] = [P][V^2] = [ML^{-1} T^{-2}] \cdot [L^3]^2 = [ML^{-1} T^{-2}][L^6] = [ML^5 T^{-2}]. $$

Question 4

4. A scientist measures the diameter of a thin wire using a screw gauge of least count 0.001 cm. The successive readings are 0.052 cm, 0.053 cm, and 0.051 cm. The mean absolute error in these readings evaluates to:

A) 0.052 cmB) 0.001 cmC) 0.002 cmD) 0.00067 cm

Answer

✓ Correct Answer: Option D

Explanation: Solution: Arithmetic Mean value $\bar{x} = \frac{0.052+0.053+0.051}{3} = 0.052cm$. Individual Absolute Errors calculations: $|\Delta x_1| = |0.052-0.052|=0cm$ $|\Delta x_2| = |0.053-0.052|=0.001cm$ $|\Delta x_3| = |0.051-0.052|=0.001cm$ $$Mean Absolute Error = \frac{0 + 0.001 + 0.001}{3} = \frac{0.002}{3} \approx 0.00067cm.$$

Question 5

5. If Planck's constant (h), the speed of light in vacuum (c), and Newton's gravitational constant (G) are taken as foundational base fundamental units, which of the following expressions correctly represents the dimensions of Length?

A) $\sqrt{\frac{hG}{c^5}}$B) $\frac{hG}{c^{3/2}}$C) $\sqrt{\frac{hG}{c^3}}$D) $h^{1/2} G^{1/2} c^{5/2}$

Answer

✓ Correct Answer: Option C

Explanation: Solution: Let $L \propto h^x c^y G^z$. Substituting standard base dimension definitions: $[L] = [ML^2 T^{-1}]^x \cdot [LT^{-1}]^y \cdot [M^{-1} L^3 T^{-2}]^z$. Equating exponents for M, L, T: x - z = 0 ⇒ x = z 2x + y + 3z = 1 -x - y - 2z = 0 Solving this linear system yields x = 1/2, z = 1/2, y = -3/2. Thus: $$L = \sqrt{\frac{hG}{c^3}}$$

Question 6

6. A Vernier calliper has 20 divisions on the vernier scale which coincide precisely with 19 divisions on the main scale. If each main scale division is 0.5 mm, what is the least count of the instrument?

A) 0.05 mmB) 0.025 mmC) 0.01 mmD) 0.005 mm

Answer

✓ Correct Answer: Option B

Explanation: Solution: Least Count (LC) = 1 MSD - 1 VSD. Given that $20 VSD = 19 MSD ⇒ 1 VSD = \frac{19}{20} MSD$. $$ LC = 1 MSD - \frac{19}{20} MSD = \frac{1}{20} MSD = \frac{0.5 mm}{20} = 0.025 mm. $$

Question 7

7. Which of the following pairs features completely different dimensional properties?

A) Density and Relative densityB) Impulse and Linear momentumC) Planck's constant and Angular momentumD) Work and Torque

Answer

✓ Correct Answer: Option A

Explanation: Solution: **Density** possesses dimensions of $[ML^{-3}T^0]$, whereas **Relative Density** is a ratio of identical quantities and is entirely dimensionless ($[M^0L^0T^0]$). All other given options consist of pairs sharing identical dimensional formats.

Question 8

8. The universal value of $G = 6.67 \times 10^{-11} Nm^2 kg^{-2}$. What is its magnitude value in CGS units ($g^{-1} cm^3 s^{-2}$)?

A) $6.67 \times 10^{-5}$B) $6.67 \times 10^{-13}$C) $6.67 \times 10^{-7}$D) $6.67 \times 10^{-8}$

Answer

✓ Correct Answer: Option D

Explanation: Solution: Converting basic fundamental units to CGS dimensions: $1 N = 10^5 dyne$, $1m = 10^2cm$, $1kg = 10^3g$. $$ G = 6.67 \times 10^{-11} \times \frac{(10^5 dyne) \times (10^2cm)^2}{(10^3g)^2} $$ $$ G = 6.67 \times 10^{-11} \times \frac{10^5 \times 10^4}{10^6} = 6.67 \times 10^{-11} \times 10^3 = 6.67 \times 10^{-8} CGS units. $$

Question 9

9. The number of significant figures present within values 0.0006032 and $6.032 \times 10^4$ are respectively:

A) 7 and 4B) 4 and 4C) 4 and 5D) 8 and 4

Answer

✓ Correct Answer: Option B

Explanation: Solution: Based on significant criteria rules: leading zeros before non-zero numbers are placeholders and not significant. Thus, 0.0006032 contains 4 significant digits ('6032'). For numbers in scientific notation, exponential factors do not contribute, giving $6.032 \times 10^4$ exactly 4 significant figures.

Question 10

10. The frequency of vibration f of a stretched string depends on its length L, the tracking tension force T, and its mass per unit length \mu. Using structural dimensional metrics, the proportional relation resolves to:

A) $f \propto \frac{1}{L} \sqrt{\mu T}$B) $f \propto L \sqrt{T/\mu}$C) $f \propto \frac{1}{L} \sqrt{T/\mu}$D) $f \propto \frac{L}{\sqrt{T\mu}}$

Answer

✓ Correct Answer: Option C

Explanation: Solution: Let $f = k L^a T^b \mu^c$. Substituting dimensional formulas: $[M^0 L^0 T^{-1}] = [L]^a \cdot [M^1 L^1 T^{-2}]^b \cdot [M^1 L^{-1}]^c$. System components matching breakdown: b + c = 0 ⇒ c = -b a + b - c = 0 -2b = -1 ⇒ b = 1/2. This yields b = 1/2, c = -1/2, a = -1. Therefore: $$f \propto \frac{1}{L} \sqrt{\frac{T}{\mu}}$$

Question 11

11. A screw gauge with a pitch of 0.5 mm and 100 individual divisions on its circular scale layout is tracking a plate width. If the circular scale reading points to 48 and main scale records 2.5 mm, what is the thickness?

A) 2.74 mmB) 2.98 mmC) 2.548 mmD) 3.00 mm

Answer

✓ Correct Answer: Option A

Explanation: Solution: $Least Count (LC) = \frac{Pitch}{Total Divisions} = \frac{0.5 mm}{100} = 0.005 mm$. $$Total Thickness Reading = Main Scale Reading + (Circular Scale Reading \times LC)$$ $$Thickness = 2.5 mm + (48 \times 0.005 mm) = 2.5 + 0.24 = 2.74 mm.$$

Question 12

12. If force (F), velocity (V), and time (T) are adopted as fundamental structural entities, what is the correct dimensional expression for Surface Tension?

A) $[FV^{-1}T^{-2}]$B) $[FV^{-1}T^{-1}]$C) $[FV^{-2}T^{-1}]$D) $[F^2 V T^{-1}]$

Answer

✓ Correct Answer: Option B

Explanation: Solution: Surface Tension (\sigma) is defined as Force per unit length (F/L). Since $Velocity = \frac{Length}{Time} ⇒ L = V \times T$. Substituting this dimension profile yields: $$\sigma = \frac{F}{V \cdot T} = [FV^{-1}T^{-1}].$$

Question 13

13. The measured tracking percentage error for a sphere's radius is 1.5%. What will be the maximum percentage error calculation inside its total Volume?

A) 1.5%B) 3.0%C) 4.5%D) 6.0%

Answer

✓ Correct Answer: Option C

Explanation: Solution: Volume of a sphere $V = \frac{4}{3}\pi r^3$. Expressing this via fractional errors gives: $\frac{\Delta V}{V} = 3 \times \left(\frac{\Delta r}{r}\right)$. $$Percentage Error in Volume = 3 \times 1.5% = 4.5%.$$

Question 14

14. What are the base dimensions for the solar constant (defined as incident energy received per unit surface area per unit tracking time)?

A) $[ML^2 T^{-2}]$B) $[MT^{-2}]$C) $[MLT^{-3}]$D) $[MT^{-3}]$

Answer

✓ Correct Answer: Option D

Explanation: Solution: $$Solar Constant = \frac{Energy}{Area \times Time}$$ Substituting the corresponding dimensions: $$Dimensions = \frac{[ML^2 T^{-2}]}{[L^2] \cdot [T]} = [M^1 L^0 T^{-3}] = [MT^{-3}].$$

Question 15

15. Inside the wave layout expression $y = a \sin(\omega t - kx)$, where t maps time metrics and x marks dimensional distance, the structural profile parameters \omega/k match the dimensions of:

A) VelocityB) AccelerationC) WavelengthD) Frequency

Answer

✓ Correct Answer: Option A

Explanation: Solution: The trigonometric arguments inside a sine function must always remain dimensionless. $[\omega t] = [1] ⇒ [\omega] = [T^{-1}]$ $[kx] = [1] ⇒ [k] = [L^{-1}]$ Therefore, dividing these parameters yields: $$\left[\frac{\omega}{k}\right] = \frac{[T^{-1}]}{[L^{-1}]} = [LT^{-1}], which matches the dimensions of Velocity.$$

Question 16

16. A complex physical property P scales via the expression $P = \frac{A^3 B^{1/2}}{C^{-4} D^{3/2}}$. If variables A, B, C, D all share identical percentage measurements error factors, which individual element contributes the absolute MINIMUM error fraction to P?

A) AB) CC) BD) D

Answer

✓ Correct Answer: Option C

Explanation: Solution: Expanding the relative error equation yields: $$ \frac{\Delta P}{P} = 3\left(\frac{\Delta A}{A}\right) + 0.5\left(\frac{\Delta B}{B}\right) + 4\left(\frac{\Delta C}{C}\right) + 1.5\left(\frac{\Delta D}{D}\right) $$ The variable tracking the absolute smallest numerical power exponent multiplier (coefficient) acts as the lowest overall contributor. Here, B maps an index of 0.5, providing the minimum overall error impact.

Question 17

17. If a clock pendulum length measures 20.0 cm using a tracking scale of 1 mm accuracy, while the collection duration for 100 cycles resolves to 90 s via a 1 s resolution watch, the net calculation error inside g evaluates to:

A) 1.5%B) 2.7%C) 3.4%D) 5.0%

Answer

✓ Correct Answer: Option B

Explanation: Solution: The acceleration due to gravity is modeled via $g = \frac{4\pi^2 L}{T^2}$, which gives the error relation: $\frac{\Delta g}{g} = \frac{\Delta L}{L} + 2\left(\frac{\Delta T}{T}\right)$. Substituting the measured properties (converting units to match): $\frac{\Delta L}{L} = \frac{0.1cm}{20.0cm} = 0.005$ $\frac{\Delta T}{T} = \frac{1 s}{90 s} \approx 0.0111$ $$\frac{\Delta g}{g} = 0.005 + 2(0.0111) = 0.005 + 0.0222 = 0.0272 \approx 2.7%.$$

Question 18

18. The systemic dimensions representing expression $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ match exactly with:

A) Velocity of lightB) ImpedanceC) InductanceD) Capacitance

Answer

✓ Correct Answer: Option A

Explanation: Solution: From classical Maxwell electrodynamics equations, the transmission propagation speed of electro-magnetic waves inside open vacuum media is explicitly defined as $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$. Thus, it shares identical dimensions with the velocity of light ($[L^1 T^{-1}]$).

Question 19

19. What is the correct standard SI unit representing thermal conductivity metrics?

A) $J m^{-1} K^{-1}$B) $W m K^{-1}$C) $W m^{-1} s^{-1}$D) $W m^{-1} K^{-1}$

Answer

✓ Correct Answer: Option D

Explanation: Solution: The baseline rate equation for linear thermal heat flow is: $\frac{dQ}{dt} = KA\left(\frac{dT}{dx}\right)$. Isolating conductivity parameter $K = \left(\frac{dQ}{dt}\right) \times \left(\frac{dx}{A \cdot dT}\right)$. $$Units = (Watt) \times \frac{m}{m^2 \cdot K} = W m^{-1} K^{-1}.$$

Question 20

20. What is the fundamental dimensional configuration formula representing magnetic flux (\Phi)?

A) $[ML^2 T^{-2} A^{-2}]$B) $[ML T^{-2} A^{-1}]$C) $[ML^2 T^{-2} A^{-1}]$D) $[ML^2 T^{-1} A^{-1}]$

Answer

✓ Correct Answer: Option C

Explanation: Solution: Magnetic flux matches the relation \Phi = B \cdot A. From the Lorentz force relation $F = qvB ⇒ B = \frac{F}{qv}$. $$\Phi = \frac{F \cdot A}{qv} = \frac{[MLT^{-2}] \cdot [L^2]}{[AT] \cdot [LT^{-1}]} = [ML^2 T^{-2} A^{-1}].$$

Accepted Answer

✓ Correct Answer: Option B

Explanation: Solution: Dimensional formula of Force is $[M^1 L^1 T^{-2}]$. Using the relation $n_1 u_1 = n_2 u_2$: $$ 1 \cdot [1kg \cdot 1m \cdot (1s)^{-2}] = n_2 \cdot [10kg \cdot 100m \cdot (60s)^{-2}] $$ $$ n_2 = 1 \times \left(\frac{1}{10}\right) \times \left(\frac{1}{100}\right) \times \left(\frac{1}{60}\right)^{-2} = \frac{1}{1000} \times 3600 = 3.6 units. $$

Grade 11 Physics Units Quiz

1. A hypothetical physical quantity X is defined by the relation $X = \frac{a^2 b}{c \sqrt{d}}$. If the percentage errors in the measurement of a, b, c and d are 1%, 2%, 3% and 4% respectively, what is the maximum possible percentage error in the calculation of X?

2. In a new system of units, the unit of Mass is 10 kg, the unit of Length is 100 m, and the unit of Time is 1 minute. What is the magnitude of 1 Newton in this new system?

3. The Van der Waals equation for a real gas is given by $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$. What are the dimensional components of the constant a?

4. A scientist measures the diameter of a thin wire using a screw gauge of least count 0.001 cm. The successive readings are 0.052 cm, 0.053 cm, and 0.051 cm. The mean absolute error in these readings evaluates to:

5. If Planck's constant (h), the speed of light in vacuum (c), and Newton's gravitational constant (G) are taken as foundational base fundamental units, which of the following expressions correctly represents the dimensions of Length?

6. A Vernier calliper has 20 divisions on the vernier scale which coincide precisely with 19 divisions on the main scale. If each main scale division is 0.5 mm, what is the least count of the instrument?

7. Which of the following pairs features completely different dimensional properties?

8. The universal value of $G = 6.67 \times 10^{-11} Nm^2 kg^{-2}$. What is its magnitude value in CGS units ($g^{-1} cm^3 s^{-2}$)?

9. The number of significant figures present within values 0.0006032 and $6.032 \times 10^4$ are respectively:

10. The frequency of vibration f of a stretched string depends on its length L, the tracking tension force T, and its mass per unit length \mu. Using structural dimensional metrics, the proportional relation resolves to:

11. A screw gauge with a pitch of 0.5 mm and 100 individual divisions on its circular scale layout is tracking a plate width. If the circular scale reading points to 48 and main scale records 2.5 mm, what is the thickness?

12. If force (F), velocity (V), and time (T) are adopted as fundamental structural entities, what is the correct dimensional expression for Surface Tension?

13. The measured tracking percentage error for a sphere's radius is 1.5%. What will be the maximum percentage error calculation inside its total Volume?

14. What are the base dimensions for the solar constant (defined as incident energy received per unit surface area per unit tracking time)?

15. Inside the wave layout expression $y = a \sin(\omega t - kx)$, where t maps time metrics and x marks dimensional distance, the structural profile parameters \omega/k match the dimensions of:

16. A complex physical property P scales via the expression $P = \frac{A^3 B^{1/2}}{C^{-4} D^{3/2}}$. If variables A, B, C, D all share identical percentage measurements error factors, which individual element contributes the absolute MINIMUM error fraction to P?

17. If a clock pendulum length measures 20.0 cm using a tracking scale of 1 mm accuracy, while the collection duration for 100 cycles resolves to 90 s via a 1 s resolution watch, the net calculation error inside g evaluates to:

18. The systemic dimensions representing expression $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ match exactly with:

19. What is the correct standard SI unit representing thermal conductivity metrics?

20. What is the fundamental dimensional configuration formula representing magnetic flux (\Phi)?