units and measurements class 11 questions answers
Chapter 1: units and measurements class 11 questions answers
1. Multiple Choice Questions
- i) [L1 M1 T-2] is the dimensional formula for:
(A) Velocity
(B) Acceleration
(C) Force
(D) Work - ii) The error in the measurement of the sides of a rectangle is 1%. The error in the measurement of its area is:
(A) 1%
(B) ( )%
(C) 2%
(D) None of the above - iii) Light year is a unit of:
(A) Time
(B) Mass
(C) Distance
(D) Luminosity - iv) Dimensions of kinetic energy are the same as that of:
(A) Force
(B) Acceleration
(C) Work
(D) Pressure - v) Which of the following is not a fundamental unit?
(A) cm
(B) kg
(C) centigrade
(D) volt
2. Answer the following questions
i) Star A is farther than star B. Which star will have a larger parallax angle?
Answer: Star B will have a larger parallax angle because the parallax angle is inversely proportional to the distance of the star from the observer. Hence, the closer star (Star B) will show a larger parallax angle.
ii) What are the dimensions of the quantity √(l/g), l being the length and g the acceleration due to gravity?
Answer: The dimensions of √(l/g) are derived as follows: l has the dimension [L], and g has the dimension of acceleration [LT-2]. Thus, the dimensional formula for √(l/g) is [T], which represents time.
iii) Define absolute error, mean absolute error, relative error, and percentage error.
- Absolute Error: The difference between the measured value and the true value of a quantity.
- Mean Absolute Error: The average of the absolute errors of several measurements.
- Relative Error: The ratio of the absolute error to the true value of the quantity.
- Percentage Error: The relative error expressed as a percentage of the true value.
iv) Describe what is meant by significant figures and order of magnitude.
- Significant Figures: The digits in a measured quantity that are known with certainty plus the first uncertain digit, reflecting the precision of a measurement.
- Order of Magnitude: The power of ten used in scientific notation, providing a rough estimate of a quantity's size or scale.
v) If the measured values of two quantities are A ± ΔA and B ± ΔB, what is the maximum possible error in A ± B? Show that if Z = A/B, ΔZ/Z = ΔA/A + ΔB/B.
Answer: The maximum possible error in A ± B is given by the sum of the absolute errors: Δ(A ± B) = ΔA + ΔB.
For Z = A/B, the maximum relative error is:
ΔZ/Z = ΔA/A + ΔB/B.
This can be derived by differentiating the equation Z = A/B, assuming small errors relative to the quantities themselves.
vi) Derive the formula for kinetic energy of a particle having mass m and velocity v using dimensional analysis.
Answer: Kinetic energy (K.E) has the dimensional formula [ML2T-2]. Using dimensional analysis, we assume K.E ∝ mavb. The mass (m) has dimension [M], and velocity (v) has dimension [LT-1]. By matching the dimensional formula, we obtain the relation: K.E = ½ mv2, which is the formula for kinetic energy.
vii) Find the percentage error in kinetic energy of a body having mass 60.0 ± 0.3 g moving with a velocity 25.0 ± 0.1 cm/s.
Answer: To find the percentage error in kinetic energy:
The formula for kinetic energy is K.E = ½ mv2.
The percentage error in K.E is given by:
ΔK.E/K.E = Δm/m + 2Δv/v.
Substituting values, we find:
ΔK.E/K.E = (0.3/60.0) + 2*(0.1/25.0) = 0.005 + 0.008 = 0.013 or 1.3%
viii) In Ohm's experiments, the values of the unknown resistances were found to be 6.12 Ω, 6.09 Ω, 6.22 Ω, 6.15 Ω. Calculate the mean absolute error, relative error, and percentage error in these measurements.
Answer: The mean resistance is:
(6.12 + 6.09 + 6.22 + 6.15) / 4 = 6.145 Ω
Mean absolute error = |6.12 - 6.145| = 0.025 Ω (using 6.12 as a reference for calculation)
Relative error = Mean absolute error / Mean resistance = 0.025 / 6.145 = 0.00407 Ω
Percentage error = Relative error × 100 = 0.00407 × 100 = 0.407%
Answer: Mean absolute error = 0.04 Ω, Relative error = 0.0065 Ω, Percentage error = 0.65%.
ix) An object is falling freely under the gravitational force. Its velocity after travelling a distance h is v. If v depends upon gravitational acceleration g and distance, prove with dimensional analysis that v = k√(gh) where k is a constant.
Answer: We assume v = k(gh) with dimensions:
[v] = [L T-1], [g] = [L T-2], [h] = [L].
The dimensions of √(gh) are:
[√(gh)] = [√(L T-2 * L)] = [√(L2 T-2)] = [L T-1], matching the dimension of velocity, hence confirming v = k√(gh).
x) The equation v = v0 + at is a dimensionally valid equation. Obtain the dimensional formula for a, b, and c where v is velocity, t is time, and v0 is initial velocity.
Answer: From the equation:
v = v0 + at
Since both v and v0 have dimensions [LT-1]:
- For a: at has to match the dimensions of v, hence [a] = [LT-1] / [T] = [LT-2].
- For b: v0 has the same dimensions as v, so [b] = [LT-1].
- For c: Initial velocity v0 also has the dimensions of velocity [LT-1].
The dimensional formulas are:
a: [L1M0T-2], b: [L1M0T0], c: [L1M0T1].
xi) The length, breadth, and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2.01 cm respectively. Give the area and volume of the sheet to correct significant figures.
Answer: The area of the sheet is:
Area = length × breadth = 4.234 m × 1.005 m = 4.255 m2
The volume of the sheet is:
Volume = length × breadth × thickness = 4.234 m × 1.005 m × 0.0201 m = 8.552 m3
Thus, the area and volume to correct significant figures are:
Area = 4.255 m2, Volume = 8.552 m3.
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