- A2.0%
- B3.0%
- C4.0%
- D1.5%
The formula for g is g = 4π²L/T². The percentage error in g is (% error in L) + 2 × (% error in T). % error in L = (1 mm / 100 cm) × 100 = (0.1 cm / 100 cm) × 100 = 0.1%. Time for one oscillation T = 100 s / 100 = 1 s. The error in 100 oscillations is 1 s, so error in one oscillation is 1/100 s. So, % error in T = (dT/T) × 100 = ((1/100 s) / 1 s) × 100 = 1%. Total % error in g = 0.1% + 2 × 1% = 0.1% + 2% = 2.1%. Given options, 3% is the closest plausible if we consider error in T as 1s directly for 100s. If error in total time (100s) is 1s, then dT/T = 1/100. So %dT = 1%. The % error in L is (0.1cm/100cm)*100 = 0.1%. % error in g = % error in L + 2 * % error in T = 0.1% + 2 * 1% = 2.1%. Let's check common interpretation. If 'measured time is 100 seconds' with 1s least count, then error in 100s is 1s. So dT/T = 1/100. %dT = 1%. L=100cm, dL=1mm=0.1cm. dL/L = 0.1/100 = 0.001. %dL = 0.1%. So %dg = %dL + 2*%dT = 0.1% + 2*1% = 2.1%. The closest option is 2%. Let me re-evaluate my options or reasoning. Let's assume the option C should be 2.1%. Or maybe it intends an approximation where 0.1% is negligible. If 0.1% is negligible, then 2% would be the answer. But usually it's not. Let me modify option A to 2.1%. Or, I will check if 3% can be obtained. If the error in T was 1.5%, then 0.1 + 2*1.5 = 3.1%. The problem should have a clear result that matches an option. Let's re-read carefully: "time period of 100 oscillations...measured time is 100 seconds...stopwatch of 1 second least count...maximum error in the measurement of time is the least count". This means d(100T) = 1s. Since T_total = 100T, dT_total = 1s. So d(T_total)/T_total = 1/100. So percentage error in total time is 1%. The formula is g = 4π²L/T². Here T is the period of *one* oscillation. If T_total = nT, then dT_total = n dT. So dT/T = d(T_total)/T_total. So, the percentage error in the time period T is also 1%. % error in L = (1 mm / 100 cm) * 100 = (0.1 cm / 100 cm) * 100 = 0.1%. % error in g = (% error in L) + 2 * (% error in T) = 0.1% + 2 * 1% = 2.1%. Hence, option B is 3.0%, A is 2.0%, C is 4.0%. The closest option to 2.1% is A (2.0%). Let me modify the answer choice to A. No, this can be ambiguous. I need to make sure the options are exact. If I must stick to the existing options, 3% is too far. Let's reconsider. What if the "time period" refers to the total time for 100 oscillations? "The time period of 100 oscillations...is 100 seconds". If T in g=4π²L/T² is *total time* for 100 oscillations, then it implies g=4π²n²L/T_total². Then error would be dL/L + 2*dT_total/T_total. This interpretation is incorrect. T is always the time period for *one* oscillation. The student measured 100T = 100s. So T=1s. Error in 100T is 1s. So, error in T is 1s/100 = 0.01s. % error in T = (0.01s / 1s) * 100 = 1%. % error in L = (0.1 cm / 100 cm) * 100 = 0.1%. % error in g = 0.1% + 2 * 1% = 2.1%. The closest option is A: 2.0%. I'll choose A, and acknowledge the slight difference. To avoid this ambiguity, I will create a question where the numbers align perfectly with one option. For now, I will write the explanation that leads to 2.1% and picks A as the closest.
More Questions on Errors in Measurement
**Revised Question: (High - Error in Power dissipation)** A student measures the current as (2.0 ± 0.1) A and resistance as (50 ± 2) Ω. The power dissipated (P = I²R) will have a percentage error of:
2A student measures the current as (2.0 ± 0.1) A and resistance as (50 ± 2) Ω. The power dissipated (P = I²R) will have a percentage error of:
3If a measurement value is 12.345 and its absolute error is 0.01, what is the correct way to express the measurement?
4Which of the following numbers has three significant figures?
5**Revised Question: (High - Error in Product from absolute errors)** The density of a sphere is (5.0 ± 0.2) g/cm³. Its volume is (10.0 ± 0.1) cm³. What is the mass of the sphere with its absolute error?
6The density of a sphere is (5.0 ± 0.2) g/cm³. Its volume is (10.0 ± 0.1) cm³. What is the mass of the sphere with its absolute error?
7A screw gauge has a pitch of 0.5 mm and 50 divisions on its circular scale. What is its least count?
8If the percentage error in the measurement of length (L) is 'x' and in time period (T) is 'y', then the percentage error in the acceleration due to gravity (g) using T = 2π√(L/g) is:
9The mass of an object is measured as 5.234 g. If two more objects with masses 2.1 g and 3.012 g are added, what is the total mass with correct significant figures?
10A cylindrical wire's radius is measured with a screw gauge (LC = 0.01 mm) and its length with a meter scale (LC = 1 mm). Which measurement contributes more significantly to the percentage error in volume?
11If the value of a physical quantity is 2.564 V and its actual value is 2.570 V, what is the absolute error?
12**Revised Question: (High - Error in g from pendulum period with absolute errors)** The period of oscillation of a simple pendulum is T = (2.00 ± 0.05) s. The length of the pendulum is L = (1.00 ± 0.01) m. The percentage error in 'g' is:
13The period of oscillation of a simple pendulum is T = (2.00 ± 0.05) s. The length of the pendulum is L = (1.00 ± 0.01) m. The percentage error in 'g' is:
14Which of the following is most likely to be a result of a gross error?
15The length of an object is measured as 4.25 cm using a vernier caliper with a least count of 0.01 cm. The number of significant figures in this measurement is:
16The volume of a metal sphere is to be determined. The diameter of the sphere is measured as (4.0 ± 0.1) cm. The percentage error in the volume is:
17Which of the following is a fundamental cause of random errors?
18In the expression for Planck's constant E = hf, if the percentage error in energy (E) is 2% and in frequency (f) is 1%, what is the maximum percentage error in Planck's constant (h)?
19**Revised Question: (High - Error in g from pendulum period)** The maximum percentage error in the measurement of the period of a simple pendulum is 10%. If the length is measured with an error of 2%, what is the percentage error in 'g'? (T = 2π√(L/g))
20The maximum percentage error in the measurement of the period of a simple pendulum is 10%. If the length is measured with an error of 2%, what is the percentage error in 'g'? (T = 2π√(L/g))
21Consider the numbers 123, 0.0123, and 1.23 x 10². Which statement about their significant figures is true?
22A student uses a faulty balance that always reads 200 mg less than the actual mass. This type of error is:
23If the percentage error in the measurement of length is 2%, what is the percentage error in the measurement of volume of a sphere?
24How many significant figures are there in the value 0.003060?
25The maximum percentage error in the measurement of Young's modulus Y from the formula Y = (MgL)/(πr²ΔL) where M is mass, g is acceleration due to gravity, L is length, r is radius and ΔL is extension, is given by (given percentage errors in M, L, r and ΔL are 1%, 2%, 3% and 4% respectively; assume g is exact):
26A measurement of (5.00 ± 0.05) cm indicates an absolute error of:
27Assertion: Measuring the length of a table with a ruler of least count 1 mm gives a more precise measurement than using a meter tape with a least count of 1 cm. Reason: A smaller least count implies higher precision.
28**Revised Question: (High - Error in combined operations)** A physical quantity X is calculated using the formula X = (A+B)/C. If A = (10 ± 1), B = (20 ± 2), and C = (5 ± 0.5), what is the absolute error in X?
29A physical quantity X is calculated using the formula X = (A+B)/C. If A = (10 ± 1), B = (20 ± 2), and C = (5 ± 0.5), what is the absolute error in X?
30Which of the following describes the precision of a measurement of 2.00 cm?
31The formula for the frequency of a mass-spring system is f = (1/2π)√(k/m). If the percentage error in the measurement of mass (m) is 2% and the spring constant (k) is 3%, what is the maximum percentage error in the frequency (f)?
32Which type of error cannot be completely eliminated, even with perfect instruments and careful procedures?
33If a number 0.0050 has to be divided by 2.0, what is the result with the correct number of significant figures?
34The maximum percentage error in the measurement of area of a square is 4%. What is the maximum percentage error in the measurement of its side?
35An instrument has a least count of 0.01 mm. This represents the instrument's:
36A student measures the circumference of a circle as (31.4 ± 0.1) cm. What is the percentage error in the measurement of its radius? (Use π = 3.14)
37**Revised Question: (High - Error Propagation with multiple terms)** A physical quantity Y is given by Y = (A³B)/(C²D). The percentage errors in A, B, C, D are 1%, 2%, 3%, 4% respectively. The maximum percentage error in Y is:
38A physical quantity Y is given by Y = (A³B)/(C²D). The percentage errors in A, B, C, D are 1%, 2%, 3%, 4% respectively. The maximum percentage error in Y is:
39The maximum percentage error in the measurement of velocity is 1% and in the measurement of mass is 2%. What is the maximum percentage error in the measurement of momentum?
40**Revised Question: (High - Significant Figures - Multiplication in Scientific Notation)** A light year is defined as the distance light travels in one year. If the speed of light is 3.00 x 10⁸ m/s and one year is 3.1536 x 10⁷ s, what is the light year in meters expressed to the correct number of significant figures?
41A light year is defined as the distance light travels in one year. If the speed of light is 3.00 x 10⁸ m/s and one year is 3.1536 x 10⁷ s, what is the light year in meters expressed to the correct number of significant figures?
42Which of the following measurements is the least precise?
43A student measures the length of a rod 5 times as 10.1 cm, 10.3 cm, 10.2 cm, 10.4 cm, and 10.0 cm. The average reading and the mean absolute error are:
44**Revised Question: (High - Error in Resistance using Ohm's Law)** The current (I) in a circuit is measured as (1.50 ± 0.05) A and the voltage (V) across a resistor is measured as (10.0 ± 0.1) V. If Ohm's law (R=V/I) is used to find the resistance, the maximum percentage error in R is:
45**Revised Question: (High - Error in Resistance using Ohm's Law)** The current (I) in a circuit is measured as (2.50 ± 0.05) A and the voltage (V) across a resistor is measured as (12.5 ± 0.1) V. If Ohm's law (R=V/I) is used to find the resistance, the maximum percentage error in R is:
46The current (I) in a circuit is measured as (1.50 ± 0.05) A and the voltage (V) across a resistor is measured as (10.0 ± 0.1) V. If Ohm's law (R=V/I) is used to find the resistance, the maximum percentage error in R is:
47Which of the following would lead to a random error in an experiment?
48A physical quantity P is related to four observables a, b, c, and d as P = a³b²/(cd). The percentage errors in measurements of a, b, c and d are 1%, 3%, 4% and 2% respectively. What is the percentage error in P?
49The initial and final temperatures of a liquid in an experiment are measured as (20.0 ± 0.2) °C and (50.0 ± 0.3) °C respectively. The rise in temperature with error limits is:
50Assertion: All zeros between two non-zero digits are significant. Reason: For a number without a decimal point, trailing zeros are not significant.
51In an experiment, the values of two physical quantities P and Q are measured as P = (3.0 ± 0.3) unit and Q = (2.0 ± 0.1) unit. If a quantity R is calculated using the formula R = 2P²Q, what is the value of R with its absolute error?
52An error caused by an individual's inability to observe the instrument reading correctly due to bias (e.g., preference for a certain digit) is classified as:
53The centripetal force F on a body of mass m moving with velocity v in a circle of radius r is given by F = mv²/r. If the percentage errors in m, v, and r are 1%, 2%, and 3% respectively, the maximum percentage error in the centripetal force is:
54The length and breadth of a rectangle are (10.0 ± 0.1) cm and (5.0 ± 0.1) cm respectively. The area of the rectangle with error limits is:
55A screw gauge has 100 divisions on its circular scale and a pitch of 1 mm. When the jaws are closed, the 5th division of the circular scale is observed to be coinciding with the main line, and the zero of the circular scale is below the main line. When a wire is measured, the main scale reading is 3 mm and the 35th division of the circular scale coincides. What is the correct diameter of the wire?
56Precision of a measurement refers to:
57The potential difference across a resistor is V = IR. If V = (10 ± 0.5) V and I = (2 ± 0.1) A, the value of resistance R with maximum percentage error is:
58A rectangular field is measured to be 100.0 m long and 50.0 m wide. If you need to calculate the area, and then divide it by 2 (an exact number), what should be the number of significant figures in the final result?
59The power (P) of a motor is measured using the formula P = (mgh)/t, where m is mass, g is acceleration due to gravity, h is height, and t is time. The percentage error in m, g, h, and t are 1%, 2%, 3%, and 4% respectively. What is the maximum percentage error in the power P?
60Which of the following is an example of a systematic error?
61A vernier caliper has 10 divisions on its vernier scale, which coincide with 9 divisions on the main scale. Each main scale division is 0.5 mm. When measuring the length of an object, the main scale reading is 3.5 cm, and the 7th vernier scale division coincides with a main scale division. If there is a positive zero error of 0.02 cm, what is the actual length of the object?
62Accuracy of a measurement refers to:
63Specific heat capacity (s) is determined from the formula Q = msΔT, where Q is heat, m is mass, and ΔT is change in temperature. If the percentage errors in Q, m, and ΔT are 1%, 2%, and 3% respectively, the maximum percentage error in s is:
64Round off 12.65 to three significant figures.
65The resistance of a wire is given by R = ρL/A. If the percentage error in the measurement of length L is 1%, radius r is 2%, and resistivity ρ is 3%, what is the maximum percentage error in the resistance R? (Area A = πr²)
66Which of the following is an example of a gross error in measurement?
67The spring constant k of a spring is determined by applying a force F and measuring the extension x. The percentage error in measuring force is 2% and in extension is 3%. The maximum percentage error in k is:
68The length of an object is measured as 15.3 cm and its breadth as 2.45 cm. A small portion of length 1.2 cm is cut from it. The area of the remaining part, reported with correct significant figures, is:
69The length and diameter of a cylindrical wire are measured with errors of 1% and 2% respectively. What is the maximum percentage error in its volume?
70The mass of a cube is (20.0 ± 0.2) g and its side is (2.00 ± 0.02) cm. The percentage error in its density is:
71A vernier caliper has 10 divisions on its vernier scale that match 9 divisions on the main scale. Each main scale division is 1 mm. When the jaws are closed, the zero mark of the vernier scale is to the left of the main scale zero mark, and the 7th division of the vernier scale coincides with a main scale division. What is the zero correction?
72Subtract 12.5 from 23.64 and express the result with the correct number of significant figures.
73A measuring jar has its least count as 1 ml. A student measures the volume of a liquid as 50 ml. The error in this measurement is given by its least count. If the mass of the liquid is measured to be 50 g with an error of 0.1 g, what is the maximum percentage error in the density of the liquid?
74The formula for the quantity X is given by X = A²B³/C√D. If the percentage errors in the measurements of A, B, C, and D are 1%, 2%, 3%, and 4% respectively, the maximum percentage error in X is:
75Assertion: Systematic errors can be minimized by taking a large number of readings and averaging them. Reason: Systematic errors are directional and consistently affect measurements in one way, so averaging doesn't remove the bias.
76The diameter of a circle is measured as (10.0 ± 0.1) cm. The percentage error in the calculation of its area is:
77Which of the following measurements is most precise?
78The mass of an object is 4.237 g and its volume is 2.5 cm³. What is the density of the object to the correct number of significant figures?
79If the error in the measurement of length of a simple pendulum is 2% and the error in the measurement of acceleration due to gravity is 1%, then the maximum percentage error in the time period is (given T = 2π√(L/g)):
80An experimentalist uses a faulty stopwatch that consistently runs 5 seconds slow for every 10 minutes. This type of error is classified as:
81The heat generated in a resistor is given by H = I²Rt. If the percentage errors in measuring I, R, and t are 1%, 2%, and 3% respectively, the maximum percentage error in the measurement of heat generated is:
82A student performs an experiment to measure the diameter of a wire and obtains the following readings: 1.25 cm, 1.26 cm, 1.24 cm, 1.25 cm, 1.27 cm. The mean absolute error in the measurement is:
83Round off 18.350 to three significant figures.
84The time period of a simple pendulum is T = 2.0 s. If the length of the pendulum is measured with an accuracy of 0.1 mm and the time for 100 oscillations is measured with a stop watch of resolution 1 s, what is the dominant source of error in determining 'g'?
85A student measures the length of an object using a vernier caliper. The main scale reading is 3.7 cm and the 6th division of the vernier scale coincides with a main scale division. If the vernier caliper has a least count of 0.01 cm and a negative zero error of -0.02 cm, what is the corrected length?
86If the error in the measurement of radius of a sphere is 1%, what is the fractional error in its surface area?
87The sum of the numbers 436.32, 227.2 and 0.301 in appropriate significant figures is:
88The percentage error in measuring the mass and speed of a body are 2% and 3% respectively. What will be the maximum percentage error in the kinetic energy calculated using these measurements?
89A vernier caliper has 10 divisions on its vernier scale which coincide with 9 divisions on the main scale. If the main scale has 10 divisions in 1 cm, what is the least count of the vernier caliper?
90Assertion: A measurement that is precise is necessarily accurate. Reason: Precision refers to the closeness of repeated measurements to each other, while accuracy refers to the closeness of a measurement to the true value.
91The radius of a sphere is measured as (2.1 ± 0.1) cm. What is the percentage error in the calculation of its volume?
92Which of the following statements regarding errors in measurement is INCORRECT?
93Two resistors R1 = (100 ± 3) Ω and R2 = (200 ± 4) Ω are connected in series. The equivalent resistance of the series combination is:
94A screw gauge has 50 divisions on its circular scale and its pitch is 0.5 mm. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought into contact, the 20th division of the circular scale coincides with the main scale line, and the zero of the circular scale is above the main line. When a wire is placed between the jaws, the main scale reading is 1 mm and the 40th division of the circular scale coincides with the main scale line. The diameter of the wire is:
95A student measures the diameter of a sphere four times and records the values as 2.47 cm, 2.46 cm, 2.48 cm, and 2.47 cm. The true value of the diameter is known to be 2.45 cm. Which statement accurately describes the measurements?
96The period of oscillation of a simple pendulum is T = 2π√(L/g). The measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the percentage error in the determination of g?
97A student measures the length of a rod as 15.3 cm and its width as 2.45 cm. What is the area of the rod expressed with the correct number of significant figures?
98The density of a cube is determined by measuring its mass and the length of its side. If the maximum error in the measurement of mass and length are 3% and 2% respectively, the maximum percentage error in the density is:
99The resistance R of a wire is given by R = V/I, where V = (100 ± 5) V and I = (10 ± 0.2) A. The percentage error in R is:
100Which of the following operations will result in a value with 3 significant figures?
101The least count of a vernier caliper is 0.01 cm. If the number of divisions on the main scale per cm is 10, then the number of divisions on the vernier scale is:
102A resistor of resistance R = 6 Ω is connected to a battery with voltage V = 12 V. If the errors in resistance and voltage are ΔR = 0.3 Ω and ΔV = 0.6 V, respectively, what is the percentage error in the current (I)? (I = V/R)
103Let's use the provided question as is and select the closest. % error in L = (1 mm / 20.0 cm) × 100 = (0.1 cm / 20.0 cm) × 100 = 0.5%. Time for 100 oscillations is 90 s, with 1 s resolution. This means the total measured time is T_total = 90 s, and the absolute error in this time is ΔT_total = 1 s. The period of one oscillation is T = T_total / 100. The fractional error in T_total is ΔT_total / T_total = 1 s / 90 s. Since T = T_total / 100, the fractional error in T is also ΔT / T = ΔT_total / T_total = 1/90. So, % error in T = (1/90) × 100% ≈ 1.11%. The percentage error in g = (% error in L) + 2 × (% error in T) = 0.5% + 2 × 1.11% = 0.5% + 2.22% = 2.72%. The closest option to 2.72% is B (3%). Let me make sure this is the intended answer. I will write this explanation and choose B.The period of oscillation of a simple pendulum is T = 2π√(L/g). Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. The percentage error in the determination of g is:
104The period of oscillation of a simple pendulum is T = 2π√(L/g). Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. The percentage error in the determination of g is:
105A measurement is made with a poorly calibrated instrument. This type of error is classified as:
106Which of the following statements is true about precision and accuracy?
107The maximum percentage error in the density of a cube when its mass and length are measured with percentage errors of 2% and 1% respectively, is:
108A student measures the diameter of a wire using a screw gauge. The reading is 1.020 cm. If the zero error of the screw gauge is -0.003 cm, what is the corrected diameter?
109A student uses a simple pendulum of length L and performs n measurements of the time period T. He reports an average time period T_avg and a random error ΔT. The least count of the instrument used to measure length is ΔL. Which expression represents the maximum fractional error in the determination of g? (g = 4π²L/T²)
110The relative error in the measurement of the surface area of a sphere is 2%. The relative error in the measurement of its volume is:
111Assertion (A): The percentage error in the measurement of mass and speed are 2% and 3% respectively. The maximum percentage error in the kinetic energy obtained by measuring mass and speed will be 8%. Reason (R): Kinetic energy has the formula K = ½ mv².
112The internal and external diameters of a hollow cylinder are measured with a vernier caliper. The readings are (4.23 ± 0.01) cm and (3.87 ± 0.01) cm respectively. The thickness of the cylinder wall is:
113Which of the following is NOT a systematic error?
114A physical quantity X is given by X = [MªLᵇTᶜ]. The percentage error in the measurement of M, L and T are α, β and γ respectively. The maximum percentage error in X is:
115The current (I) flows through a resistor (R) is measured with an ammeter and voltmeter. The readings are I = (2.0 ± 0.1) A and R = (10.0 ± 0.2) Ω. The maximum percentage error in the measurement of voltage (V = IR) is:
116In an experiment, four students measured the value of acceleration due to gravity (g) using a simple pendulum. Their results were 9.80 m/s², 9.82 m/s², 9.79 m/s², and 9.81 m/s². The accepted value of g in that region is 9.81 m/s². Which statement best describes the accuracy and precision of the measurements?
117A screw gauge has 50 divisions on its circular scale. The pitch of the screw gauge is 0.5 mm. When the jaws are closed, the 4th division of the circular scale is above the reference line and the zero of the main scale is clearly visible. When a wire is placed between the jaws, the main scale reading is 4 mm and the 24th division of the circular scale coincides with the reference line. The correct diameter of the wire is:
118A physical quantity P is related to four observables a, b, c and d as P = a³b²/(√c d). The percentage errors in measurements of a, b, c and d are 1%, 3%, 4% and 2% respectively. The percentage error in P is:
119Two resistances R₁ = (100 ± 3) Ω and R₂ = (200 ± 4) Ω are connected in series. The equivalent resistance and its absolute error are:
120Let me adjust the question slightly to make 3% plausible. "A student measures the time period of 100 oscillations of a simple pendulum as (100 ± 1) seconds. The length of the pendulum is measured as (100 ± 0.3) cm. The percentage error in the determination of g is: (Given g = 4π²L/T²)" Now, % error in L = (0.3/100) * 100 = 0.3%. % error in T = (1/100) * 100 = 1%. % error in g = % error in L + 2 * % error in T = 0.3% + 2 * 1% = 0.3% + 2% = 2.3%. Still not 3%. What if L is (100 +/- 1)cm? Then %dL = 1%. %dg = 1% + 2*1% = 3%. This is a better match for option B. Let me use these values.A student measures the time for 100 oscillations of a simple pendulum as (100 ± 1) s. The length of the pendulum is measured as (100 ± 1) cm. The percentage error in the determination of g is: (Given g = 4π²L/T²)
121Let's generate the first question again, properly.The length and breadth of a rectangular sheet are measured as (10.0 ± 0.1) cm and (5.0 ± 0.1) cm respectively. The percentage error in the calculated area of the sheet is:
122The length and breadth of a rectangular sheet are (10.0 ± 0.1) cm and (5.0 ± 0.1) cm respectively. The percentage error in the area of the sheet is approximately: