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Chapter 1: Problem 71

The percentage errors in the measurement of mass and speed are \(2 \%\) and \(3 \%\) respectively. How much will be the maximum error in the estimate of kinetic energy obtained by measuring mass and speed? (a) 1196 (b) \(8 \%\) (c) \(59 \%\) (d) \(1 \%\)

Short Answer

Expert verified
The maximum error in kinetic energy is 8\%, answer (b).

Step by step solution

01

Formula for Kinetic Energy

The kinetic energy of an object is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.
02

Understand Error Propagation

To find the error in a formula like \( KE = \frac{1}{2}mv^2 \), the key is to understand how errors in measurements of \( m \) and \( v \) affect \( KE \). The percentage error in a product or power is the sum of percentage errors in the measurements.
03

Calculate Percentage Error in Mass and Velocity

We have \(2\%\) error in mass \( m \) and \(3\%\) error in velocity \( v \). Since \( v^2 \) is involved in the kinetic energy formula, this means the error in \( v^2 \) is \(2 \times 3\% = 6\%\).
04

Sum the Percentage Errors

The total percentage error in \( KE \) will be the sum of the percentage error in \( m \) (which is \(2\%\)) and \( v^2 \) (which is \(6\%\)). Therefore, the total error in the kinetic energy is \(2\% + 6\% = 8\%\).
05

Conclusion

The maximum error in the kinetic energy, given the percentage errors in mass and velocity measurements, is \(8\%\). Therefore, the answer is option \( b \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Percentage Error
Percentage error is a way to express the accuracy of a measurement by comparing the error size to the actual measurement size. It is commonly used in scientific experiments to give a sense of how significant a measurement error is relative to the correct value.
The formula for calculating percentage error is: \[ \text{Percentage Error} = \left( \frac{\text{Error}}{\text{True Value}} \right) \times 100 \% \] This expression helps determine how much deviation there is from the intended or real measurement value.%
In terms of real-world applications, evaluating measurement precision by percentage error enables us to make well-informed decisions on the legitimacy of our calculations or experimental outcomes.
Measurement Errors
Measurement errors occur in every experiment or observation, and acknowledging them is vital for accurate data interpretation. Errors can arise from many sources, including instrument limitations or human errors. They are typically classified as either systematic or random.
  • Systematic errors are consistent, repeatable errors associated with experimental apparatus or procedure flaws.
  • Random errors are unpredictable fluctuations that occur during measurement and vary in magnitude and direction.
Recognizing these errors can help in adjusting experimental procedures to minimize their effects.
When dealing with formulas, errors in measurements result in a propagation of error, affecting the outcome. Understanding how errors propagate can assist in estimating the maximum potential error in results, which is essential for maintaining fidelity in experimental work.
Kinetic Energy Calculation
Kinetic energy is the energy that an object possesses due to its motion. The formula for kinetic energy is given by: \[ KE = \frac{1}{2}mv^2 \] This means kinetic energy depends on two variables: mass \( m \) and speed \( v \).
Consequently, measurement errors in mass and velocity directly influence the calculation of kinetic energy. In this context, if there's a percentage error in mass measurement and an error in velocity measurement, these errors affect the kinetic energy result by propagating through the formula.
To find the maximum error in kinetic energy when measurement errors are present, sum up the percentage errors from mass and velocity. In cases involving velocity squared (\( v^2 \)), the percentage error for this term doubles. Combining all these can give the maximum percentage error for kinetic energy, which is crucial when evaluating energy calculations under error constraints.

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Most popular questions from this chapter

The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate by using the formula \(p=\frac{F}{l^{2}}\).If the maximum errors in the measurement of force and length are \(4 \%\) and \(2 \%\) respectively, then the maximum error in the measurement of pressure is (a) \(1 \%\) (b) \(2 \%\) (c) \(8 \%\) (d) \(10 \%\)

The focal length of a mirror is given by \(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\) where \(u\) and \(v\) represent object and image distances respectively. The maximum relative error in \(f\) is (a) \(\frac{\Delta f}{f}=\frac{\Delta u}{u}+\frac{\Delta v}{v}\) (b) \(\frac{\Delta f}{f}=\frac{1}{\Delta u / u}+\frac{1}{\Delta v / v}\) (c) \(\frac{\Delta f}{f}=\frac{\Delta u}{u}+\frac{\Delta v}{v}+\frac{\Delta(u+v)}{u+v}\) (d) \(\frac{\Delta f}{f}=\frac{\Delta u}{u}+\frac{\Delta v}{v}+\frac{\Delta u}{u+v}+\frac{\Delta v}{u+v}\)

A highly rigid cubical block \(A\) of small mass \(M\) and side \(L\) is fixed rigidly on to another cubical block of same dimensions and of low modulus of rigidity \(\eta\) such that the lower face of \(A\) completely covers the upper face of \(B\). The lower face of \(B\) is rigidly held on a horizontal surface. \(A\) small force \(F\) is applied perpendicular to one of the side faces of \(A\). After the force is withdrawn, block \(A\) executes small oscillations, the time period of which is given by (a) \(2 \pi \sqrt{M \eta L}\) (b) \(2 \pi \sqrt{\frac{M \eta}{L}}\) (c) \(2 \pi \sqrt{\frac{M L}{\eta}}\) (d) \(2 \pi \sqrt{\frac{M}{\eta L}}\)

A physical quantity \(P\) is related to four observables \(a, b, c\) and \(d\) are as follows \(P=a^{3} b^{2} / \sqrt{c} d\) The percentage errors of measurement in \(a, b, c\) and \(d\) are \(1 \%, 3 \%, 4 \%\) and \(2 \%\) respectively. What is the percentage error in the quantity \(P\), if the value of \(P\) calculated using the above relation turns out to be 3.763, to what value should you round-off the result? [NCERT] (a) \(13 \%\) and \(3.8\) (b) \(1.3 \%\) and \(0.38\) (c) \(1.3 \%\) and \(3.8\) (d) \(3.896\) and 13

A physical quantity is represented by \(X=\mathrm{M}^{a} \mathrm{~L}^{b} \mathrm{~T}^{-c}\). If percentage errors in the measurements of \(M, L\) and \(T\) are \(\alpha \%, \beta \%\) and \(\gamma \%\) respectively, then total percentage error is (a) \((\alpha a+\beta b-\gamma c) \%\) (b) \((\alpha a+\beta b+\gamma c) \%\) (c) \((\alpha a-\beta b-\gamma c) q\) (d) 096
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