Question

If \(g\) is to be determined by dropping an object through a distance of (exactly) \(10 \mathrm{~m}\), how accurately must the time be measured to obtain a result good to \(0.1 \%\) ? Calculate a percent error and an absolute error, in milliseconds.

Short answer

The time must be measured with an accuracy of 1.02 ms for the result for g to be good to 0.1% precision. The absolute error in the time measurement is 1.02 ms and the percent error is 0.1%.

Step-by-step solution

Understanding the Basic Concepts

Acceleration due to gravity (g) can be calculated using the formula used in the second equation of linear motion i.e. \(h = 0.5 * g * t^2\). Here, h is the height from which the object is dropped, t is the time it takes to fall and g is the acceleration due to gravity.

Find the Time for the Fall

Rearranging the equation to solve for the time, it's found that \(t = \sqrt{\frac{2h}{g}}\). Using 10 m as h and 9.81 m/s^2 as the approximate value for g, we find \(t = \sqrt{\frac{2*10 \mathrm{~m}}{9.81 \mathrm{~m/s^2}}}\) which gives t approximately 1.02 s.

Calculate the Allowable Variation in Time

Since it's required that the result for g is accurate to 0.1%, the same level of precision is required for the time measurement. The allowable variation in the time measurement would then be 0.1% of the calculated time which gives \( \Delta t = 0.001*1.02s = 0.00102s = 1.02ms\), where \(\Delta t\) is the variation in time.

Calculate the Absolute and Percent Error in Time Measurement

The absolute error is equal to the allowable variation in the time measurement, which we found to be 1.02 ms. The percent error will be given by the ratio of the absolute error to the exact value expressed as a percentage, i.e., \( \text{Percent error} = \frac{\Delta t}{t} * 100 = \frac{1.02ms}{1.02s} * 100 = 0.1% \)

Key concepts

Physics Error Analysis

Understanding the intricacies of error analysis in physics is essential to gaining precise and accurate results during experiments. In essence, error analysis involves evaluating the uncertainties in measurements and their potential impact on the conclusions of an experiment or calculation.

Error sources can be classified as either systematic, which consistently affect the measurement in the same way each time, or random, which are unpredictable and can fluctuate in magnitude and direction with each measurement. In the given exercise, it's critical to understand that the timing must be measured with such precision that the calculation for acceleration due to gravity (\(g\)) is within 0.1% of the true value, signifying meticulous attention to error analysis to ensure the reliability of results.

In practical scenarios, achieving such a degree of precision involves not only selecting appropriate measuring instruments but also accounting for human reflexes in starting and stopping the timer. Calibrating the equipment and performing multiple trials can help in reducing random errors, thereby contributing to an increase in the precision of the measurements. Calculating both the percent error and the absolute error, as shown in the solution, further contextualizes the degree of accuracy required in the time measurement when determining \(g\).

Time Measurement in Physics

The measurement of time is a fundamental aspect of physics that plays a pivotal role in experimental and theoretical applications. Time measurements are critical when it comes to kinematics and dynamics, where they often dictate the accuracy of other derived quantities, such as velocity, acceleration, and forces.

In the context of our exercise, the time taken for an object to fall through a known distance under the influence of gravity is a direct measure that is used to determine \(g\), the acceleration due to gravity. The precision of time measurement is paramount—hence, devices like electronic timers or photocells can be used for better precision as opposed to mechanical stopwatches, which are more prone to human error.The process of measuring time must account for factors such as the response time of the measuring device, environmental conditions that may affect the device's operation, and operator response time, especially if the process isn't fully automated. For problem-solving, it is crucial to understand the capabilities and limitations of the timing method employed to ensure the results obtained are within the acceptable range of error.

Equations of Motion

The equations of motion are a set of mathematical formulas that describe the relationships between displacement, velocity, acceleration, and time. These equations are vital in predicting the final state of an object in motion, provided certain initial conditions are known.

In the exercise provided, the equation \(h = 0.5 * g * t^2\) is used, which is one of the four kinematic equations for uniformly accelerated motion. This particular equation connects the distance \(h\) an object has traveled with the time \(t\) it has been in motion and the acceleration due to gravity \(g\), assuming the only force acting on the object is due to gravity and the initial velocity is zero.

Aside from pinpoint accuracy in time measurement for calculating \(g\), one should also take into consideration that these equations assume that there is no air resistance and the motion is in a straight line. In practical scenarios, these ideal conditions may not hold, which can introduce additional error into the calculated values. Thus, while equations of motion present a simplified model, it's crucial to recognize their limitations when applying them to real-world situations.