02 Determining g, with precision

Aim¶

This demonstration from ShowthePhysics Pols, 2024 introduces students to the ideas of measurement uncertainty.

Subjects¶

• 1A20 (Error and Accuracy)

Diagram¶

Equipment¶

• Balloon

• Long ruler

• Phone with Phyphox app

• Hammer

• Rope

• Ladder

• Awl

• Tall tripod

Presentation¶

Inflate the balloon and hang the hammer from it with a string. Hang the whole setup on the framework and ensure the wooden plank is directly beneath the hammer. Use the acoustic chronometer in the Phyphox app with a threshold value of 0.35 and a delay of ~0.3 s.

Start the demonstration by discussing what an accurate measurement entails and why an accurate value can be important. Explain the experiment: Using a single measurement and the equation, we will determine the gravitational acceleration $g$ as accurately as possible. Popping the balloon starts the acoustic chronometer, and the hammer’s BANG on the ground stops the timer.

• Is it necessary for a good measurement to have the largest or smallest possible falling height? Are there any conditions on the extreme values you choose?

• Where should you hold the phone? At the top, bottom, middle, or does it not matter? Why?

• How accurately do we measure the time and height?

Have students brainstorm answers to the questions in pairs, write down the answers, and then invite students to share their answers (think, pair, share).

After measuring the falling height (with an estimate for the uncertainty) and determining the fall time, you can start a discussion about the uncertainty in the time measurement. Is it 1.0 s, 0.1 s, 0.01 s, or 0.001 s? Or is it something in between?

Normally, you determine the uncertainty in time by repeating the experiment, but what numbers do you expect to see when you repeat the measurement? Or said differently: Which decimal digit will likely vary?

By using the code provided below, you can calculate the final value of g with its associated uncertainty. Click the rocket at the top of the page to make the Python code cell active.

# Import libraries
import numpy as np

# Record your measurements here. H in cm, dt in s.
H = /100        #measured height  in cm
u_H = /100      #the estimated uncertainty in the height
dt =            #measured fall time in s (using phyphox)
u_dt =          #the estimated uncertainty in the fall time

# Function for determining the acceleration due to gravity
def g(H,dt):
    return 2*H/dt**2

print('In this experiment, g is determined to be: ', g(H,dt))

err_in_g = np.sqrt((u_H/H)**2+4*(u_dt/dt)**2)*g(H,dt)

print('In this experiment, g is determined to be: %.1f +/- %.1f m/s^2' %(g(H,dt), err_in_g))

Explanation¶

Every measurement can only be performed with a certain precision. Sometimes the precision is determined by the equipment, and sometimes by the measured property. In science, the answer is always given with the associated uncertainty, where the chance that a repeated measurement falls within the value with its associated uncertainty is equal to 68% ($P(\bar{x}-\mu_x<x<\bar{x}+\mu_x)=0.68$). When multiple quantities influence the uncertainty, as in the determination of the acceleration due to gravity, each measurement contributes to the uncertainty. The final value of the uncertainty in this equation is given by:

The uncertainty is always given with only one significant figure. The final answer is presented with the same decimal digit (e.g., $9.8 \pm 0.3 \text{m}/\text{s}^2$).

Remarks¶

• This demonstration is shown in our Introductory Laboratory Course.