This is a classic calculation, and one that’s surprisingly tricky: suppose you could bore a hole all the way through the Earth. If you jumped in, how long would it take you to fall through and come out of the other side?
If you’re taking A-level physics you know pretty much everything you need to do the calculation for yourself, but there are a few fiddly little problems you need to deal with along the way. Most importantly: if you’re falling through the Earth, only some of it is below your feet. The rest of it is above your head, and the force of gravity from that part is pulling you not downwards but upwards. So working out the forces isn’t as simple as you might hope, but you do get something back: when you start picking away at the maths, you’ll find that what happens when you reach the centre of the planet is pretty important.
Have a think about how you’d tackle the problem yourself, then take a look at this video from MinutePhysics.
Tip of the hat to Alom Shaha for pointing us to this.
https://nustem.uk/wp/wp-content/uploads/2016/05/How-Long-To-Fall-Through-The-Earth.jpeg10801920Jonathanhttps://nustem.uk/wp/wp-content/uploads/2017/02/logo-banner.pngJonathan2016-05-26 08:00:282016-05-25 14:51:56How long would it take to fall through the Earth?
As a project with ‘Physics’ in our title, it hardly seems possible not to be talking about gravitational waves in the office this morning. We read the reports avidly, we got all excited, and we also realised that we’re hardly the experts on this. So here’s our brief run-down of the really useful stuff we’ve found from better journalists than ourselves and more informed cosmologists:
First up, an excellent film from the New York Times, which sets out what the LIGO experiment in Louisiana and Washington has done:
The rest of the Times’ report is a good solid overview of what’s happened. Through the arms-length reporting you can glimpse the level of excitement and the significance of the work.
If animation is more your style, this primer from PhD Comics will spin you through the bumpy landscape of gravitational waves:
“Space and time became distorted, like water at a rolling boil. In the fraction of a second that it took for the black holes to finally merge, they radiated a hundred times more energy than all the stars in the universe combined. They formed a new black hole…
The waves rippled outward in every direction, weakening as they went. On Earth, dinosaurs arose, evolved, and went extinct. The waves kept going. About fifty thousand years ago, they entered our own Milky Way galaxy, just as Homo sapiens were beginning to replace our Neanderthal cousins as the planet’s dominant species of ape. A hundred years ago, Albert Einstein, one of the more advanced members of the species, predicted the waves’ existence, inspiring decades of speculation and fruitless searching.”
It’s a beautifully-written piece, and it really captures the human aspect of this – of hundreds of physicists around the world experiencing that moment of discovery. It’s an image that’s ingrained in popular conceptions of how science works, of Archimedes leaping out of his bathtub and exclaiming ‘Eureka!’ The reality, of course, is usually very different. Science tends to proceed in small steps, miniature breakthroughs in labs and desks and computers around the world, inching forwards piece by piece. But the LIGO work appears to be a genuine breakthrough, and the excitement is both real and hard-earned.
“It is the cleanest signal you can imagine… you have to feel fantastic for those 800 scientists, who have been spending – some of them – decades of their careers working towards this first detection.”
— Dr. Andrew Pontzen, UCL
The programme also hears from the leading UK scientist on the project, Prof. Sheila Rowan of the University of Glasgow. You can get a good sense of how giddy everyone is about this by listening to her impression of the signal ‘chirp.’
A-Level Physics Required Practicals:
Measuring g using a free fall method
A-level specifications from all the exam boards include “measure the acceleration due to gravity of a freely falling body” as one of the practicals. Students might be a bit uninterested in measuring the value of a constant with which they are already familiar. However, this practical is likely to be undertaken close to the beginning of an A level course. As such, it can be used to make a number of valuable points, each of which is worth introducing our students to at this stage:
By comparing more than one method, students can practice thinking critically about experimental methods.
Learn to assess and reduce uncertainty.
Consider how to present and process data.
Discuss what is meant by ‘constant.’
What’s in the Film
The film shows four different methods of measuring g using a falling object:
Drop a ball and time its fall with a stopwatch.
Drop a ‘g-ball’, which times its own fall.
Drop a ball through light gates.
Use an electromagnetic switch to release a ball bearing, with triggered timing.
Which of these methods you use might depend on your students’ skill, your own preference, apparatus availability, ease of data collection or processing, and class size. It’s good to compare at least two methods (even if one is shown as a demonstration only), to prompt and inform discussion about precision, accuracy and uncertainty.
What’s not in the film
Since we anticipate this practical being used early in the A level course, we’ve not included comments about how to do a full error analysis. There’s more error handling in some of the other films in the series, and we’re planning to complete a film specifically about error as the series continues.
A measurement is precise if values cluster closely.
In the film, at about 2:22, the word ‘precision’ is used to mean the timer’s smallest scale division. This is us showing our age! A better term might be ‘resolution’.
Other ASE definitions:
Accuracy: a result is accurate if it is close to the true value.
Uncertainty: the interval within which the true value can be expected to lie.
Calculating g from h and t
Methods 1, 2 and 4 give values of the time \(t\) for a ball to fall from rest at a height \(h\). From the equation:
\(s = ut + \frac{1}{2}at^2\)
we have:
\(g = \frac{2h}{t^2}\)
Measuring \(t\) for different values of \(h\) allows a graph to be drawn of \(h\) against \(t^2\). The gradient of that graph is \(g/2\).
In method 3 the ball falls through two light gates separated by a distance \(h\). Each gate gives a value for the average speed of the ball as it passes through, so we can use \(v^2 = u^2 + 2as\) to find its acceleration between the gates.
Uncertainties
Uncertainties arise in four ways:
Starting the timer.
During the ball’s descent, due to air resistance or other factors.
Stopping the timer.
Determining the height of the fall.
Students can think about these as they compare the different methods. They can try to assess whether each uncertainty will make the values of \(t\) and \(h\) too big or too small, or just more uncertain. They can also discuss how each factor will affect the value of \(g\).
For example: if the measured value of \(t\) is too big, the calculated value of \(g\) will be too small, because t is on the bottom line of \(g = 2h/t^2\).
Error in g
Eventually, students must be able to assess the overall uncertainty in their measurements. For example, they might state their measured value as \(g = 8.7 \pm 1~\mathrm{ms^{-2}}\). For now, however, it is a good start to be able to calculate the percentage error, e.g.:
\(\frac{9.8 – 8.7}{9.8} \times 100\% = 11\%\)
Method 1: Dropping a ball
Drop a ball from a measured height \(h\); start and stop a timer to find \(t\).
Students should be able to comment on the problems with starting and stopping the timer at the correct instants. Reaction time here is not the same as the human response to a random stimulus, since we can watch the ball falling and anticipate the correct moment to stop the timer. It’s still, however, a poor way to measure \(g\)!
Repeated measurements for a fixed value of \(h\) will give a range of values for \(t\). Some students may be better at using a consistent technique, which will give a smaller spread in the values of \(t\). This reduces the random error, but there may still be systematic error, which itself may be revealed by repeating the experiment for different values of \(h\).
Questions to ask your students:
What’s a better way to release the ball?
What’s a better way to measure the time?
Why don’t we try to reduce the uncertainty in measuring \(h\)?
Method 2: The g-ball
These cost about £20 + VAT from education suppliers, one of which is Timstar (as of 2020-08-03 Timstar appear to have stopped selling the G Ball. We’ve found other UK suppliers: Philip Harris, Better Equipped, Breckland Scientific. Search for ‘G Ball’ to find others. There appear to be at least two manufacturers, Unilab and Mollic).
The g-ball starts timing when released and stops timing as soon as it hits the floor. Like most stopclocks, it measures to a resolution of 0.01 seconds. The switch release can limit accuracy, but overall the g-ball is a quick way to collect a large number of data points. In the film, Alom and Christina use an L-shaped bracket clipped to a metre rule to press the release switch, to aid a clean release.
Questions to ask your students:
How is this better than using a stopwatch?
How can they ‘average’ their data?
Your students could just repeatedly drop the g-ball from the same height and average all their data, then substitute their average \(t\) into \(s = \frac{1}{2}gt^2\) to find \(g\). Much better would be to exclude anomalous values first, and better still would be to plot a graph of \(h\) against \(t^2\) to spot those anomalies.
If you’re going to plot a graph, however, you might as well have two meaningful variables. In the film, Christina and Alom drop the ball from different heights and collect a lot of (not very good!) data quickly.
This approach:
Helps to spot anomalies visually – an indication of random error.
Shows that \(g\) is (more-or-less!) constant: the graph is a straight line.
Allows \(g\) to be determined from a gradient, an important skill.
Might lead on to a discussion about systematic errors (should the graph pass through the origin?).
Systematic errors can often be eliminated by plotting a graph. For example, if the height is measured 1 cm too short each time, the line on the graph will be shifted downwards. The gradient will be unchanged, and the systematic error should be easily detected.
Another approach, incidentally, would be to plot \(2s\) against \(t^2\), giving a simpler gradient of \(g\). This is a matter of personal preference.
Language:
Random error: a measurement error due to results varying in an unpredictable way.
Systematic error: A measurement error where results differ from the true value by a consistent amount each time.
Method 3: Light gates & data logger
Using light gates is a great way to get students familiar with data loggers. Watch out for the common misconception that using a computer will automatically give a better result! In this case, the timing typically does have a higher resolution, and it’s possible to collect lots of data quickly – both good reasons for using the apparatus.
With two light gates, there are three possibilities:
Display values of \(u\) and \(v\), and calculate g using \(v^2 = u^2 + 2as\).
Display values of \(u\), \(v\) and \(t\), and calculate g using \(v = u – at\).
Position the top light gate just below where the ball is released so the initial velocity is close to zero, then proceed as above.
Note that the measured speeds are always average values, because the ball is accelerating during the time it takes to pass through the light gate.
Questions to ask your students:
Can we be sure that the data logger determines accurate times?
If we vary \(h\), what graph should we plot to determine \(g\)?
How can a graph help to reveal problems with the experimental technique?
Apparatus
If you don’t have multiple sets of data loggers, you could have one set up and have students use it in turn. However, the experience of setting up the apparatus is itself valuable, as it prompts the student to think through the role of each piece of equipment rather than to approach the configured apparatus as a ‘black box.’
Ronan’s version addresses several subtle issues, including that of releasing the ball cleanly, and using a plumb-line to confirm that \(h\) is being measured vertically. We’ll update this article with more details of Ronan’s apparatus when we have them.
Questions to ask your students
Between which two points should we measure the height of fall \(h\)?
Can we be sure that the timer starts and stops at the exact moments we want it to?
If we vary \(h\), what graph should we plot to determine \(g\)?
Assessment
Common Practical Assessment Criteria
At the time of writing, the exam boards appear to agree that this practical might be used to address, in whole or in part:
CPAC 2: Applies investigate approaches and methods when using instruments and equipment.
CPAC 4: Makes and records observations.
Apparatus & Techniques
Each exam board has published a list of apparatus and techniques with which students much be familiar, along with suggestions as to which elements might be addressed by each practical. For example, Edexcel’s guidance for this practical suggests:
1. Use appropriate analogue apparatus to record a range of measurements (to include length/distance, temperature, pressure, force, angles, volume) and to interpolate between scale markings
2. Use appropriate digital instruments, including electrical multimeters, to obtain a range of measurements (to include time, current, voltage, resistance, mass).
4. Use stopwatch or light gates for timing.
11. Use ICT such as computer modelling, or datalogger with a variety of sensors to collect data, or use of software to process data.
Check your exam board’s resources: there should be a mapping document to help you decide which criteria to assess on each practical.
Student Worksheet
We’ve drafted a student worksheet for this practical, which you may find useful:
As ever, no single film can encompass everything one might wish to say about a practical. Please, leave comments with your thoughts about the approach we’ve taken, and your suggestions for alternatives or improvements.
https://nustem.uk/wp/wp-content/uploads/2016/01/g-via-free-fall.jpg10801920Jonathanhttps://nustem.uk/wp/wp-content/uploads/2017/02/logo-banner.pngJonathan2017-06-02 15:37:472020-08-03 09:47:22Measuring g via Free Fall
They may be great fun, but there is some serious physics involved in designing a roller coaster. This workshop looks at some of the different specialists who are required to bring a ride from the drawing board to the theme park. Scientists and engineers need to be experts at team work so that they can bring complex projects to realisation; in this workshop we focus on how scientists have to work together in order to solve problems.
Structural engineer
Using K’nex, we look at how structural engineers use clever design to create strong, safe and economical structures. We discuss how different shapes can confer different properties on a structure and the concepts of tension, compression, shear and torsion forces. We then construct a model roller coaster and look at the design elements that have been used to create a loop structure.
Dynamics Expert
Once our roller coaster loop is constructed we need to make sure that our carriage will make it all the way around. This involves being able to calculate gravitational potential energy, kinetic energy and centripetal force. A great example of how important these calculations can be is shown here by Greg Foot’s larger version.
In our workshop, we compare our experimental results to the calculated results, and discuss the disparity.
Materials Scientist
Our next expert lets us take a look at the different types of materials that are needed. How can we match up the correct material to the specific role? Materials scientists have to consider strength, durability, flexibility, cost, density, how changes in weather will effect it, and its environmental impact – basically they need to think of every possible detail from the paint on the supports to the padding on the chairs. They will have to work closely with the designers and safety regulators to ensure that the materials they choose will do the job they want.
Safety regulator
We also need to think carefully about how to make our roller coaster safe for the riders. Part of this is working out the g forces that will be experienced. We look at ways of measuring these experimentally and through calculations. You can see how important g forces are by watching this video about the forces experienced by astronauts and fighter pilots.
Roller Coaster Games
There are loads of great games dedicated to roller coaster design; this is one of my favourites. It shows the relationship between gravitational potential energy and kinetic energy and lets you experiment with different designs. Be warned though, it’s not as easy as it appears at first.
For a more in depth look at some of the different variables involved, try out this game.
This morning, I was back at Kenton for some more K’nex rollercoaster building. There’s a lot that goes into building a rollercoaster, and we only just scratched the surface. For more details, head over to our workshop notes page where you’ll find videos, games, and more information than you can shake a (K’nex) stick at. Now if you’ll excuse me, I have six rollercoasters to dismantle and put back into the correct boxes. Sigh.
https://nustem.uk/wp/wp-content/uploads/2015/12/Rollercoaster-2.jpg12381650James Brownhttps://nustem.uk/wp/wp-content/uploads/2017/02/logo-banner.pngJames Brown2015-12-09 14:13:512015-12-09 14:13:519th December: More Rollercoasters
This morning 22 year 9 students from Kenton visited Think Lab. They tried out a range of simple activities, all of which encouraged them to ‘Do Physics!’ Some of the explanations used physics ideas that they had met before, but some required them to think about topics which they won’t meet until A-level physics (or beyond). All were chosen to make the students go ‘Hmmm’ and think hard about what they had observed.
One of the activities was dropping different balls to see which reached the ground first. They had two tennis balls, one of which was filled with water. Lots of the students predicted that the heavier ball would fall faster.
It didn’t. Although it was hard to spot visually, when we listed for the balls hitting the ground, they bounced at the same time. This was quite surprising! Part of the reason for this is that ‘common sense’ tells us that heavier things fall faster. And if air resistance becomes a significant factor, then we do find that heavy things fall faster. The tennis balls take air resistance out of the experiment, and then the balls fall at the same rate.
A slightly larger demonstration of this phenomenon was done by Professor Brian Cox using a giant vacuum chamber.
Another very popular activity was looking at water beads, or hydrogel beads. These are a water absorbing polymer. When they are dropped into water, they disappear.
You can buy water beads online, or from florists shops.
https://nustem.uk/wp/wp-content/uploads/2014/11/Surface-effects-1-scaled.jpg14402560Carolhttps://nustem.uk/wp/wp-content/uploads/2017/02/logo-banner.pngCarol2015-04-01 12:13:142015-04-01 17:50:16Physics things to make you go ‘Hmmm’
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