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Forces and motion guidance notes
- Solving problems – force or energy?
- Using speed-time graphs to find an equation
- Newton's laws of motion
- Discussion leading to Newton's second law
- Multiflash photography
- Classroom management in semi-darkness
- Two dimensional motion
- A language for measurements
- Straight line graphs
- The automatically straight-line graph
Forces and motion guidance notes
Forces and motion guidance notes.
Physicists aim to understand, and sometimes to predict, physical interactions. They use two abstract concepts a lot: force and energy.
A force is something that can change an object's shape or how it is moving (or not moving). A force can have any size and acts in a particular direction. Forces are something to think about when analyzing things such as:
- turning effect
The concept of force is used to explain what causes things to happen. You can analyze the forces acting on macroscopic objects or systems, but also on microscopic objects such as the particles in a gas. When forces acting on an object are in equilibrium (balanced), the velocity of the object does not change. If the object is at rest, it stays there. If the object is moving it doesn’t speed up, slow down or change direction.
Energy is a numerical value that we calculate for an object, or a system, that quantifies the amount of change that has taken, or will take, place. What is important about this quantity is that, in every event or process, there is the same amount of it at the end as there was at the beginning. Energy does not explain why things happen, though we can use it to explain why some things do NOT happen.
The concept of energy has much wider use than the concept of force. The concept of energy can provide insights into movement and materials. It can also be used to analyze electrical and magnetic behaviour, wave behaviour, changes inside the atom, engines that burn fuels … and anything else.
In some situations, you can think in terms of either energy or force. For example, when trying to improve vehicle safety in the event of a crash, you could calculate the energy absorbed in the collision, or calculate the forces acting on the vehicle’s occupants.
Physicists are resourceful and will draw on whatever thinking tools help them understand a particular situation. They prefer to understand things quantitatively.
Using speed-time graphs to find an equation
Teaching Guidance for 14-16
Imagine a graph plotted with SPEED on the vertical axis against TIME on the horizontal axis.
For an object moving along with constant speed v, the graph is just a horizontal line at height v above the axis. You already know that s, the distance travelled, is speed multiplied by time, vt; but on your graph v x t is the AREA of the shaded block of height v and length t.
Sketch a graph for an object starting from rest and moving faster and faster with constant acceleration. The line must slant upwards as v increases. And if the acceleration is constant the line must be a
straight slanting line.
Take a tiny period of time from T to T' on the time axis when the speed was, say, v1 . Look at the pillar that sits on that and runs up to the slanting graph line (Graph III). The area of that pillar is its height v1 multiplied by the short time TT'. That area is the distance travelled in that short time.
How big is the distance travelled in the
whole time, t, from rest to final v? It is the area of all the pillars from start to finish. That is the area of the triangle (in Graph IV) of height final v and base t, the total time.
The area of any triangle is ½ (height) x (base).
So distance s is ½(height, v) x (base, t) s = ½vt.
Suppose the object does not start from rest when the clock starts at 0 but is already moving with speed u. It accelerates to speed v in time t. Then the graph is like graph V below; and the distance travelled is given by the shaded area. That is made up of two patches, a rectangle and a triangle (Graph VI).
The rectangle's area is ut, the triangle's is ½(v-u)t.
Then s = ut + ½(v-u)t
- = ut + ½vt-½ut
- = ½vt + ½ut
- = (v + u2)t
Alternatively, since v-u = at
- s = ut + ½(v-u)t can be expressed as
- = ut + ½(at)t
- = ut + ½at 2
These formulae are only true for constant acceleration. Look at Graph VII. Is the acceleration constant? Which part of the area for s is different now?
What part of ut = ½at 2 is no longer safe for calculating s?
Newton's law of motion
First and second laws
If you are considering the forces acting on just one body, either law I or law II will apply.
The first law describes what happens when the forces acting on a body are balanced (no resultant force acts) – the body remains at rest or continues to move at constant velocity (constant speed in a straight line).
If a book is placed on a table, it stays at rest. This is an example of Newton’s first law. There are two forces on the book and they happen to balance owing to the elastic properties of the table. The table is slightly squashed by the book and it exerts an elastic force upwards equal to the weight of the book. You can show this by placing a thick piece of foam rubber on a table and placing a book on top of it. The foam rubber squashes.
Galileo was the first person to challenge the common sense notion that steady motion requires a steady force. He looked beyond the obvious and was able to say if there was no friction
then an object would continue to move at constant velocity. In other words, he put forward a hypothesis. He could see that a motive force is generally needed to keep an object moving in order to balance frictional forces opposing the motion.
The motion of air molecules is a good example to consider with students. When air temperature is constant, no force is applied to keep air molecules moving, yet they do not slow down. If they did, in a matter of minutes the air would condense into a liquid.
The second law describes what happens when the forces acting on a body are unbalanced (a resultant force acts). The body changes its velocity, v, in the direction of the force, F, at a rate proportional to the force and inversely proportional to its mass, m. The rate of change of v is proportional to F / m. And rate of change of velocity is acceleration, a.
So if the table mentioned above were in an upwardly accelerated lift, an outside observer would see that the two forces acting on the book were unequal. The resultant force would be sufficient to give the book the same upward acceleration as the lift. Put some bathroom scales between the book and the table. If the book is accelerating downwards, its weight would be greater than the reaction force from the table. The book would, however, appear to be weightless.
Mass is measured in kilograms and acceleration in m /s2. With an appropriate choice of unit for force, then the constant of proportionality, k, in the equation F = k ma is 1. This is how the newton is defined, giving F = ma or a = F / m.
This can also be expressed as F = rate of change of momentum or F = Δ p / Δ t.
Newton wanted to understand what moves the planets. He realized that a planet requires no force along its orbit to move at constant speed, but it does require a force at right angles to its motion (gravitational attraction to the Sun) to constantly change direction.
The third law
Newton’s third law can be stated as ‘interactions involve pairs of forces’. Be careful in talking about third law pairs (often misleadingly called ‘action’ and ‘reaction’). Many students find this law the most difficult one to understand.
Returning to the book on a table, there are three bodies involved: the Earth, the book, and the table. In this example, the interaction pairs of forces are:
- The weight of the book and the pull of the book on the Earth (gravitational forces)
- The push of the book on the table and the push of the table on the book (contact forces)
In general, action and reaction pairs can be characterized as follows:
- They act on two different bodies
- They are equal in magnitude but opposite in direction
- They are the same type of force (e.g. gravitational, magnetic, or contact)
Discussion leading to Newton's second law
Teaching Guidance for 14-16
Students will have discovered that:
- a constant force accelerates a given mass with constant acceleration;
- doubling the force doubles the acceleration, i.e. the acceleration is directly proportional to the force for a given mass. F is proportional to a;
- the force, F, needed for a given acceleration is inversely proportional to the mass, m
- for a given force, F, the acceleration, a, is inversely proportional to the mass, m.
(many students find inverse proportion a problem).
Considering these points together leads to F is proportional to ma or F = a constant x ma.
Mass is measured in kg and acceleration in m/s/s but what of force? If the constant is equated to unity, then we are defining a unit of force. In the SI system the force is measured in newtons (symbol N), leading to F = ma.
The tendency of a body to resist any change in its motion (speed or direction) – in other words, to resist any change in its acceleration – is called its ‘inertia’. Mass can be thought of as a measure of a body’s inertia.
Inertia means ‘reluctance to change’. Inertia reduces a rate of change but cannot stop it. Inertia can take many forms, e.g.
- Electromagnets have electrical inertia: they resist changes of current through their coils
- Flutes and organ pipes have acoustic inertia: their vibrations take time to diminish after the forces causing them stop
You might help students develop a feeling for inertia by asking them:
- Can you stop a moving railway carriage that is smoothly running along a line, having just been shunted? Yes, but can you stop it easily, or at once?
- What keeps a spaceship going once it is far out in space, well away from the gravitational pull of the Earth and Sun, and the rocket motors are turned off? Is there anything to stop it moving?
In both cases, inertia keeps the object moving. A force is needed to change its velocity, but even the smallest resultant (net) force will do so.
Masses of objects can be compared in principle by seeing how their velocity changes compare, in response to the same force or in the same interaction.
An object in free fall is said to be ‘weightless’ but is better described as ‘apparently weightless’.
Questions about weightlessness are likely to come up when discussing satellite motion. In free fall, no forces other than gravity act. To someone in a satellite, or (hypothetically) in an ordinary lift after the cables have been cut, objects appear to have no weight. Something placed in mid-air will just float there. Video clips of astronauts show this vividly.
A stationary observer watching from outside the satellite (or lift) will see all objects falling in exactly the same way. They are all in free fall.
An object can only be truly weightless if there is no gravitational field. This would have to be infinitely far away from any other body. Or it could be at a point between two bodies, such the Earth and the Sun, where the pull of the Earth exactly balances the pull of the Sun. Or at the centre of the Earth, where an object would be pulled equally in all directions.
Teaching Guidance for 14-16
Multiflash photography creates successive images at regular time intervals on a single frame.
Method 1: Using a digital camera in multiflash mode
You can transfer the image produced direct to a computer.
Method 2: Using a video camera
Play back the video frame by frame and place a transparent acetate sheet over the TV screen to record object positions.
Method 3: Using a camera and motor-driven disc stroboscope
You need a camera that will focus on images for objects as near as 1 metre away. The camera will need a B setting, which holds the shutter open, for continuous exposure. Use a large aperture setting, such as f3.5. Digital cameras provide an immediate image for analysis. With some cameras it may be necessary to cover the photocell to keep the shutter open.
Set up the stroboscope in front of the camera so that slits in the disc allow light from the object to reach the camera lens at regular intervals as the disc rotates.
Lens to disc distance could be as little as 1 cm. The slotted disc should be motor-driven, using a synchronous motor, so that the time intervals between exposures are constant.
You can vary the frequency of ‘exposure’ by covering unwanted slits with black tape. Do this symmetrically. For example, a disc with 2 slits open running at 300 rpm gives 10 exposures per second.
The narrower the slit, the sharper but dimmer the image. Strongly illuminating the objects, or using a light source as the moving object, allows a narrower slit to be used.
Illuminate the object as brightly as possible, but the matt black background as little as possible. A slide projector is a good light source for this purpose.
Method 4: Using a xenon stroboscope
This provides sharper pictures than with a disc stroboscope, provided that you have a good blackout. General guidance is as for Method 3. Direct the light from the stroboscope along the pathway of the object.
In multiflash photography, avoid flash frequencies in the range 15-20 Hz, and avoid red flickering light. Some people can feel unwell as a result of the flicker. Rarely, some people have photosensitive epilepsy.
General hints for success
You need to arrange partial blackout. See guidance note
Use a white or silver object, such as a large, highly polished steel ball or a golf ball, against a dark background. Alternatively, use a moving source of light such as a lamp fixed to a cell, with suitable electrical connections. In this case, place cushioning on the floor to prevent breakage.
Use the viewfinder to check that the object is in focus throughout its motion, and that a sufficient range of its motion is within the camera’s field of view.
Place a measured grid in the background to allow measurement. A black card with strips of white insulating tape at, say, 10 cm spacing provides strong contrast and allows the illuminated moving object to stand out.
As an alternative to the grid, you can use a metre rule. Its scale will not usually be visible on the final image, but you can project a photograph onto a screen. Move the projector until the metre rule in the image is the same size as a metre rule held alongside the screen. You can then make measurements directly from the screen.
Use a tripod and/or a system of clamps and stands to hold the equipment. Make sure that any system is as rigid and stable as possible.
Teamwork matters, especially in Method 3. One person could control the camera, another the stroboscope system as necessary, and a third the object to be photographed.
- Switch on lamp and darken room.
- Check camera focus, f 3.5, B setting.
- Check field of view to ensure that whole experiment will be recorded.
- Line up stroboscope.
- Count down 3-2-1-0. Open shutter just before experiment starts and close it as experiment ends.
Classroom management in semi-darkness
Teaching Guidance for 14-16
There are some experiments which must be done in semi-darkness, for example, optics experiments and ripple tanks. You need to plan carefully for such lessons. Ensure that students are clear about what they need to do during such activities and they are not given unnecessary time. Keep an eye on what is going on in the class, and act quickly to dampen down any inappropriate behaviour before it gets out of hand.
Shadows on the ceiling will reveal movements that are not in your direct line of sight.
Two dimensional motion
Galileo was the first to realize that a moving body can have several separate motions, which are independent of each other. His thinking provides a foundation for Newton's treatment of acceleration and force.
A body moving at constant velocity can be described by a sum of velocities in two directions, typically x and y co-ordinates.
The path of a projectile may combine constant speed in the horizontal direction with acceleration due to gravity in the vertical direction. This independence of vertical and horizontal motions is counter-intuitive, and only careful teaching combined with demonstration experiments will convince students.
Accelerations have the same additive properties. They too are vectors that can be added by constructing a parallelogram. Forces too are vectors and obey the same addition rule. In other words: when several forces act on a body, each produces its own effect on motion. One force does not interfere with the motion produced by another force.
A language for measurments
What is a measurement?
A measurement tells you about a property of something you are investigating, giving it a number and a unit. Measurements are always made using an instrument of some kind. Rulers, stopclocks, chemical balances and thermometers are all measuring instruments.
Some processes seem to be measuring, but are not, e.g. comparing two lengths of string to see which one is longer. Tests that lead to a simple yes/no or pass/fail result do not always involve measuring.
The quality of measurements
Evaluating the quality of measurements is an essential step on the way to sensible conclusions. Scientists use a special vocabulary that helps them think clearly about their data. Key terms that describe the quality of measurements are:
- Precision (repeatability or reproducibility)
- Measurement uncertainty
Validity: A measurement is ‘valid’ if it measures what it is supposed to be measuring. What is measured must also be relevant to the question being investigated.
If a factor is uncontrolled, the measurements may not be valid. For example, if you were investigating the heating effect of a current ( P = I 2R) by increasing the current, the resistance of the wire may change as it is heated by the current to different temperatures. This would skew the results.
Correct conclusions can only be drawn from valid data.
Accuracy: This describes how closely a measurement comes to the true value of a physical quantity. The ‘true’ value of a measurement is the value that would be obtained by a perfect measurement, i.e. in an ideal world. As the true value is not known, accuracy is a qualitative term only.
Many measured quantities have a range of values rather than one ‘true’ value. For example, a collection of resistors all marked 1 kΩ. will have a range of values, but the mean value should be 1 kΩ.. You can have more confidence in a number of measurements of a sample rather than an individual measurement. The variation enables you to identify a mean, a range and the distribution of values across the range.
Precision: The closeness of agreement between replicate measurements on the same or similar objects under specified conditions.
reproducibility (precision): The extent to which a measurement replicated under the same conditions gives a consistent result. Repeatability refers to data collected by the same operator, in the same lab, over a short timescale. Reproducibility refers to data collected by different operators, in different laboratories. You can have more confidence in conclusions and explanations if they are based on consistent data.
Measurement uncertainty: The uncertainty of a measurement is the doubt that exists about its value. For any measurement – even the most careful – there is always a margin of doubt. In everyday speech, this might be expressed as ‘give or take…’, e.g. a stick might be two metres long ‘give or take a centimetre’.
The doubt about a measurement has two aspects:
- the width of the margin, or ‘interval’. This is the range of values one expects the true value to lie within. (Note this is not necessarily the range of values one might obtain when taking measurements of the value, which may include outliers.)
- confidence level’, i.e. how sure the experimenter is that the true value lies within that margin. Discussion of confidence levels is generally appropriate only in advanced level science courses.
Uncertainty in measurements can be reduced by using an instrument that has a scale with smaller scale divisions. For example, if you use a ruler with a centimetre scale then the uncertainty in a measured length is likely to be ‘give or take a centimetre’. A ruler with a millimetre scale would reduce the uncertainty in length to ‘give or take a millimetre’.
It is important not to confuse the terms ‘error’ and ‘uncertainty’. Error refers to the difference between a measured value and the true value of a physical quantity being measured. Whenever possible we try to correct for any known errors: for example, by applying corrections from calibration certificates. But any error whose value we do not know is a source of uncertainty.
Measurement errors can arise from two sources:
- a random component, where repeating the measurement gives an unpredictably different result;
- a systematic component, where the same influence affects the result for each of the repeated measurements.
Every time a measurement is taken under what seem to be the same conditions, random effects can influence the measured value. A series of measurements therefore produces a scatter of values about a mean value. The influence of variable factors may change with each measurement, changing the mean value. Increasing the number of observations generally reduces the uncertainty in the mean value.
Systematic errors (measurements that are either consistently too large, or too small) can result from:
- poor technique (e.g. carelessness with parallax when sighting onto a scale);
- zero error of an instrument (e.g. a ruler that has been shortened by wear at the zero end, or a newtonmeter that reads a value when nothing is hung from it);
- poor calibration of an instrument (e.g. every volt is measured too large).
Whenever possible, a good experimenter will try and correct for systematic errors, thus improving accuracy. For example, if it is known that a balance always reads 2 g greater than the true reading it is perfectly possible to compensate for that error by simply subtracting 2 g from all readings taken.
Sometimes you can only find a systematic error by measuring the same value by a different method.
Errors that are not recognized contribute to measurement uncertainty.
ASE/Nuffield booklet: The Language of Measurement
In 2010, following a series of meetings with Awarding Organisations, the ASE and Nuffield Foundation jointly published a booklet to enable teachers, publishers, awarding bodies and others in England and Wales to achieve a common understanding of key terms that arise from practical work in secondary science. Order a copy or see extracts from the booklet
This webpage is based on the National Physical Laboratory's Good Practice Guide: A Beginner's Guide to Uncertainty of Measurements written by Stephanie Bell.
Straight line graphs
Drawing straight line graphs
Once you have plotted the points of a graph, checked for any anomalies and decided that the best fit will be a straight line:
- To select the best fit straight line, take a weighted average of your measurements giving less weight to points that seem out of line with the rest.
- Use a ruler to draw the line.
Interpreting straight line graphs
Proportionality: A straight line through the origin represents direct proportionality between the two variables plotted, y = mx. If the plotted points (expressing your experimental results) lie close to such a line, then they show the behaviour of your experiment is close to that proportionality.
Linear relationships: In many experiments the best straight line fails to go through the origin. In that case, there is a simple linear relationship, y = mx + c. Historically, one of the most far-reaching examples is the graph of pressure of gas in a flask (constant volume) against temperature. The intersect on the temperature axis gives an absolute zero of temperature, and an estimate of its value.
Identifying systematic errors: In some experiments, all measurements of one quantity are wrong by a constant amount. This is called a ‘systematic error’. (For example, in a pendulum investigation of T against l all the lengths may be too small because you forgot to add the radius of the bob. Plotting T2 against l will still give a straight line if every value of l is too short by the radius but the line does not pass through the origin.) In such cases, the intersect can give valuable information.
Checking for constancy: Consider the acceleration of a trolley. If you plot s against t2, where s is the distance and t is the total time of travel from rest, then you hope to get a straight line through the origin. [A straight line through the origin shows that s = constant t2]
In fact we know that s is proportional to t2 for any case of constant acceleration from rest. Simple mathematics lead from the statement that Δv / Δt = acceleration, giving s = 1/2at2 providing a is constant. [Δv = change of velocity, Δt = time taken.]
IF a is constant, THEN s = 1/2at2 because logic does that. So why might you plot the graph? To find out whether the trolley moved with constant acceleration.
The automatically straight-line graph
Teaching Guidance for 14-16
Examples of using a straight line graph to find a formula.
Example 1: To show that πR 2 gives the area of a circle.
For any circle π is the number 3.14 in the equation:
circumference = 2π x radius or π x diameter
So π is circumferencediameter
Starting from that (as a definition of π) we can show that the area of a circle is πR 2.
Draw a large circle with centre 0 and radius R. Plot a graph of 2πr upwards against r along.
Then the graph
must be a straight line and its
slope will be 2π.
The end-point A, of the graph belongs to a big circle of radius R. Each other point of the graph: line 0A belongs to a smaller circle, of radius r.
Sketch III shows two small circles close together with radii (r) and (r + tiny bit of radius).
What is the
area of the shaded ring between them? The ring has width (tiny bit of radius) and
length 2πr (its circumference). Its
area is 2πr x (tiny bit of radius).
On the Graph IV the shaded pillar shows just that same
area, 2πr x (tiny bit of radius).
Now ask about all such rings from the centre 0 out to radius R. Their total area is the same as the area of all the pillars in Graph V. That is the triangle of height2πR and base R.
AREA = ½2πR x R = πR 2.
Therefore area of circle is πR 2.
Example 2: To show that s = ½at 2 for constant acceleration from rest.
Plot a graph of at upwards against t along. Then with a constant the graph
must be a straight line; and its slope will be a (Graph VI).
Choose a tiny bit of time on the t-axis and draw a pillar up to the line (Graph VII). The area of the pillar is: height x width,
(at) x (tiny bit of time)
and that is (v) x (tiny bit of time), since acceleration x time is speed.
And that is (tiny bit of time travelled).
Then total distance travelled, s, is given by the total area of all such pillars (Graph VIII).
s = area of triangle OAB = ½at x t = ½ at 2.