Forces

2. Forces Have you ever wondered: What’s it like to walk on the moon? or Why does my stomach lurch in a lift? The answers are both linked to gravity and the following activities will help to answer these questions and others.

3. Before we start... Two words that we need to distinguish between are ‘mass’ and ‘weight’. In everyday language people talk about: • weighing ingredients • their own weight • the weight of a baby In each of these cases they are actually talking about ‘mass’, which is measured in kilograms. Weight?... we’ll get to that...

4. What happens? Take two objects of differing masses, a pen and a book would work well. One person hold them at exactly the same height. Everyone watch and listen carefully... Release the two objects at exactly the same time. What happened?

5. What happened? How many noises did you hear? What does this tell you? There is an equation linking initial velocity (u), time (t), displacement (s), and acceleration (a). s = ut + ½at2 What can you say about the acceleration for each object? This acceleration is called the acceleration due to gravity.

6. A bit of history Galileo Galilei (1564-1642) is said to have done essentially this same experiment, dropping balls of the same material but different masses from the Tower of Pisa. He concluded that they reached the ground at the same time. Until then, it was believed that heavier objects accelerated faster. Historical note: there is some doubt about whether or not it was Pisa, and even whether or not it was Galileo, but it was around this time that Aristotle’s belief was being questioned.

7. What is Gravity? Isaac Newton (1642-1726) devised a theory that between any two bodies is an attractional force which is related to each of their masses and to the distance between them. The bigger their masses, the stronger the attraction. The closer they are, the stronger the attraction.

8. What is Gravity? Newton observed the planets, whose motion seemed to suggest the following formula, known as Newton’s Law of Universal Gravitation: F=G F is the force between the two objects (N) m1 and m2 are the masses (kg) of the two objects r is the distance (m) between the two objects, measured from their centres G is a constant 6.67 x 10-11 (Nm2kg-2)

9. Acceleration due to Gravity Imagine a 1kg mass placed on the Earth’s surface. Using Newton’s Law of Universal Gravitation, work out the force F between the Earth and the mass. F=G Gis 6.67 x 10-11 Nm2kg-2 Mass of the Earth is 5.97 x 1024 kg Radius of the Earth is 6.37 x 106 m

10. Acceleration due to Gravity You should have found that the force is 9.81N. If we use Newton’s second law f = ma, a formula linking force (f), mass (m) and acceleration (a), we can find out the acceleration due to gravity at the Earth’s surface. In this case, f = 9.81N and m=1kg so a = 9.81ms-2 which we often approximate to 9.8ms-2

11. Weight Weight is a force and is measured in Newtons (N). To calculate the weight of an object we use Newton’s second law f = ma. Acceleration due to gravity is 9.8ms-2 So, to find the weight of a person whose mass is 60kg we use: Weight = 60 x 9.8 Weight =588N

12. Real Weights Find the weights of the following masses: • A 1kg bag of flour • A 7kg baby • A 15kg suitcase

13. Is gravity always the same, everywhere on Earth? What do you think? Is the Earth a perfect sphere?

14. Is gravity always the same, everywhere on Earth? The Earth isn’t a perfect sphere; it’s slightly flatter at the poles, where the radius is 6.36 x 106 m (and 6.38 x 106 m at the equator) What weight would a 1kg mass have at the poles? What is the acceleration due to gravity here? F=G Mass of the Earth is 5.97 x 1024 kg G is 6.67 x 10-11Nm2kg-2 F=ma

15. Is gravity always the same, everywhere on Earth? At 8 848m, Mount Everest is the highest mountain on Earth. What weight would a 1kg mass have at the top of Everest? What is the acceleration due to gravity here? F=G Mass of the Earth is 5.97 x 1024 kg Radius of the Earth is 6.37 x 106 m Gis 6.67 x 10-11Nm2kg-2 f=ma

16. Is gravity always the same, everywhere on Earth? As with many ‘real life’ contexts, things aren’t ‘ideal’ so we use a good approximation that will serve our purposes reasonably well. Acceleration due to gravity isn’t the same everywhere on Earth because it’s not a perfect sphere. In calculations we use an average value of 9.8ms-2 or 9.81ms-2. A level explanation, click here

17. How much would I weigh on the moon? Use the data on the next slide to find out how much you would weigh on the moon, or on some of the other bodies in our Solar system. You will still need to use: F=G G is 6.67 x 10-11Nm2kg-2 f=ma

18. How much would I weigh on the moon?

19. Walking on the moon So what do you think it would feel like to walk on the moon? On which planet would it feel most similar to walking on Earth?

20. Forces in a lift Think about being in a lift. If the lift is accelerating upwards, do you feel heavier or lighter? What about when it’s accelerating downwards?

21. What’s the Reaction? Push your hand against a desk top. Push it harder. What do you feel?

22. Normal Reactions Assuming you didn’t break the desk(!), you should have felt that the harder you pushed, the more the desk seemed to ‘push back’. You’d have felt a ‘push back’ force if you had pushed against a wall instead. This is called the normal reaction force. (‘normal’ as in ‘at 90° to the surface’, not as in ‘usual’). This is what we feel.

23. At rest When stood still on the ground, we feel a normal reaction (NR) which, in the case when we are stood still, is equivalent in to our weight (W). The forces acting are equal and opposite so we can say the system is ‘in equilibrium’. NR W

24. Reaction Forces In fact, it’s a bit more complicated than it first seems. Newton’s third law says that for every force there is an equal and opposite force, sometimes known as an action-reaction pair. Beware! Just because the two forces acting are equal and opposite, doesn’t mean that W and R are an example of the action-reaction pair mentioned in Newton’s third law! Find out more

25. Lift Off! When a person is stood in a lift which is accelerating downwards, the forces acting on the person are not equal and opposite. Newton’s second law,f=ma, says that to have an acceleration, there must be a resultant force, f. R a W

26. Lift Off! Supposing a 45kg child is in a lift which is accelerating at 2ms-2 downwards. Using Newton’s second law, f=ma, we have: W-R = ma 441 - R = 45 x 2 R = 441-90 = 351N So the child feels lighter. R a W Weight doesn’t change 45 x 9.8 =441N

27. Lift Off! Can you work out what the 45kg child feels in a lift which is accelerating at 2ms-2 upwards? R a W

29. A level explanation If we consider an object of mass a small distance, , from the Earth’s surface using Newton’s Law of Universal Gravitation, with being the mass of the Earth and being the radius of the Earth: Using the binomial approximation we obtain:

30. A level explanation Since is very small and especially considering the ratio we have: Substituting our expression back into our first equation we can express the force of gravitational attraction as: and using values discussed earlier we can see how little the change is, even on top of Everest! Back to the activity

31. More Reactions Newton’s Law of Universal Gravitation tells us that two masses (e.g. a person and the Earth) attract each other with the same force. This is an action-reaction pair. And in Australia........ W W’

32. More Reactions Newton’s Law of Universal Gravitation tells us that two masses (e.g. a person and the Earth) attract each other with the same force. This is an action-reaction pair. Read on... W’ W

33. More Reactions W If the person is in contact with the Earth’s surface then, in addition to their weight W, they will experience a Normal Reaction force, NR. NR The person exerts an equal and opposite Normal Reaction force on the Earth. The forces on the Earth are W’ and NR’ NR’ W’ NR and NR’ are another action-reaction pair.

34. Action- Reaction Pair The W and NR are NOT an action-reaction pair. The fact that they are equal and opposite, such as the case where the person is stood still, often leads people to think they are. W NR W’ NR’

35. Teacher notes: Forces This issue looks at forces, particularly gravity. This content will be compulsory in the new Mathematics A level, and is currently studied within A level Mechanics modules. In GCSE Mathematics, students don’t need to know about gravity and other forces, however, in GCSE Science, they do. In GCSE Science, students also learn about Newton’s laws and the equations of constant acceleration, so using them in maths may help students connect their learning in different subjects. The activities simply require the use of familiar and unfamiliar equations, so should be accessible to both GCSE and A level students, and provide opportunities for practising efficient calculator use as well as consolidating understanding of algebraic rules.

36. Teacher notes: Forces • Students should have the opportunity to discuss this with a partner or in a small group • Students should sketch or calculate (as appropriate)

37. What happens?/ What happened Slide 4 If the items are dropped at the same time, from the same height, they should hit the ground at the same time. • There may be some air resistance which will slow down the object with the larger surface area, but over this distance, the effect should be negligible. Slide 5 If the distance is the same, and the time is the same (one sound), and they both start from rest, then the acceleration must be the same for both.

38. What is Gravity? Slide 7 Note: Newton’s law of gravitation has been superseded by Einstein’s theory of general relativity, but we still use Newton’s law as it gives a very good approximation to the force of attraction in most cases and is much simpler to use. Where more precision is required, or where the masses are particularly large or the distances (relatively) close, Einstein’s theory should be used. This would make a good discussion point: using something that we know is wrong, just because it’s easier and is good enough, for now. Similarly, in mathematics we sometimes overlook things, for now. For example, KS3 or KS4 students might be told “you can’t find the square root of a negative number”.

39. What is Gravity? Slide 8 It’s worth spending some time understanding the formula and knowing ‘which bits you do first’. Note: the value of the Gravitational constant G varies according to the source. Slide 9 The calculation should result in 9.81N Slide 12 Flour: 9.8N Baby: 68.6N Suitcase: 147N

40. Is Gravity always the same? Slide 13 No, it’s not a perfect sphere. There are mountains and valleys as well as deep trenches in the ocean. The main shape is not spherical, it’s slightly flattened at the poles and slightly bulgy around the equator. Slide 14 At the poles acceleration is 9.84ms-2 At the equator acceleration is 9.78ms-2 Slide 15 Students will need to add Everest’s height to the radius of the Earth to obtain r. At the top of Everest, acceleration is 9.79ms-2

41. How much would I weigh on the moon? Slide 18 It all depends on the mass of the person. Values given below are for a 1kg mass, so simply multiply by the number of kg required.

42. Walking on the moon Slide 19 What’s it like to walk on the moon? Your guess is as good as anyone’s. On the moon, gravity is about 1/6 of Earth’s gravity. This means that objects will accelerate to the surface more slowly. When walking on Earth, gravity helps pulls our feet back down to the surface, this force would be greatly reduced on the moon, perhaps resulting in a slower pace and a lighter feeling. The planet with gravity most similar to ours is Saturn, however, since this is a Gas Giant, probably consisting of liquids and gases, there is likely to be no solid surface to walk on! In terms of gravity, Venus is next closest to our own – and has a solid surface. However, with a mean temperature of 462°C, humans are unlikely to walk there in the near future.

43. Lift Off! Slide 20 It’s important to emphasise that this is about when a lift is accelerating (positive or negative acceleration). When the lift is travelling at a constant velocity, the weight and reaction forces acting on the person are balanced, as they are when it is at rest. Slide 24 The ‘find out more’ content is recommended for A level students, but is equally accessible to others. Slide 25 The idea of a resultant force is important. In this case the two forces are in opposite directions, and there is an excess in one direction. Refer to directed numbers: -2+2=0 this is balanced, the numbers are ‘equal and opposite’ -2+3=+1 this has a result in a specific direction and is different to +2+-3=-1

44. Lift Off! Slide 27 Accelerating upwards: using Newton’s second law, f=ma, we have: R-W= ma R - 441= 45 x 2 R = 441+90 = 531N So the child feelsheavier. In the lift activities, action-reaction pairs are mentioned. These are a very common source of confusion for students and will be explored in more details in an M4 classroom resource next academic year.

45. Acknowledgements https://en.wikipedia.org/wiki/Earth_radius#Polar_radius https://en.wikipedia.org/wiki/Isaac_Newton https://en.wikipedia.org/wiki/Galileo_Galilei Thank you to Simon Clay and Sharon Tripconey for providing the ideas for these activities. If you would like to explore these themes further you may be interested in the Get Set for Mechanics course which will be run by MEI in preparation for the new A level. See the webpage for details.