ON GIANTS\' SHOULDERS: GREAT SCIENTISTS AND THEIR DISCOVERIES FROM ARCHIMEDES TO DNA

MEI Mechanics 1 Forces and Newton’s Laws of Motion Section 1: Force diagrams and motion Study plan Background So far you have looked at changing velocities of particles, without considering what causes them to change. If the velocity of a particle changes, then some force must have acted upon it. This is an informal way of stating Newton’s First Law, which is stated formally on page 45 of your textbook. This section and the next look in more detail at the effect of forces acting on a particle, usually in ideal situations. (In ‘ideal situations’, forces remain constant, strings have negligible weight and factors not mentioned explicitly in the question do not have any influence – real life is rarely so simple!) Sir Isaac Newton was mainly responsible for the theories behind the work in this chapter. There is a brief historical note about him in the textbook on page 56, but it is worth finding out more about him if you have time. ‘On Giant’s Shoulders’, by Melvyn Bragg, ISBN 0 340 712597 has a good biographical chapter about him, as well as many other important scientists and mathematicians. Newton’s work holds true for the vast majority of physical situations. Only when particles become astronomically large or exceedingly small do his theories need to be modified. For virtually all engineering applications, ‘Newtonian Mechanics’ is an adequate model. The extreme situations in which Newtonian Mechanics needs to be modified are beyond the scope of this course.

Detailed work plan 1. Look at the introductory section on pages 27 – 38. In the situation described on page 37, drawing in all the forces makes a fairly simple situation appear more complicated than it really is. The diagram on the bottom of page 38 is all that is really needed. 2. Read through pages 39 to 41. This first section concentrates on the forces of weight, contact (also called ‘reaction’), friction and resistance. The Notes and examples give an idea of when to look for certain types of force and their basic properties. 3. Exercise 3A Attempt all the questions in this exercise. The answers in the back of the book are good examples of how simple the diagrams should be, even if they are too small! Solutions are given fully in the textbook and so do not appear on the web site.

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MEI M1 Newton’s Laws Section 1 Study plan 4. The section on Forces and Motion (from page 44) looks at slightly more difficult situations involving forces .The concept of equilibrium is introduced. Equilibrium is when there is no resultant force in a particular direction causing the particle to change speed and/or direction. Some more explanation and examples of Newton’s laws are given in the Notes and examples 5. Exercise 3B Attempt all the questions in this exercise. Solutions are given fully in the textbook and so do not appear on the web site. 6. From page 47 to 51, various applications of forces are considered. Read through them carefully to make sure you understand them properly. These concepts are fundamental to the rest of the work in mechanics. An extra example is included in the Notes and examples. 7. Newton’s 3 laws of motion are re-stated together in the chapter summary on page 56, and in the Notes and examples . You should do your best to learn and understand these three laws, which underpin the whole of the mechanics you will study at A/S and A level. 8. Exercises 3C Attempt all the questions in this exercise. 9. Read the short section on pulleys on pages 52 – 54. 10. Exercise 3D Attempt all the questions in this exercise.

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MEI Mechanics 1 Forces and Newton’s Laws of Motion Section 1: Force diagrams and motion Notes and Examples These notes contain subsections on  Understanding Newton’s Laws of Motion  Types of force  Force diagrams

Understanding Newton’s Laws of Motion It will help you to understand and remember Newton’s laws of Motion if you can relate them to your own experience. There are good examples in the textbook, but here are some more. Try discussing them with other people to help clarify your understanding. Newton’s first law: “Every object continues in a state of rest or uniform motion in a straight line unless acted on by a resultant external force.” 1. Think about going quickly round a corner in a car. You feel that you are being thrown across the car, but this is not really what is happening. When you go around a corner, your velocity is changing because your direction of motion is changing. If you were not held in the car, by the friction between your body and the car seat, the car doors and your seat belt, you would continue to travel in a straight line, so the sensation of being thrown across the car is an illusion. What is actually happening is that the car is changing direction and if you are to change direction with it, a force must push you round the corner too. This force is provided by the friction between your body and the car seat, the car doors or your seat belt, which prevent you from continuing in a straight path and force you around the corner. The sensation of being thrown to one side is caused by being forced to change direction. 2. Think about doing an emergency stop in a car. You feel as though you are being thrown against the seatbelt (or through the windscreen if you are not wearing a seatbelt). What is actually happening is that your body will obey Newton’s first law and continue to move forward with your original velocity, unless a force acts upon you body to slow it down. This force is provided by the seatbelt (or the windscreen). You are not really being thrown forward, you are being forced back. 3. Look at the necks of Formula 1 racing drivers. You will notice that they appear unusually thick in relation to most people. This is because racing cars turn corners extremely quickly and their drivers need to keep their heads as still as possible so that they can see clearly what is happening. To achieve this, their neck muscles must be very strong to force their

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MEI M1 Newton’s Laws Section 1 Notes and Examples heads around the corners without them being thrown to one side. This is why racing drivers have thick necks! You have probably heard of whiplash injuries, which can happen when a person’s head is thrown backwards. They can occur when a car crashes into the back of a stationary car. People in the stationary car often suffer whiplash injuries, though these can be prevented if the car’s seats are provided with head rests. Can you explain this using Newton’s first law? Newton’s second law: “Resultant force = mass  acceleration or F  ma .” 1. It takes a heavy lorry much more distance to stop than a car, even though its brakes can probably provide a greater braking force. This is because F (from Newton’s second law), so for a given force, the larger m is, the a m smaller a is, so a heavy lorry will slow down more slowly than a light car. Another similar example is oil tankers at sea. They can literally take miles to stop because they are so massive. 2. Try throwing a tennis ball as hard as you can, then try to throw a brick as far as you can. The distance something you throw (a projectile) will travel is dependent upon its initial velocity and the faster it is thrown (at a given angle), the further it will go (you will meet this in detail in chapter 6). With your maximum throwing force you will be able to throw the tennis ball at a greater initial speed than the brick because the brick is much heavier than the tennis ball, so the maximum force from your throwing arm will give it a smaller acceleration and hence a smaller initial speed when you let it go. The tennis ball will therefore go much further. (we have ignored air resistance and assumed the same angle of throw for both the brick and the tennis ball – both are reasonable assumptions) Can you use Newton’s second law to explain why rugby players tend to be big and heavy? Newton’s third law: “When one object exerts a force on another there is always a reaction which is equal and opposite in direction to the acting force.” 1. If this were not the case, you would fall through the floor. When you are standing on the floor, the force of your weight acts vertically downwards. From Newton’s first law, without a balancing force you would accelerate down through the floor. When you are standing stationary on the floor, the reaction force from the floor is exactly balancing your weight, so that the forces acting upon you are in equilibrium. 2. To feel a reaction force directly, try punching the wall! Boxers often break their hands by hitting their opponents.

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MEI M1 Newton’s Laws Section 1 Notes and Examples There is a video (in three parts) at http://openlearn.open.ac.uk/mod/oucontent/view.php?id=398660§ion=1.1 which will help you to understand how Newton’s ideas revolutionised thinking about force and motion.

Types of force There are a number of types of force you need to consider and whose properties you need to be familiar with. Weight

All particles have weight, unless they are defined as being ‘light’, in which case their weight does not effect the situation significantly and can be ignored. Weight is a force that always acts vertically downwards (towards the centre of the earth).

Resistance

This is a force that opposes the motion of a particle. Its direction is always opposite to the direction of motion. It can vary in size in some cases.

Reaction

When two objects come into contact with each other, each exerts a force on the other object. The direction of the force is perpendicular to the surface of contact.

Friction

This is a special type of resistance force. For a stationary object, any frictional force is always at exactly the correct size and direction to keep the object stationary. However, the frictional force has a maximum value. When the resultant force on an object exceeds this maximum, the object will move. The model of friction we use in Mechanics 1 assumes that whilst and object is moving, the frictional force is constant at this maximum value.

External forces These are when objects are pushed or pulled. Driving forces and braking forces are examples of external forces. Tension

Tension is a force that prevents two objects moving away from each other. It is often found in strings and rods. The cross-piece in a pair of step ladders is in tension as it prevents the two sides from separating. Tension forces ‘pull’.

Thrust

Thrust is a force which prevents two objects coming together. It can be found in a rod but not in a string or rope. The legs of a chair provide a thrust force that prevents the seat of the chair falling to the floor. A rod between two objects can provide the thrust force required to keep them apart. A string or rope would just go slack. Thrust forces ‘push’.

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MEI M1 Newton’s Laws Section 1 Notes and Examples It is important to remember that strings and ropes can exert only tension forces, whereas rods can exert both tensions and thrusts.

Force Diagrams These are the fundamental tools for making sense of mechanics questions. Unfortunately many students (usually unsuccessful ones) are reluctant to draw them. Try not to fall into this category! Keep them simple – use plain rectangles – no fancy artwork required! (See the example below) The idea of a force diagram is to show where forces are said to act, their line of action and, if possible, their direction. A force diagram should be quite large (10cm) and show all forces acting. If necessary, do a number of diagrams showing the forces acting on the whole system and on different parts of the system separately. The example below illustrates this idea.

Example 1 A team of husky dogs is pulling a ‘sledge train’ over the ice. The ‘sledge train’ consists of the dog driver’s sledge, total mass (including the driver) 200kg and two trailer sledges strung behind it carrying supplies. Each of these trailer sledges has mass 300kg. The resistive forces experienced by each part of the sledge train are shown on the diagram below.

150N

Dog driver’s sledge 200kg

Sledge A 300kg

Sledge B 300kg

Driving force from dogs

100N

150N The sledge train is moving at constant velocity.

(a) By considering the forces on the system as a whole, calculate the driving force from the dogs. (b) By considering the forces on the dog driver’s sledge, calculate the tension in the coupling between the dog driver’s sledge and sledge A. (c) What is the tension in the coupling between sledge A and sledge B?

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MEI M1 Newton’s Laws Section 1 Notes and Examples Solution (a) RN

Whole sledge train 800 kg

FN

Since the sledge train is moving at constant velocity, the resultant force upon it must be 0, so R = F. R = 150 + 150 + 100 = 400 (the combined total resistive force on the sledge train) So the driving force from the dogs is R = F = 400 N.

(b) TN

Dog driver’s sledge 200 kg

400 N

100 N The whole sledge train is moving with constant velocity, so each part of it must be moving with constant velocity, so the forces on the dog driver’s sledge must be in equilibrium. T + 100 = 400  T = 300 The tension in the coupling between the dog driver’s sledge and sledge A is 300 N. (c)

T1 N

Sledge A 300 kg

300 N

150 N Sledge A must also be in equilibrium, so T1 + 150 = 300  T1 = 150. The tension in the coupling between sledges A and B is 150 N

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Make sure you can get the same answer by considering the forces on sledge B

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Mechanics 1 Forces and Newton’s laws of motion Section 1: Force diagrams and motion Crucial points 1. Don’t confuse mass and weight Mass is a scalar quantity. It is a measure of the amount of matter in an object. The mass of an object is the same on the moon as on the earth. Weight is a vector quantity. It is the force that acts upon objects subject to gravity. On the earth, the force of weight is always directed downwards, towards the centre of the earth. From Newton’s second law, W = mg, where W is the weight of an object m is its mass and g is the acceleration due to gravity. On the moon, the acceleration due to gravity is less, so objects with the same mass weigh less on the moon than on the earth. Students often forget to multiply the mass of objects by g to give their weight.

2. Make sure that you know Newton’s laws thoroughly Newton’s three laws are fundamental to the whole of mechanics. If you know and understand them well, they can really help you to avoid mistakes in mechanics problems. You should learn them carefully.

3. Always draw large, simple force diagrams Force diagrams do not need to be artistic, they do need to be large enough to label easily and show all relevant forces (see the example in the Notes and Examples). Students often either do not draw diagrams at all, or draw small, confusing ones. Try to get into the habit of drawing good force diagrams; they will help you in all of your Mechanics work.

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Mechanics 1 Forces and Newton’s laws of motion Section 1: Force diagrams and motion Exercise 1. A car of mass M pulls a trailer of mass m along a straight level road at constant speed. The car experiences a resistance to motion of 5000 N and the trailer feels a resistance to motion of 1000 N. (i) Draw diagrams showing all of the forces acting on the car and on the trailer. (ii) By considering the forces on the whole system of car and trailer, find the driving force in the engine. (iii) By considering the forces on the trailer only, find the tension in the towbar. m1

2.

m2

m3

In the diagram above, mass m1 lies on a rough horizontal table and is connected to masses m2 and m3, where m3 > m2, by light inextensible strings passing over smooth pulleys. (i) Draw diagrams showing all of the forces acting on each of the masses. (ii) Assuming that the frictional force is large enough to prevent the system from moving, find expressions for the tension in each string and the frictional force.

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Mechanics 1 Forces and Newton’s laws of motion Section 1: Force diagrams and motion Solutions to Exercise 1. (i) D is the driving force of the car engine. T is the tension in the towbar.

R1

R2 1000

5000

T

Trailer

Car

T

mg

D

Mg

(ii) By considering the whole system, the tension in the towbar is not included. Since the system is moving at constant speed, there is no resultant force. Horizontally: D − 5000 − 1000 = 0 D = 6000 The driving force in the engine is 6000 N.

T − 1000 = 0 T = 1000

(iii) For the trailer:

Since m3 > m2 ,

The tension in the towbar is 1000 N.

the mass m1 will tend to move to the right, so the frictional force will oppose this.

2. (i)

T1

T2

R T2

T1 F m2g

(ii) Considering mass m1: Considering mass m3: Considering mass m1:

m1g

m3g

T 1 = m2 g T 2 = m3 g T 2 −T 1 − F = 0 F = T 2 − T 1 = m3 g − m2 g = (m3 − m2 )g

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MEI Mechanics 1 Forces and Newton’s laws of motion Section 1: Force diagrams and motion Multiple Choice test For questions 1 to 7 use the diagrams below. A B R

W C

W R

D T

P F W E

P

W F

P

R

W

W

R stands for Reaction, W stands for Weight, T stands for Tension, P is an additional force and F stands for Friction.

1) Which of the force diagrams could represent a ball in flight, if there is no air resistance? (a) D (c) E (e) I don’t know

(b) A (d) B

2) Which two force diagrams could represent a suitcase standing at rest on a smooth floor? (a) C and F (c) A and C (e) I don’t know

(b) E and A (d) A and F

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MEI M1 Forces Section 1 MC test 3) Which force diagram could represent a suitcase being pushed across a rough floor? (a) D (c) E (e) I don’t know

(b) C (d) F

4) Which force diagram could represent a glider flying at constant velocity, assuming negligible air resistance? (a) F (c) B (e) I don’t know

(b) E (d) A

5) Which force diagram could represent a rocket, just after the engines have ignited, but before it begins to leave the ground? (a) F (c) D (e) I don’t know

(b) A (d) C

6) Which force diagram could represent a spider hanging by a thread? (a) D (c) B (e) I don’t know

(b) A (d) E

7) Which force diagram could represent a puck sliding across perfectly smooth ice, assuming no air resistance? (a) C (c) A (e) I don’t know

(b) B (d) E

For questions 8 to 10 use the diagram below. The pulleys are assumed frictionless. The acceleration due to gravity is g. T2

T1 10 kg

T2 7 kg

T1 4 kg

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MEI M1 Forces Section 1 MC test 8) If the system is in equilibrium, what is the size of the tension T2? (a) 4g N (c) 7N (e) I don’t know

(b) 3g N (d) 7g N

9) If the system is in equilibrium, there must also be a friction force acting between the 10 kg block and the table. This friction force must be: (a) 3g N to the right (c) 10g N to the right (e) I don’t know

(b) 4g N to the right (d) 7g N to the left

10) If the maximum frictional force is F and it is not sufficient for the system to remain in equilibrium, the resultant force on the 10 kg mass is: (a) T2 + T1 (c) T1 − T2 (e) I don’t know

(b) T2 − T1 + F (d) T2 − T1 − F

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