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Learning outcomes

After studying this course, you should be able to:

  • appreciate the concept of force, and understand and model forces such as weight, tension and friction

  • model objects as particles or as rigid bodies, and the forces that act on an object in equilibrium

  • use model strings, rods, pulleys and pivots in modelling systems involving forces

  • understand and use torques

  • model and solve a variety of problems involving systems in equ
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1.5 Exercises

Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

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1.1: Converting to component form

In some applications of vectors there is a need to move backwards and forwards between geometric form and component form; we deal here with how to achieve this.

To start with, we recall definitions of cosine and sine. If P is a point on the unit circle, and the line segment OP makes an angle θ measured anticlockwise from the positive x-axis, then cos θ is the x-coordinate of P and sin θ is the y-coordinate of P (
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Learning outcomes

After studying this course, you should be able to:

  • convert a vector from geometric form (in terms of magnitude and direction) to component form

  • convert a vector from component form to geometric form

  • understand the use of bearings to describe direction

  • understand the difference between velocity and speed

  • find resultant displacements and velocities in geometric form, via the use of components.


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Acknowledgements

All materials included in this course are derived from content originated at the Open University.

Course image: fdecomite in Flickr made available under Creative Commons Attribution 2.0 Licence.

Don't miss out:

If reading this text has inspired you to learn m
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1.4.4 O is for Objectivity

One of the characteristics of ‘good’ information is that it should be balanced and present both sides of an argument or issue. This way the reader is left to weigh up the evidence and make a decision. In reality, we recognise that no information is truly objective.

This means that the onus is on you, the reader, to develop a critical awareness of the positions represented in what you read, and to take account of this when you interpret the information. In some cases, authors may
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1.3.1 Introduction

You can find a lot of information about the maths and statistics on the internet.

To find this information you might choose to use:

  • search engines and subject gateways;

  • books and electronic books;

  • databases;

  • journals;

  • encyclopedias

  • internet resources


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Studying mammals: The social climbers
Monkeys have long fascinated us because of their similarities to the human race. In this free course, Studying mammals: The social climbers, you will find out about some of the characteristics that make them so like us: their physiology, complex social interactions, large brains and intelligence. This is the ninth course in the Studying mammals series.Author(s): Creator not set

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Except for third party materials and otherwise stated (see http://www.open.ac.uk/conditions terms and conditions), this content is made available under a http://creativecommons.org/licenses/by-nc-sa/2

4.4.1 Partnerships for sustainable consumption

Moderate NGOs, progressive businesses and government all have a stake in seeing roundtable partnerships come up with practical steps that can bring sustainability closer. One area that has attracted the attention of all these players is consumption. Directing or limiting consumption is politically difficult for even the NGOs to promote. Similarly, ‘voluntary simplicity’ of the sort lived at Findhorn eco-village (Author(s): The Open University

4.3.3 Pipe dreams?

The idea underlying complementary currencies – that there is a great well of social capital waiting to be drawn upon to make society more sustainable – is an idea that is becoming quietly influential. ‘Social capital’ is a term frequently used by those mainstream politicians and civil servants tasked with addressing the widening gap between rich and poor people within societies throughout the world. Indeed, investing in and enhancing social capital is now the starting point in
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3.2.1 What is the difference between government and governance?

Governance is from the Greek words kybenan and kybernetes, meaning ‘to steer’ and ‘pilot’ or ‘helmsman’. It is the process whereby ‘an organization or a society steers itself, and the dynamics of communication and control are central to the process’ (Rosenau and Durfee, 1995, p. 14). Of course, you could read these words as a pretty sound definition of government but that would be missing the point. Government describes a more rigid and narrower set
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Introduction

Human societies have to take urgent action to end their dependence on fossil fuels. They also have to prepare to adapt to the uncertainties inherent in global environmental changes, particularly climatic ones. We have to alter the whole path of our development and decision making in order to make our societies both environmentally adaptable and sustainable. This unit takes on the task of trying to chart some of the ways in which this might come about.

The context for these changes by g
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3.1 Greenland's snowfall

Greenland snowfall differs depending on whether it falls in summer (when snow is comparatively warm and moist) or winter (when snow is cold and dry). These differences mean that as the snow is turned to ice, annual layers are formed that are in many ways similar to tree rings: thick annual layers mean high snowfall and thin annual layers low snowfall. The accumulation of snowfall on the summit of Greenland – and most importantly what is trapped within the crystals as it turns to ice – can
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6.2 The role of individuals and households

You've been considering how to reduce your own carbon footprint to help tackle the worst effects of climate and other environmental changes. To that extent, ‘I’ as an individual consumer has a role to play.

But unless you live alone, you share your household with other people, a group that could be called ‘we’. Everyone in the household may have similar views on living lightly. But, even within a household, there may be different views and priorities about what, if anything, sho
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6.1 ‘I’, ‘we’ or ‘they’?

We must abandon the conceit that individual, isolated, private actions are the answer. They can and do help. But they will not take us far enough without collective action.

(Al Gore, 2007)

There are some things that we can do as individuals: making this an energy-efficient house and making smart transport choices. Then there a
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5.2 Lighter living costs and constraints

The costs of ‘light living’ actions need, of course, also to be considered. Some actions involve no cost or save money, for example, less flying, shopping or meat eating, or can even make money, such as letting out a spare room to increase household occupancy. Others are low cost with a rapid payback time; for example, replacing an incandescent light bulb with a low-energy compact fluorescent lamp (CFL) should pay back the new lamp's cost in lower electricity bills in about
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4.3 Technical and behavioural actions

The numbers generated by the carbon calculator use a computer model based on some of the best information available. However, as I mentioned earlier, the results are not exact because calculators typically require you to enter broad categories of information about yourself and your household. And there are always uncertainties about some of the data on which the calculator is based. Nevertheless, the calculator allows you to explore the important actions needed to lighten your carbon load and
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2.2.2 Including other greenhouse gases

The above calculations count only carbon dioxide from fossil fuel use. But as you've seen, other greenhouse gases, especially methane and nitrous oxide converted into their CO2 equivalents, should be included in the carbon footprint. This means that the above calculations underestimate the carbon footprint, especially of food supply, which generates large amounts of methane.

The most thorough attempt to include other greenhouse gases, as well as imports and exports, in the ca
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2.5.1 Physical and weather-related indicators

The indicators collected in Table 4 have been observed to change over large regions of the Earth during the 20th century. According to the TAR, there is now a good level of confidence that what is being recorded is the result of long-term change rather than short-term natural fluctuations. As we noted earlier (Section 2.2.2), the most recent period of warming has been almost global in extent, but particularly marked at high latitudes. So are the changes in Table 4 consistent with rising tempe
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2.2.2 Temperature changes over the past millennium

One of the most striking images in the IPCC TAR is reproduced (in adapted form) in Figure 24. Together, these two temperature records tell a compelling story, crystallised in our earlier quotes from the SPM. So let's just pause to take a closer look at each of them.

Figure 24
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