Vector and scalar quantities

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Vector and scalar quantities A scalar quantity

quantities e.g. 2 metres + 3 metres = 5 metres!

its magnitude (or size) and direction for

direction must be taken into account when

1 Vector and scalar quantities A scalar quantity is defined only by its magnitude (or size) for example: distance, speed, time. It is easy to combine two or more scalar quantities e.g. 2 metres + 3 metres = 5 metres! A vector quantity is defined by stating both its magnitude (or size) and direction for example: displacement, velocity, force The direction must be taken into account when combining two or more vector quantities e. 2 metres + (-3 metres) = -1 metre

2 Introduction to Forces I Sharon Tripconey

3 Combining vector quantities A sledge is being pulled with forces of 5N east and 12N west, what single force would have the same effect? 12N 5N What single force would have the same effect if the forces were 5N north and 12N west? 12N 5N

4 Combining vector quantities What single force would have the same effect?

5 Resultant vectors Three methods are available to find the resultant of a set of vectors. Scale drawing Geometry This is cumbersome and not always as accurate as is necessary Possibly most useful when a triangle of forces can be drawn and hence the sine/cosine rules or Pythagoras can be used. Working with components Very useful when more than two or three forces are involved. But the positive direction has to be made clear and a sketch is essential to prevent mistakes.

6 Converting between the two main ways to define a vector Use Pythagoras theorem and basic trigonometry Component form e.g. 3i +4j or Magnitude and direction form o This process is called resolving the vector

7 Resultant force Extract from Mechanics in Action (1990) A fantastic free resource! This is available from the Stem Centre; you have to set up a free account or log in to access it. Mechanics%20in%20action.pdf

8 Forces You cannot actually see forces, but you can often see their effect. In most cases a body will be under the influence of several forces. Forces are vector quantities with the newton (N) as unit. They have magnitude, direction and a line of action.

9 Resultant force: what direction? Sketch the velocity time graph for the motion on the ball.

10 What is gravity? It is the means by which objects with mass attract one another. Modern physics describes gravitation using the general theory of relativity, in which gravitation is a consequence of the curvature of spacetime which governs the motion of inertial objects. The simpler Newton's law of universal gravitation provides an accurate approximation for most calculations. mm F F G r G = constant

11 Let m 1 be the mass of the earth = kg mm F F G r Let r = radius of the earth = m G = m 3 kg -1 s -2 Calculate the gravitational force of attraction that would act on a mass of 1 kg (at the earth s surface)

12 Identifying forces Ref: A Resource for Free-standing Mathematics Qualifications

13 Identifying forces Ref: A Resource for Free-standing Mathematics Qualifications

14 Force diagrams θ

15 Force diagrams Force diagrams are an essential tool in helping to visualise the situation under investigation convey information such as which direction is positive, which forces are being equated in the solution etc. Force diagrams should be large enough to be helpful show all forces clearly and with unambiguous labels show and mark significant points indicate the positive direction in some way.

16 Bathroom scales and a broom If you stand on the scales and press down on the floor with the broom, what happens to the reading on the scales?

17 Bathroom scales and a broom If I press up on the ceiling with the broom, why does the reading on the scales go up? If I press down on the scales with the broom, the reading is unchanged. Why?

18 About MEI Registered charity committed to improving mathematics education Independent UK curriculum development body We offer continuing professional development courses, provide specialist tuition for students and work with employers to enhance mathematical skills in the workplace We also pioneer the development of innovative teaching and learning resources

Vector and scalar quantities

Vector and scalar quantities A scalar quantity is defined only by its magnitude (or size) for example: distance, speed, time. It is easy to combine two or more scalar quantities e.g. 2 metres + 3 metres