What is a Vector? ![]() In adding vectors we have to consider their directions too. If they act in the same direction or in exactly opposite they can be simply added considering their plus or minus signs. If they are acting at different angles we can use ‘vector diagrams’. Fig. 1. Vector diagram for the boat.
The figure shows a vector diagram to get the direction of a boat. It is sailing to East but a cross current is acting to North. The velocity of the boat is 4 meters per second while the current is flowing at 3 meters per second. The red arrow gives the velocity. The magnitude is the length which can be obtained by measurement or by using the Pythagoras theorem. This also gives the direction of the resultant force. R 2 = 32 + 42 ........> R = √ ( 9 + 16).......> R = 5 ms-1 Using trigonometry to get the angle giving the direction:-
Taking the Tangent according to fig. 1 we get:- Tan ϴ = N/S = ¾ =0.75
A Force has a direction and as such it is a Vector. A force has to be represented by a line in length and direction. Consider the two forces in the given example. The wind F-1, is acting towards East by a value of 800 N. The water is pushing it to North by a force of 600N. We cannot draw lines of 800 or 600 cm. Fig.3 Adding two forces F1 and F2.
Subtracting vectors In subtracting a vector the same method can be used after taking the quantity to be subtracted as a negative quantity. When the changing angle is a right angle the magnitude of the difference will be the same as the sum, but the direction will change. Example 1 The flight path of a bird is shown in red. It flies at a constant speed and takes a right angle turn. Find the change in velocity. Answer. The change in velocity is = Final velocity - Initial velocity. The diagram on the right shows the vector diagram obtained by changing the direction of the flight path. Δ,V = Vf - Vi = √ 62 - (-62) ms-1 = √ 36 + 36 = √ 72 = 8.5 ms-1Example 2 Fig.4 Parallelogram method to subtract. A fish is swimming to east with a speed of 30 kms-1 while the river is flowing in a north westerly direction at 20s-1. Find the change in these two vectors. Calculation Fig.5. h/20= sine 45° = 0.71 h = 0.71x 20 = 14.2 y is also = 14.2 ( two sides of an isosceles triangle) As x + y = 30 ........ x = (30 – 14.2) = 15.8 ...... Using Pythagoras theorem we get R2 = x2 + x2...............R2 = 14.2 2 +15.8 2 Therefore R = √(249 + 201.64) =√ 450.64.............R = 21.2 Resolving forces When a force acts in a certain direction, we very often have to resolve the vertical and horizontal effects of this.
Horizontal force = F cos 30° and the Vertical force = F sin 30°. If the pulling force is 100 N the horizontal force will be = 100 cos 30°. = 100 x0.87 = 87 N. An example. A man is pulling a log exerting a force of 40 N towards East. His son is pulling at an angle of 30 ° to his line. Find the resulting force. Answer
When you complete the parallelogram , the diagonal becomes the resultant. Or you can use trigonometry by extending the section shown in red. Sin 30 = h/20........... h/20 = 0.5 Therefor h= 10 Tan 30 = h/x .............h/x = 0.58 Therefore x= 10/x = 0.58 = 17.2 Now consider the big triangle with the base 40 + 17.2 Tan ϴ = 10/ 57.2...........0.17 Therefre ϴ = Question 1. 0 Fig. Airplane in flight. 1. to 4. Name the forces shown by the coloured arrows. Select the answers from the given words. 2.0 Which one out of the following cannot be considered as a force? a. Pull b. Kick c. Magnet d. Repulsion e. Attraction (10 marks)
4.0 John and Jerry are pulling a boat. 4.1 Find the total force on the boat. (6 marks) 4.2 using the same forces suggest a way to increase the total pull on the boat ( 6 marks). 5.0Fig.4 Ali and Nelly pulling a cow. Look at this picture. Ali is pulling the cow by a force of 600N. Nelly is using a force of 400N. The angle between the two forces is 60º.
If the cow does not wish to be dragged, what force should it use? For Answers visit .. Answers |
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