Engineering Tutorials

Torque due to a force: As discussed earlier, torque about a point due to a force is obtained as the vector product

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Equilibrium of Bodies

In the static part when we say that a body is in equilibrium, what we mean is that the body is not moving at all even though there may be forces acting on it. (In general equilibrium means that there is no acceleration i.e., the body is moving with constant velocity but in this special case we take this constant to be zero).

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Derivative of a vector: After reviewing the vector algebra, we would now like to introduce you to the idea of differentiating a vector quantity. Here we take a vector as depending on one parameter, say time t , and evaluate the derivative . This is similar to what we do for a regular function. We evaluate the vector at time (t+ Δt) , subtract from it, divide the difference by Δt and then take the limit Δ t → 0 .

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Geometric interpretation of cross product : The magnitude of the cross-product , which is equal to , is the area of a parallelogram formed by vectors . This is shown in figure.

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Vector or cross product: In defining the scalar product above, we have used three out of the nine possible products of the components of two vectors. From the six of these that are left i.e. , if we define the vector

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Scalar or dot product:

Now it is easy to show that is a scalar quantity. To show this we calculate this quantity in a rotated frame (rotation could be about the x, y or the z axis) that is obtain and show that it is equal to . As an example we show it for a frame rotated about the z-axis with respect to the other one. In this case

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Analytical Method for Addition and Subtraction of Vectors

Although graphical way is nice to visualize vectors in two dimensions, it becomes difficult to work with it in three dimensions, and also when many vectors and many operations with them are involved. So vector algebra is best done by representing them in terms of their components along the x, y & z axes in space. We now discuss how to this is done.

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Example1: A person walks 300m to the east and 400m to the north to reach his friend’s house. What is the total displacement of the person, and what is the total distance traveled by him?

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Adding and subtracting two vectors (Graphical Method):

When we add two vectors and by graphical method to get , we take vector , put the tail of on the head of .Then we draw a vector from the tail of to the head of . That vector represents the resultant (Figure 4). I leave it as an exercise for you to show that . In other words, show that vector addition is commutative.

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Introduction and Equality of Vectors

You have learnt in the past is that vectors are quantity which have both a magnitude and a direction in contrast to scalar quantities that are specified by their magnitude only. Thus a quantity like force is a vector quantity because when I tell someone that I am applying X- amount of force, by itself it is not meaningful unless I also specify in which direction I am applying this force. Similarly when I ask you where your friend’s house is you can’t just tell me that it is some 500 meters far. You will also have to tell me that it is 500 meters to the north or 300 meters to the east and four hundred meters to the north from here. Without formally realizing it, you are telling me a about a vector quantity. Thus quantities like displacement, velocity, acceleration, force are vectors. On the other hand the quantities distance, speed and energy are scalar quantities. In the following we discuss the algebra involving vector quantities. We begin with a discussion of the equality of vectors.

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