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Revision:Vectors In Mechanics

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Vectors in Mechanics

These notes are based on the requirements of the M5 A Level mathematics module.


Contents

Solution of simple vector differential equations

It’s the same as in previous modules (M3 and M4), where you had to solve scalar differential equations, but here you have vectors: all what you have to do is substitute

\displaystyle \mathbf{v} = u\mathbf{i} + w\mathbf{j}

where u an w are scalars, so

\displaystyle \frac{d\mathbf{v}}{dt} = \frac{du}{dt} \mathbf{i} + \frac{dw}{dt} \mathbf{j}, and

\displaystyle \frac{d^2\mathbf{v}}{dt^2} = \frac{d^2u}{dt^2}\mathbf{i} + \frac{d^2w}{dt^2}\mathbf{j}.

Now equate coefficients of i and j, solve a if it is a scalar equation, hence you can find u and w.


Work done by a constant force

\displaystyle \mathsf{Work\ done} = \mathbf{F.d}

All what you have to do is to find the scalar product of the Force vector and the Displacement vector. Of course to find the distance: position vector of final point - position vector of initial point.


Vector moment of a force

\displaystyle \mathsf{Vector\ moment\ of\ a\ force} = \mathbf{r} \times \mathbf{F}

Where r is the position vector of any point on the line of action of F relative to the point where the moment is to be taken about.


Resultant Force and Couples

\displaystyle \mathsf{Resultant\ force} = \sum_{i = 1}^n F_i

\displaystyle \mathsf{Couples\ of\ moment\ G} = \sum_{i = 1}^n r_i \times F_i


If a system of forces can be reduced to a resultant force, then G = 0

If a system of forces can be reduced into a couple of moment, then FR = 0


A system s I equilibrium when both G and FR are equal to zero.

It doesn’t matter about what point the resultant force is about. It’s the same about any point.


Comments

Originally written by yazan_l on TSR forums.

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