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Medicine: Physics: Equilibrium

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### Equilibrium

1.4 Equilibrium

A particle or body is said to be in equilibrium when all the forces acting on it balance and it is no in motion so to say, its resultant moment is zero. Algebraically, this just means that the vector sum of the forces is zero:$\sum&space;F=0$or, equivalently, the components of the vectors in three directions (which must be linearly independent, of course, but not necessarily orthogonal) sum to zero. Geometrically, this means that the vectors representing the forces (in both direction and magnitude) can be joined to form a closed polygon.

In order to determine whether a particle is in equilibrium (or, given that it is in equilibrium, to determine an unknown force) we have to check that the vector sum of the forces, i.e. the resultant force, is zero. That means that the resultant force should have no non-zero component in any direction. Normally, the way to check this is to check the components of the resultant force in three independent directions, which need not be orthogonal but are usually, for convenience, orthogonal. This process is called resolving forces. It can best be understood in a concrete example.

Example

A particle of weight 3W lies on a fixed rough plane inclined at angle α to the horizontal. It is held in position by a force of magnitude T acting up the line of greatest slope of the plane. Find the frictional force, F, in terms of W, α and T

The strategy for all similar problems is to determine the equations of equilibrium by resolving (i.e. taking components of the vectors) forces in two directions and equating to zero. It helps tochoose the directions carefully in order to reduce the number of terms in each equation. Clearly, for our problem, it is a good plan to resolve parallel and perpendicular to the plane.We have, respectively

T = F +W $\sin&space;\alpha$                    (1)

R = W$\cos&space;\alpha$                           (2)

Thus F = T −W $\sin&space;\alpha$, using only the first equation. Normally, we are interesting in finding the value of T that will support the particle on the plane. To accomplish this, we have to know something about the frictional force. The experimental result relating the frictional force to the normal reaction

F = μR                          (3)

is generally used. Here μ is the coefficient of friction, the value of which depends on the surfaces involved. When the equality holds, the friction is said to be limiting.In our example, combining equations (1) and (2) with the experimental law (3) gives

$T\leq&space;W(\sin&space;\alpha$$\cos&space;\alpha&space;)$

Note that in the case of limiting friction, T is determined by this equation. If, instead of assuming that the particle is tending to slip up the plane, we assume that it is tending to slip down the plane, then the frictional force would act up the plane. In this case (check this!) we find

$T\geq&space;W(\sin&space;\alpha&space;-$μ$\cos&space;\alpha&space;)$

and combining the two results gives the range of values of $T$ for which equilibrium is possible, for a given value of μ:

$W(\sin&space;a-$μ$\cos&space;\alpha&space;)\leq&space;T\leq&space;W(\sin&space;\alpha&space;+$μ$\cos&space;\alpha&space;)$

Not surprisingly, in order for the particle to remain in equilibrium  i.e. not move T cannot be too big or too small. Note that if $T$ were given (for example, if it were the tension in a string that passes over a pulley and has a weight dangling on the other end) the above equation would give bounds on the values of α allowed for equilibrium.

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