The weight of PQ and the tensions at P,Q.
The relation between the three forces acting on any particle, viz, the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a single force-diagram, as shown in fig.
These latter lines measure the tensions in the successive portions of string.
The resistance to slipping of a flat belt on a pulley may be obtained by considering the equilibrium of a small arc of the pulley surface subtending an angle dB at the centre, and having tensions T and T+dT at its extremities.
We note that the tensions have now the same horizontal projection (represented by the dotted line in fig.
The tensions in the successive portions of the string are therefore proportional to the respective lengths, and thelinesBH,CK.., are all equal.
The problem of a rod suspended by strings attached to two points of it is virtually identical, the tensions of the strings taking the place of the reactions of the planes.
9 is balanced by three determinate tensions (or thrusts) in the three links, provided the directions of the latter are not concurrent.
-- T.r-OT If T, T + aT be the tensions at P, Q, and 4 be the angle between the directions of the curve at these points, the components Q of the tensions along the tangent at P give (T + T) cos T,
In the case of a string stretched over a smooth surface, but in other respects free from extraneous force, the tensions at the ends of a small element s must be balanced by the normal reaction of the surface.
The motion being assumed to be small, the tensions of the two strings may be taken to have their statical values Mgb/(a+b), Mga/(a+b), where a, b are the distances of G from the two threads.
Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT =dF =fTdO; which equation, being integrated throughout the entire arc of contact, gives the following formulae:
F = T1 Ti = T1 (I ef9) Ta(ef 1)j When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged.
(20) Hence the tension of a thick film is equal to the sum of the tensions of its two surfaces as already calculated (equation 7).
The tension of the surface separating two liquids which do not mix cannot be deduced by any known method from the tensions of the surfaces of the liquids when separately in contact with air.
The three angles between the tangent planes to the three surfaces of separation at the point 0 are completely determined by the tensions of the b o a three surfaces.
For if in the triangle abc the side ab is taken so as to represent on a given scale the tension of the surface of contact of the fluids a and b, and if the other sides be and ca are taken so as to represent on the same scale the tensions of the surfaces between b and c and between c and a respectively, then the condition of equilibrium at 0 for the corresponding tensions R, P and Q is that the angle ROP shall be the supplement of abc, POQ of bca, and, therefore, QOR of cab.
When three films of the same liquid meet, their tensions are equal, and, therefore, they make angles of 120 with each other.
When a drop of one liquid, B, is placed on the surface of another, A, the phenomena which take place depend on the relative magnitude of the three surface-tensions corresponding to the surface between A and air, between B and air, and between A and B.
If no one of these tensions is greater than the sum of the other two, the drop will assume the form of a lens, the angles which the upper and lower surfaces of the lens make with the free surface of A and with each other being equal to the external angles of the triangle of forces.
But when the surface-tension of A exceeds the sum of the tensions of the surfaces of contact of B with air and with A, it is impossible to construct the triangle of forces, so that equilibrium becomes impossible.
The edge of the drop is drawn out by the surface-tension of A with a force greater than the sum of the tensions of the two surfaces of the drop. The drop, therefore, spreads itself out, with great velocity, over the surface of A till it covers an enormous area, and is reduced to such extreme tenuity that it is not probable that it retains the same properties of surface-tension which it has in a large mass.
This drop will not spread out like the first drop, but will take the form of a flat lens with a distinct circular edge, showing that the surface-tension of what is still apparently pure water is now less than the sum of the tensions of the surfaces separating oil from air and water.
If a and b are the two fluids and c the solid then the equilibrium of the tensions at the point 0 depends only on that of thin components parallel to the surface, because the surface-tensions normal to the surface are balanced by the resistance of the solid.
As an experiment on the angle of contact only gives us the difference of the surface-tensions at the solid surface, we cannot determine their actual value.
If the tension of the surface between the solid and one of the fluids exceeds the sum of the other two tensions, the point of contact will not be in equilibrium, but will be dragged towards the side on which the tension is greatest.
(48) ° and in general the functions 0, or 4), must be regarded as capable of assuming different forms. Under these circumstances there is no limitation upon the values of the interfacial tensions for three fluids, which we may denote by T12, T23, T31.
If the above-mentioned condition be not satisfied, the triangle is imaginary, and the three fluids cannot rest in contact, the two weaker tensions, even if acting in full concert, being incapable of balancing the strongest.
P. 463) deduced relative to the interfacial tensions of three bodies.
The problem is to make the sum of the interfacial tensions a minimum, each tension being proportional to the square of the difference of densities of the two contiguous liquids in question.
The second minus the first, or the increase in the sum of tensions, is thus 2 (U n - o' n+l) (o n+ 1 - Q n +2) Hence, if an+1 be intermediate in magnitude between a,, and a71+2, the sum of tensions is increased by the abolition of the stratum; but, if a-n+1 be not intermediate, the sum is decreased.
For leather belts on cast-iron pulleys the value of may be taken as o 4, giving a ratio of the tensions on the tight and slack sides of Ti/T2= 3.514, when the angle of wrapping is 180°.
We see, then, that the removal of a stratum from between neighbours where it is out of order and its introduction between neighbours where it will be in order is doubly favourable to the reduction of the sum of tensions; and since by a succession of such steps we may arrive at the order of magnitude throughout, we conclude that this is the disposition of minimum tensions and energy.
If this takes place more rapidly on one side of the piece of camphor than on the other side, the surface-tension becomes weaker where there is most camphor in solution, and the lump, being pulled unequally by the surface-tensions, moves off in the direction of the strongest tension, namely, towards the side on which least camphor is dissolved.
November 1890) for the tensions of various watersurf aces at 18° C., reckoned in C. G.
A comparison under similar circumstances shows that there is hardly any difference in the wave-lengths of the patterns obtained with pure and with soapy water, from which we conclude that at this initial stage, the surface-tensions are the same.
If K is the height of the flat surface of the drop, and k that of the point where its tangent plane is vertical, then T = 1(K - k) 2gp. Quincke finds that for several series of substances the surfacetension is nearly proportional to the density, so that if we call Surface-Tensions of Liquids at their Point of Solidification.
When the body is twisted through an angle 0 the threads make angles aC/i, be/I with the vertical, and the moment of the tensions about the vertical through G is accordingly Kg, where K = M gab/i.