Definition
A locus (plural foci) is tha path traced out by a point which moves under given definite conditions
You may not have been aware ot it but you have met loo meny times before. One of the most common loci is that of a point which moves so that its distance from another fixed point remains constant, this produces a circle Another locus that you know is that ot a point which moves so that its distance from a line remains constant: this produces parallel lines.
Problems on loci can take several different forms. One important practical application is finding the path traced out by points on mechanisms This may be done simply to see If there is sufficient cleaiance around a mechanism or. with further knowledge beyond the scope of this book, to determine the velocity and hence the forces acting upon a component
There are very few rules to learn about loci; it is mainly a subject for common sense A locus is formed by continuous movement and you have to 'stop' the movement several times and find and plot the position of the point that you are interested in. Take, for instance, the case ol the men who was too lary to put wedges under his ladder The inevitable happened and the ladder slipped. The path that the feet ot the man took is shown in Fig 9/1.
The top of the ladder slips from T to Tr The motion of the top of the ladder has been stopped at T„ T* T* etc. end. since the length of the ladder remains constant, the corresponding positions of the bottom of the ladder. B,. B;. B y etc. can be found. The positions of the ladder T,B,. T,B,. T,Betc are drawn and the position of the man's feet 1. 2. 3, etc.. are marked. The points are joined together with a smooth curve. It is interesting to note that the man hits the ground at right angles (assuming that he remains on the ladder) The resulting jar often causes serious injury and is one of the reasons for using chocks Another simple example is the locus ol the end of a
Fig 9/1
Fig 9/1
bureau door stay. Fig. 9/2. This type of stay is also often used on wardrobe doors, lis function is to allow the door to open to a certain point, and then to support the door in that position.
The stay, of course, has two ends and the locus on one end is essily found: it is en arc whose centre is the hinge. The other end of the stay is allowed to slide through the pin but it is not allowed to move off it. As the end of the stay moves along the arc. its movement is stopped several times and the position of the other end of the stay is marked. These points are joined together with a smooth curve. Obviously the designer of such a bureau would have to plot this locus before deciding the depth of the bureau.
Loci of mechanisms
The bureau door stay is a very simple mechanism. We now look at some of the loci that can be found on the moving parts of some machines.
Velocity is speed in a given direction. It is a term usuelly reserved for inanimate objects; we talk about the muzzle velocity of a rifle or the escape velocity of a space probe When we use the word speed we refer only to the rate of motion. When we use the word velocity we refer to the rate of motion and the direction of the motion.
Linear velocity is velocity elong a straight line (a linear graph is a straight line).
Angular velocity is movement through a certain angle in a certain time. It makes no allowance for distance travel • led. If, as in Fig. 9/3, a point P moves through 60" in 1 second, its angular velocity is exactly the same as that of 0. providing that Q also travels through 6CP in 1 second. The velocity, as distinct from the angular velocity, will be much greater of course.
Constant velocity, linear or angular, is movement without accelaration or deceleration
The piston/connecting rod/crank mechanism is very widely used, pnncipelly because of its application in internal combustion engines The piston travels in a straight line: the crank rotates The connecting rod.
straight line: the crank rotates The connecting rod.
be 'stopped' several times and the positions of the centre of the connecting rod found. As with most machines that have cranks, the best policy a to plot the position of the crank in twelve equally spaced positions. This is easily achieved with a 60" set square The piston must always lie on the centre line and. of course, the connecting rod cannot change its length. It is therefore a simple matter to plot the position of the connecting rod for the twelve positions of the crank This is best done with compasses or dividers. The mid-point of the connecting rod can then be marked with dividers and the points joined together with a smooth curve.
The direction of rotation of the crank is usually given m problems of this nature. It may make no difference to plotting any of the loci but it could make a tremendous difference to the functioning of the real machine: what good is a car that does 120 km/h backwards and 10 km/h forwards 7
A trammel can consist of a piece of paper or a piece of card or even the edge of a set square It must have a straight edge and you must be able to mark It with a pencil. A trammel enables you to plot a locus more quickty than the method shown above However, if you are intending to sit the G.C.E. or C.S.E. examination check that the syllabus allows you to use trammels.
Fig. 9/5 shows a crank/rod/slider mechanism where a point | along the rod has been plotted for one revolution of the crank. The length of the rod. and the point, are marked on a piece of paper One end of the rod is constrained to travel around the crank circle and the other slides up and down the centre line of the slider. Move the trammel so that one end is always on the circle whilst the other end is always on the slider centre line, marking the required point as many times as necessary. Join the points with a smooth curve.
On mechanism» that ere more complicated, it is sometimes necessary to plot one locus to obtain another Fig. 9/6 shows a mechanism which consists of two cranks. 0,A and 0,B. and two links. AB and CD. It is required to plot the locus of P. a point on the lower end of the hnk CO
Before we can plot any positions of the link CD. end hence P. we must know the position of C at any given moment. This can only be done by plotting the locus of C. ignoring the link CD. Once this has been done, we can find the position of CD at any given moment, and hence the locus of point P.
No construction lines are shown in Fig. 9/6 since they would only make the drawing even more confusing I The locus of P could have been found by stopping' one of the cranks in twelve different places and finding the twelve new positions of the link AB This, in turn, would enable you to find twelve positions for C. end then twelve positions of the rod CD. Finally this would lead to the twelve required positions of P. Alternatively, the locus of C could have been plotted with a trammel which had the length of the link AB. and the position of C marked on it. Another trammel, with the length of the rod CD and the position of P marked on it would have given the locus of P.
Some Other Probiema in Loci A locus is defined as the path traced out by a point which moves under given definite conditions. Three examples of loci are shown below where a point moves relative to another point or to lines
To plot the locua of ■ point P which moves so thst Its distance from a point S and a line X-Y ia always the aeme (fig. 9/7)
The first point to plot is the one that lies between S and the line. Since S is 20 mm from the line, and P is equidistant from both, this first point is 10 mm from both.
If we now let the point P be 20 mm from S. It will lie somewhere on the circumference of a circle, centre S. radius 20 mm. Since the point is equidistant from the line, it must also lie on a line drawn parallel to X-Y and 20 mm away. The second point then, is the intersection of the 20 mm radius arc and the parallel line.
The third point is et the intersection of an arc, radius 30 mm and centre S. and a line drawn parallel to X-Y and 30 mm away.
The fourth point is 40 mm from both the line end the point S. This may be continued for as long as is required.
The curve produced is a parabola.
To plot the locu« of a point P which moves so that its distance from two fixad point* R and S. 50 mm apart, is always in the ratio 2:1 respectively (Fig 9/8)
As in the previous example, the first point to plot is the one that lies between R and S- Since It is twice as far away from R as it is from S. this is done by proportional division of the line RS.
If wo now let P be 40 mm from R it must be 20 mm from S. Thus, the second position of P is at the intersection of an arc. centre R. radius 40 mm and another arc. centre S and radius 20 mm.
The third position of P is the intersection of arcs, radii 60 mm and 25 mm. centres R and S respectively.
This is continued for es long as necessary. In this case, at a point 100 mm from R and 50 mm from S, the locus meets itself to form a circle.
To plot the locus of a point P which moves so that its distance from the circumference of two circles, centres 0, and 0, and radii 20 mm and 15 mm respectively, is alwaye in the ratio 2:3 respectively (Fig. 9/9)
As with the two previous exemples. the first point to plot is the one that lies between the two circles. Thus, divide the spece between the two circumferences in the ratio 3:2 by proportional division.
If we now let P be 10 mm from the circumference of the circle, Centre O,. it will lie somewhere on a circle, centre 0,. and radius 30 mm If it is 10 mm from the circumference of the circle, centre 0,, it will be 15 mm from the circumference of the circle, centre 0,. since the ratio of the distances of P from the circumferences of the circles is 3:2. Thus, the second position of P is the intersection of two arcs, radii 30 mm and 30 mm. centres O, and 0a.
Exercises 9
(All questions originally set in Imperial units) . Fig. 1 shows e door stay as used on a wardrobe door. The door is shown in the fully open position Draw. luH tit», the locus of end A of the stay as the door closes to the fully closed position The stay need only be shown diagrammatically as in Fig. 1 A. Wett Midlands Examinations Board |--15-0-J
DIMENSIONS IN *« 2 Fig. 2 shows a sketch of the working pans, sod the working parts represented by lines, of a moped engine. Using the line diagram only, and drawing in single lines only, plot, full sue. the locus of the point P for one full turn of the crank BC.
Do not attempt to draw the detail shown in the sketch
Show all construction.
The trammel method must not be used.
East Anghan Examinations Board
3. In Fig 3 the crank C rotates in a clockwise direction The rod P8 is connected to the crank at B and slides through the pivot D.
Plot, to a scale 1 { full site, the locus of P for one revolution of the crank.
South ■ East Regional Examinations Board
4. In Fig. 4 the stay BHA is pin-pointed at H and is free to rotate about the fixed point B Plot the locus of P as end A moves from A to A'. North Western Secondary School Examinations Board
DIMENSIONS IN i»m 5 In Fig. 5. rollers 1 and 2 are attached to the angled rod. Roller 1 slides along slot AB while roller 2 slides along CD end back.
Draw, fuU site, the locus of P. the end of the rod. for the complete movement of roller 1 from A to B South East Regional Examinations Board
6 As an expenment a very low gear has been fitted to a bicycle. This gear allows the bicycle to move forward 50 mm for every 15 dog roes rotation of the crank and pedal. These details are shown in Fig. 6.
(a) Oraw. half full size, the crartk and pedal in position as it rotates for every 50 mm forward motion of the bicycle up to a distance of 600 mm. The first forward position has been shown on the drawing
(b) Draw a smooth freehand curve through the positions of the pedal which you have plotted
(c) From your drawing find the angle of the crank OA to the horizontal when the bicycle has moved forward 255 mm.
Metropolitan Regional Examinations Board ^fCOAL
PosiroN or rcoA«. AND CAAN« AFT»« Ö'O» ROTTAT*
TORWMCO MCT«0»i C.eiCYCUi
7. Fig. 7 is a line diagram of a slotted link and crank of • shaping machine mechanism. The link AC is attached to a fixed point A about which it is free to move about the fixed point on the disc. The di9C rotates about centre 0. Attached to C and free to move easily about the points C and 0 is the link CO. D is also free to slide along OE.
When the disc rotates in the direction of the arrow, plot the locus ol C. the locus of P on the link CO. and clearly show the full travel of B on AC. Southern Universities'Joint Board
8. In Fig. 8, MP and NP are rods hinged at P. and A and 0 are guides through which MP and NP are allowed to move. D is allowed to move along BC. but rod NP is always perpendicular to BC. The guide A is allowed to rotate about its fixed point. Draw the locus of P above AB for all positions and when P is always equidistant from A and BC. This locus is part of a recognised curve. Name the curve and the parts used in its construction Southern Universities' Joint Board
9. In the mechanism shown in Fig 9. OA rotates about 0. PC is pivoted at P, and OB is pivoted at 0 BCDE is a rigid hnk. OA - PC - CD - DE - 25 mm. BC — 37.5 mm. QB - 50 mm and AD — 75 mm. Plot the complete locus of E.
Oxford and Cambridge Schools Examinations Board
OIMENSIONS iH mm
Fig 9
OIMENSIONS iH mm
Fig 9
10. A rod AB 70 mm long rotates at a uniform rate about end A Plot the path of a slider S, initially coincident with A. which slides along the rod. at a uniform rate, from A to B and back to A during one complete revolution of the rod
Jornt Matriculation Board
11. With a permanent base of 100 mm. draw the locus of the vertices of all the triangles with a constant perimeter ol 225 mm.
12. Three circles lie in a plane in the positions shown in Fig. 10. Draw the givon figure and plot the locus of a point which moves so that it is always equidistant from the circumferences of circles A and B Plot also the locus of a point which moves in like manner between circles A and C.
Finally draw a circle whose circumference touches the circles A. B and C and measure and state its diameter
Cambridge Local Examinations Board
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