EXAMPLES 73 



3. A balance has two pans, each of weight 8 ounces, suspended from a beam, 

 each at distance 7 inches from the pivot. A dishonest tradesman moves one 

 pan half an inch nearer to the pivot, adding weight to this same pan in order 

 that the two pans may still balance. Find how much weight he must add, and 

 by how much his profits will be increased by his dishonesty. 



4. A balance has a weight of 20 ounces suspended from one end of the beam. 

 A string is tied to the other end of the beam and at equal distance from the 

 pivot, and this string makes an angle of 45 with the horizontal. With what force 

 must it be pulled to maintain the beam of the balance in a horizontal position ? 



5. A crowbar 8 feet long is to be used to move a body, it being required to 

 apply a force of 500 pounds weight vertically upwards to this body to move it. 

 How near to the end of the crowbar must the fulcrum be placed in order that a 

 man of 140 pounds weight may be able to apply the required force by pressing 

 on the other end of the crowbar ? 



6. A table of negligible weight has any number of legs. A heavy particle is 

 placed on the table. Show that the table will tilt over if the vertical through 

 the particle meets the floor under the table, in a point outside the polygon 

 formed by joining the points of contact of the feet with the floor. 



7. A table of negligible weight has three legs, the feet forming an equilateral 

 triangle. A heavy particle is placed on the table in a position such that the table 

 does not tilt over. Find the proportion of weight which is carried by each foot. 



8. A card is suspended in a horizontal position by three equal inextensible 

 strings fastened to three points A, B, C in the card which form an equilateral 

 triangle, and also to a point P above the card. A weight is placed on the card 

 at any point Q inside the triangle ABC. Find the tensions of the strings. 



9. A card is suspended by four equal inextensible strings which pass through 

 the four points A, B, (7, D of a square in the card and are tied to four points 

 A', B', C', D' at equal heights h vertically above the points A, B, C, D. A 

 weight is placed on the card at any point P inside the square ABCD. Show 

 that the tensions in the strings cannot be determined without discussing the 

 internal stresses in these strings. 



10. If in the last question the internal stresses in the strings stretch the 

 strings very slightly, so that Hooke's law is obeyed, show that the tensions can 

 be found, and find them. 



11. A seesaw rolls on a rough circular log of radius a, fixed horizontally. ** 

 Two persons stand at distances 6, c from the middle point of the seesaw, their 

 weights being such that the seesaw is just balanced horizontally with the 

 middle point resting on the log. The first person moves a distance d towards 

 the center of the log. Through what angle will the seesaw turn ? How far 

 can the person advance before the seesaw slips off the log altogether ? 



12. A pair of wheels of radius a are connected by an axle of radius 6, and run 

 on horizontal rails. A string is wound round the axle and the end leaves the 

 axle making an angle 6 with the horizontal. If this string is pulled, show that 

 the wheels will run toward or away from the person pulling according as 

 cos 6 is greater or less than b/a. What happens when cos 6 b/a ? 



