INCLINED PLANES.] 



MECHANICAL PHILOSOPHY. DYNAMICS. 



737 



3. With what velocity must a body be projected down 

 wards from the top of a tower 150 feet high, to arrive a 

 the bottom in two seconds ? 



= r' (- Jjjrf 2 .'. v 1 = ^gt 



= - 16-1 X 2 = 42-8 feet. 

 2 



4. If a body be projected vertically downwards with a 

 velocity of 20 feet, how far will it descend in 5 seconds " 

 Ans. 505'5 feet. 



5. A body is projected upwards with a velocity of 10( 

 feet : what will be its velocity when it has ascended 10( 

 feet ? (See Form 3.) Ans. 51 '6 feet. 



6. In the preceding example, find at what time the 

 body is 100 feet from the earth, as well in its desceni 

 as in its ascent. Ans. 1-2 sec. and 4 '7 sees, after pro- 

 jection. 



MOTIONS OF BODIES DOWN INCLINED 

 PLANES. The fall of a body down an inclined plane 

 is another instance of rectilinear motion under the in- 

 fluence of gravity. When a material substance is placed 

 on au inclined plane, the force of gravity produces a 

 Fig. 1S9. certain vertical pressure 



P (Fig. 139) : if we re- 

 solve this pressure in 

 two directions, the one 

 along the plane and the 

 / ""v. other perpendicular to 



s ^ it, the former com- 



ponent will be P sin. i, 

 P taking t for the incli- 



nation of the plane to the horizon ; and to prevent the 

 body from moving down, thia is the force or pressure 

 that must be counterbalanced. 



As, therefore, P represents the pressure-force of 

 gravity on the body in a vertical direction, and P sin. i 

 the pressure in the direction of the plane, and as, when 

 motion takes the place of these statical pressures, the 

 accelerating force of gravity is proportional to them, we 

 shall have for the acceleration down the plane, 



P : P sin. i : : g : g sin. i. 



Hence the body is urged down the inclined plane by 

 the constant force, 



g'=g sin. i (1) 



And, therefore, substituting this value of g' for / in the 

 formula at page 736, those formula will then comprise 

 the entire theory of motion down an inclined plane. 

 If I represent the length of the plane, and h its height, 



then sin. i = : consequently the accelerating force down 



the plane is 

 /= 



I 



'J 



(2) 



And, therefore, the velocity acquired in descending down 

 the whole length I, that is, in descending through the 

 space s=l, by the action of this force, must be (page 736) 



..... (2) 



This expression we see involves only the height h of 

 the plane, and is independent of its length I : hence the 

 velocity acquired in descending down all planes of the 

 same height is the same, and equal to the velocity 

 acquired by falling vertically through that height. 



But the velocities of two bodies, one falling through 

 the perpendicular height, and one falling down the 

 length of the plane, are respectively 



V"gt and c'=gt' sin. i .... (4) 



where t and f are the respective times occupied in falling : 

 these expressions are therefore equal ; that is, 



., . . . t sin. i 

 gl=rjV sin. t. . _=_ - 



So that the time of falling through the height is to the 

 time of falling down the plane, as the sine of the plane's 

 inclination to unity. 



If we wish to know what extent of length down the 



VOL. L. 



plane a body will pass through, while another falls 

 through the whole height, then, referring to the expression 

 for the space (page 736), we have 



Vertical fall, h= J gf; inclined fall in time (, s= i g sin. i < 2 ; 

 .'. h : s : : 1 : sin. i, or s=h sin. i. 



If, therefore, from B we draw the perpendicular B D, 

 the length A D will be that fallen through by one body 

 moving down the plane AC, while another body falls 

 through the height A B, because 



AD=AB cos. A=ABsin. 0. 

 If we draw the vertical D B', and B B' perpendicular to 



Fig. MO. 



D B, then, by this theorem, the time of falling down the 

 oblique line D B would equal the time of falling down 

 the vertical D B', for D B = D B' cos. D ; but D B A B, 

 since A B is a parallelogram, therefore the time of falling 

 down the vertical A B is the same as the time of falling 

 down either of the oblique rig. HI. 



lines AD, D B at right 

 angles "to one another : 

 hence this remarkable 

 property of the circle, 

 namely : If from the ex- 

 tremities A, B, (Fig. 141) 

 of the vertical diameter 

 A B, chords be drawn, as 

 in the annexed diagram, 

 a body would fall down 

 either of them in the same 

 time that it would fall 

 through the vertical dia- 

 meter A B. 



The following are a few examples on the motion of 

 bodies down inclined planes : 



1. If the length of an inclined plane be 60 feet, and 

 its inclination to the horizon 30, what velocity would a 

 body acquire in falling down it for two seconds ? 



By the formula (4), since sin. 30=^-, we have vgt 

 sin. i = 32-2 X 2 X i = 32-2 feet per sec. Hence the 

 velocity is the same as would be acquired by a vertical 

 fall in one second. 



2. How long would a body be in falling down an in- 

 clined plane whose length is 100 feet, and inclination 60"? 



Substituting the acceleration g sin. t for /, and I for s, 

 in the expression for t, at page 736, and remembering 

 that sin. 60 = i J 3, we have 



_ 2 1_ /_ 200 

 0sin. i~V^ 32-2 x | */3~ 

 20 01 , 



400 



32-2 X V 3 



- seconds. 



3. A body is projected up an inclined plane whoso 

 height is ith of its length, with a velocity of 50 feet. 

 ?ind its place and velocity after six seconds have elapsed. 



In this case the force g (equation 2) retards the motion 



of the body, and must therefore be considered as nega- 

 ,ive : hence, from equation (2) page 736, we have 



- 't i h f 



= 50 X 6 16-1 x JX 30=6(50 16-1) = 203-4 feet. 



h 

 Also v = v' g j t 



= 50 32-2 = 17-8 feet, the velocity required. 



5 B 



