MIXED MATHEMATICS FORMULE IN STATICS. R=resultant of P and Q acting at an angle 0, PQR=sin QOR : sin POR : sin POQ. R=resultant of any number of forces acting upon a body in one plane. R2= {Σ (x) } 2 + {Σ (r)}2 : tan 0=2(1) ; in case of equilibrium 2 (x)=0 : 2 (r)=0. Σ(Yx-xy): R=resultant of any number of forces acting upon a l in case of equilibrium Σ (x)=0, Σ (r)=0 : Σ (z)=0 ; Σ (zy-yz)=0; Σ(xz-zx)=0; 2(xx-xy)=0. If x, y, be the coordinates of the centre of gravity of a system of bodies, x=? In the lever, Σ(P.x): y= Pressure on fulcrum=[P2+w2-2 PW cos (a+B)]. Direction of pressure=0; tan 0= In the wheel and axle Single moveable pulley, the strings. P sin a+w sin B P cos a-w cos System of pulleys each hanging by separate string 1 2n {w+(2"−1)w}, n being no. of moveable pulleys. System of pulleys, same string passing round all the pulleys, W+B=n P. B=weight of block. System of pulleys, when all the strings are attached to the weight, w=(2-1) p+(2′′ — n−1)w; n being no. of strings attached to weight. P vertical distance between two threads Screw, with friction— FORMULÆ IN DYNAMICS.. Motion uniformly accelerated v=ft; s= } ft2; v2=2fs; u=v±ƒ; When the body moves freely f=g=32·2: on inclined plane f=g sin a: if P moving on an inclined plane draw Q up an P sin a-Q sin 6 an inclined plane f=" P+Q g. Impact: A moving with velocity a, impinges on в whose velocity is b. Velocity of a after impact-Aa+Bb-Be (a+b) ̧ A+B Aа +вb+ Ae (a+b), Velocity of B after impact= A+B Projectiles in vacuo— Range at end of time t=vt cos a ; height=vt sin a- gt2. Equation to path y=x tan a x2 4h cos2 a 2g A body moving in a circle: accelerating force in Conical pendulum v2-gl sin2 a; t=2′′ COS a mv2 FORMULE IN HYDROSTATICS. Normal pressure on area a2, the depth of whose centre of gravity below surface of fluid is h=weight of volume a3 h of the fluid. Density of air in receiver of air pump after n strokes= (1+1)*P. P tr2 Bramah's press, = W a R2° In diving-bell tension of string = weight of bell-gp Ax where Aarea of top of bell, a length occupied by air, is found from the equation hb (h+a) x+x2, b=length of bell, a depth of its top. (Besant.) FORMULÆ IN PRACTICAL MECHANICS. Unit of work=pressure of 1 lb. exerted through a space of 1 foot. Work done in moving resistance of m lbs. through n feet =m n units. Unit of horse power=33000 units of work. Modulus of a machine= work yielded work expended Work done in moving a body on a horizontal plane=ƒ.ws where f=coefficient of friction, w=weight of body, s=space described. Work done in moving a body up an inclined plane=work due to friction+work due to force of gravity=wh+f. wl: h being the height, the length of plane, whose inclination is very small: otherwise work done=wh+f. w cos a × l. Work accumulated in body, moving with velocity v,= W12 2g Work done in upsetting a heavy body=work requisite to raise the body vertically through the height which its centre of gravity is raised. Work done in raising material of given form=weight of material in lbs. x number of feet through which c. G. is raised. Moment of inertia of a system of bodies =m1r2 1+m2r22+mzr32+ &c.=I. Radius of gyration =k= m 2 m1 r12 + m2 r22+mz rz2+&c. m1+m2 + m3 + &c. If 1=moment of inertia about an axis passing through C. G. I1=moment about an axis at a distance a from former, I1=I+a2m; and on similar suppositions k12=k2+a2..k12—aa is constant. Radius of gyration of wheel= radius of wheel √2 1 Radius of gyration of rod revolving about its middle point == 2√3 6 1. Prove the Triangle of Forces.' 2. A B C is an isosceles triangle, A and B the equal angles, CD a perpendicular from the vertex on the base, take G D=} CD; then if G A, G B represent two forces acting on a point at G, G C will represent the force that will keep the point G at rest. 3. Three forces in the plane of a triangular board act |