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# 2. velo & accln. tom

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### 2. velo & accln. tom

1. 1. CHAPTER – 2 : VELOCITY & ACCELERATION DIAGRAM LINEAR VELOCITY : The rate of change of linear displacement of body w.r.t. time is known as linear velocity. It is expressed by V or U. Linear velocity V = ds/dt It the displacement is along the circular path, the direction of velocity at any instant is along the tangent at that point. Fig. 2.1 LINEAR ACCELERATION : Rate of change of velocity w.r.t. time is linear acceleration. It is expressed by a. Linear acceleration a = dv/dt = d(ds/dt) dt = d2 s/dt2 Linear acceleration is also expressed as a = dv = ds x dv dt dt ds = V x dv ds ANGULAR DISPLACEMENT : It is defined as the angle described by a particle from one point to another w.r.t. time. As shown in fig. 2.2, when one line making angle 01 with x-axis moves through angle 02 with x-axis during short interval of time t2-t1 & making new position. Than 02-01 is known as the angular displacement of line. Fig. 2.2 Theory of Machine (1MEB25) Prepared By : Prof. Jignesh Dangi
2. 2. CHAPTER – 2 : VELOCITY & ACCELERATION DIAGRAM ANGULAR VELOCITY : It is defined as the rate of change of angular displacement w.r.t. time. It is expressed by w. Angular velocity = d0/dt Note : If the direction of angular displacement is constant, it is known as angular speed. ANGULAR ACCELERATION : Rate of change of angular velocity w.r.t. time is defined as angular acceleration. It is expressed by o . Angular acceleration = dw = d(d0) = d2 0 dt dt(dt) dt2 RELATION BETWEEN ANGULAR VELOCITY & ROTATIONAL SPEED : If a body is rotating at the rate of N r.p.m., then the relation between angular speed & rotational speed is given as w = 2 II N rad/sec. 60 RELATION BETWEEN LINEAR & ANGULAR MOTION : Consider a body moving along a circular path from A to B as shown in fig. 2.3. Let, r = Radius of circular path 0 = Angular displacement in radians S = Linear displacement w = Angular velocity a = Linear acceleration o = Angular acceleration Fig. 2.3 From geometry of fig. 2.3, it is know that S = r 0 V = ds = d(r0) = r d0 = r . w ----------------------- (2.1) dt dt dt Theory of Machine (1MEB25) Prepared By : Prof. Jignesh Dangi
3. 3. CHAPTER – 2 : VELOCITY & ACCELERATION DIAGRAM a = dv = d(rw) = r dw = r . o ---------------------- (2.2) dt dt dt MOTION OF PARTICLE ALONG A CIRCULAR PATH : Fig. 2.4 Consider P & Q, the two positions of particle displaced through an angle d0 in time dt. Now, V = Velocity of particle at P r = Radius of circular path V+dV = Velocity of particle at Q The change of velocity as the particle moves from P to Q obtained by drawing vector triangle OPQ as shown in fig. 2.4. Here OP represent velocity V & OQ represent velocity V+dV. The change of velocity in time dt is represented by PQ. a) Tangential component of acceleration : Velocity of P perpendicular to OP = V Velocity of Q perpendicular to OP = (V+dV)cosd0 Change of velocity = (V+dV)cosd0 – V Acceleration of P perpendicular to OP = (V+dV)cosd0 – V dt In the limit, as dt >0, cosd0 >1, So, tangential acceleration of component at = dv/dt = o . r ----------------------- (2.3) b) Centripetal or radial component of acceleration : Velocity of P parallel to OP = 0 Velocity of Q parallel to OP = (V+dV)sind0 Change of velocity = (V+dV)sind0 – 0 Acceleration of P parallel to OP = (V+dV)sind0 dt In the limit, as dt >0, sind0 >d0 So, centripetal/radial acceleration of component ar = V.d0/dt = V.w = (w.r).w = w2 .r ----------------------- (2.4) Theory of Machine (1MEB25) Prepared By : Prof. Jignesh Dangi P Q V r V+dVCOSd0 SINd0d0 d0
4. 4. CHAPTER – 2 : VELOCITY & ACCELERATION DIAGRAM RELATIVE VELOCITY : To find relative motion between two bodies moving parallel or making some angle with each other, vector method is used. Va A Vb B Va Vb o a b Vab Vba Fig. 2.5 As shown in fig. 2.5. body A & B both are moving parallel to each other in same direction. Body A moves with velocity Va & B moves with velocity Vb. Here Va>Vb. Now relative velocity of A w.r.t. B = Vab = Va – Vb or ba = oa – ob. Similarly, Vba = Vb – Va or ab = ob – oa. As shown above relative velocity of A w.r.t. B is Vab & relative velocity of B w.r.t. A is Vba. Magnitude of both Vab & Vba will be same but direction will be different i.e. Vab = -Vba. VELOCITY & ACCELERATION IN SLIDER CRANK MECHANISM : As shown in fig. 2.6 slider C is connected to connecting rod BC & connecting rod BC is connected to crank AB at the point B. Let the crank radius is r, rotating about point A with uniform angular velocity w rad/sec in clockwise direction. Slider C reciprocate along the line of stroke AD. Here the magnitude & direction of velocity of B (Vb) is known. Fig. 2.6 Velocity : Using relative velocity method, velocity of slider C (Vc) can be determined. As shown in fig. 2.6(a), the velocity diagram is drawn by following step by step procedure. Vac a,d C Ve Vab e Vbc b Fig. 2.6(a) : Velocity Diagram Theory of Machine (1MEB25) Prepared By : Prof. Jignesh Dangi E