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CHAPTER-9 Center of Mass and Linear Momentum

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Ch 9-2 The Center of Mass Center of mass (com) of a system of particles is required to describe the position and motion of the system The com of a system of particles is the point that moves as if the entire mass of the system were concentrated there and all external forces were applied there

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Ch 9-2 The Center of Mass com is defined with reference to origin of an axis For Fig. b X com =(m 1 x 1 +m 2 x 2 )/(m 1 +m 2 ) =(m 1 x 1 +m 2 x 2 )/M where M= m 1 +m 2 = m i x com = m i x i / M= m i x i / m i y com = m i y i /M; z com = m i y i /M r com = x com i +y com j +z com k = m i r i / M

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Ch 9-2 The Center of Mass (solid bodies) For a solid body having continuous distribution of matter, the particle becomes differential mass element dm x com =(1/M) x dm ; y com =(1/M) y dm ; z com =(1/M) z dm M is mass of the object and its density are related to its volume through = M/V =dm/dV Then x com =(1/V) x dV ; y com =(1/V) y dV ; z com =(1/V) z dV

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Ch-9 Check Point 1 The figure shows a uniform square plate from which four identical squares at the corners will be removed a) where is com of plate originally? b) where it is after removal of square 1 c) after removal of square 1 and 2 d) after removal of square 1 and 3 e) after removal of square 1, 2 and 3 f) All four square Answers in term of Quadrant, axes or points (a) origin; (b) fourth quadrant; (c) on y axis below origin; (d) origin; (e) third quadrant; (f) origin

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Ch 9-3 Newton’s Second Law for a System of Particles For a system of particles with com defined by: r com = m i r i / M, Newton’s Second law : F net =M a com Mr com = m i r i ; differentiating w.r.t time Mv com = m i v i ; differentiating w.r.t time Ma com = m i a i = F i = F net F net-x = Ma com-x ; F net-y = Ma com-y ; F net-z = Ma com-z

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Ch 9-4,5 Linear Momentum Linear momentum of a particle p- a vector quantity p = mv (linear momentum of a particle) Newton’s second law of motion: Time rate change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of force F net = d/dt (p) = d/dt (mv) = m dv/dt = ma Linear momentum P of a system of particles is vector sum of individuals particle’s linear momenta p P = p i = m i v i =Mv com (System of particles) F net = d/dt (P)= Ma com (System of particles)

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Ch-9 Check Point 3 The figure gives the magnitude of the linear momentum versus time t for a particle moving along an axis. A force directed along the axis acts on the particle. (a)Rank the four regions indicated according to the magnitude of the force, greatest first b) In which region particle is slowing F= dp/dt Region 1: largest slope 2: Zero slope 3: Negative slope 4: Zero slopes Ans: (a)1, 3, then 2 and 4 tie (zero force); (b) 3

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Ch 9-6 Collision and Impulse Momentum p of any point- like object can be changed by application of an external force Single collision of a moving particle-like object (projectile) with another body ( target) Ball (Projectile-R) – bat (target-L) system Change in momentum of ball in time dt, dp=F(t) dt. Net change dp= F(t) dt. Impulse J= F(t) dt. = F avg t Change in momentum P =P f -P i =J

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Ch 9-7 Conservation of Linear Momentum If F net-external = 0 then J= F(t) dt =0 P =P f -P i =J=0; then P f = P i Law of conservation of linear momentum: If no net external force acts on a system of particles, the total linear momentum of the system of particles cannot change. P fx = P ix ; P fy = P iy

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Ch-9 Check Point 4 A paratrooper whose chute fails to open lands in snow, he slightly hurt. Had he landed on bare ground, the stoppong time would be 10 times shorter and the collision lethal. Does the presence of snow increases, decreases or leaves unchanged the values of (a) the paratrooper change in momentum (b) the impulse stopping the paratrooper C) the force stopping the paratrooper Answer: (a) p =m(vf-vi) unchanged; (b) J= p ; unchanged; (c) J= F.dt ; t increase, F decrease

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Ch-9 Check Point 5 The figure shows an overhead view of a ball bouncing from a vertical wall without any change in its speed. Consider the change p in the balls’ linear momentum a) Is px positive, negative, or zero b) Is px positive, negative, or zero b) What is direction of p? x p x =p xf -p xi = 0 p y =p yf -p yi =p yf -(-p yi ) =positive Direction of P towards y-axis

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Ch-9 Check Point 6 An initially stationary device lying on a frictionless floor explodes into two pieces, which then slides across the floor. One piece slides in the positive direction of an x axis. a) What is the sum of the momenta of the two pieces after the explosion? b) Can the second piece move at an angle to the x-axis? c) What is the direction of the momentum of the second piece? a) P i = P f =0 b) No because P f = 0=P 1fx +P 2 Then P 2 =-P 1fx c) Negative x-axis

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Ch 9-8 Momentum and Kinetic Energy in collisions Elastic collision: Momentum and kinetic energy of the system is conserved then P f = P i and K f = K i Inelastic collision: Momentum of the system is conserved but kinetic energy of the system is not conserved then P f = P i and K f K i Completely inelastic collision: Momentum of the system is conserved but kinetic energy of the system is not conserved. After the collision the colliding bodies stick together and moves as a one body.

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Ch-9-9 Inelastic Collision in one Dimension One-dimensional inelastic collision For a two body system Total momentum P i before collision = Total momentum P f after collision p 1i +p 2i =p 1f +p 2f m 1 v 1i +m 2 v 2i =m 1 v 1f +m 2 v 2f One-dimensional completely inelastic collision (v 2i =0) m 1 v 1i =(m 1 +m 2 )V m 1 v 1i /(m 1 +m 2 ) Hence V v 1i [m 1 (m 1 +m 2 )]

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Ch-9-9: Velocity of the Center of Mass For One-dimensional completely inelastic collision of a two body system P=(m 1 +m 2 )v com P i =P f and m 1 v 1i =(m 1 +m 2 )V P i =(m 1 +m 2 )v com = m 1 v 1i and P f =(m 1 +m 2 )v com =(m 1 +m 2 )V v com = m 1 v 1i /(m 1 +m 2 )= V V com has a constant speed

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Sample-Problem-9-8 Ballistic Pedulum One dimensional completely inelastic collision Conservation of linear momentum mv =(m+M)V; V=mv/(m+M) Conservation of mechanical energy (K+PE) initial = (K+PE) final (m+M)V 2 /2= (m+M)gh V= (2gh); V=mv/(m+M) v= [(m+M) (2gh)]/m

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Ch-9-10: Elastic Collision in One Dimensions Elastic collision: Momentum and kinetic energy of the system is conserved then P f = P i and K f = K i Two classes: Stationary target (v 2i =0) m 1 v 1i =m 1 v 1f +m 2 v 2f m 1 v 1i 2 /2=m 1 v 1f 2 /2+m 2 v 2f 2 /2 v 1f =v 1i (m 1 -m 2 )/(m 1 +m 2 ) v 2f =2v 1i m 1 /(m 1 +m 2 )

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Ch-9-10: Elastic Collision in One Dimensions Stationary target (v 2i =0) v 1f =v 1i (m 1 -m 2 )/(m 1 +m 2 ) v 2f =2v 1i m 1 /(m 1 +m 2 ) Three cases: Equal masses: m 1 =m 2 v 1f =0 and v 2f =v 1i Massive target : m 2 >> m 1 v 1f =-v 1i and v 2f (2m 1 /m 2 )v 1i Massive projectile : m 1 >> m 2 v 1f v 1i and v 2f 2v 1i

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Ch-9-10: Elastic Collision in One Dimensions Moving target m 1 v 1i + m 2 v 2i = m 1 v 1f +m 2 v 2f m 1 v 1i 2 /2+ m 2 v 2i 2 /2 = m 1 v 1f 2 /2+m 2 v 2f 2 /2 v 1f =[v 1i (m 1 -m 2 )/(m 1 +m 2 )] +2m 2 v 2i /(m 1 +m 2 ) v 2f =2v 1i m 1 /(m 1 +m 2 ) + [v 2i (m 2 -m 1 ) / (m 1 +m 2 )]

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Ch-9 Check Point 10 What is the final linear momentum of the target if the initial linear momentum of the projectile is 6 kg.m/s and final linear momentum of the projectile is: a) 2 kg.m/s b) -2 kg.m/s c) what is the final kinetic energy of the target if the initial and final kinetic energies of the projectile are, respectively 5 J and 2 J? P i =P f a)P f2 =Pi-P f1 =6-2= 4 kg.m/s b)=6-(-2)=8 kg.m/s c) K i1 = K f1 +K f2 Then K f2 =K i1 -K f1 = 5 – 2=3 J

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Ch-9-11: Elastic Collision in Two Dimensions Solve equations: P 1i + P 2i = P 1f + P 2f K 1i + K 2i = K 1f + K 2f Along x-axis m 1 v 1i = m 1 v 1f cos 1 +m 2 v 2f cos 2 Along y-axis 0= - m 1 v 1f sin 1 +m 2 v 2f sin 2 m 1 v 1i 2 /2 = m 1 v 1f 2 /2+m 2 v 2f 2 /2

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