Integration of the equation of motion with respect to time rather than displacement leads to the equations of impulse and momentum.

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2 Integration of the equation of motion with respect to time rather than displacement leads to the equations of impulse and momentum. These equations greatl facilitate the solution of man problems in which the applied forces act during extremel short periods of time, as in impact problems, or over specified intervals of time.

3 Let s consider the general curvilinear motion in space of a particle of mass m, where the particle is located b its position vector r measured from a fixed origin O. The velocit of the particle is v r and is tangent to its path. The resultant force of all forces on m is in the direction of its a v acceleration. F

4 We ma write the basic equation of motion for the particle, as F ma mv or F mv d dt d dt mv G G Where the product of the mass and velocit is defined as the linear momentum G mv of the particle. This equation states that the resultant of all forces acting on a particle equals its time rate of change of linear momentum.

5 In SI, the unit of linear momentum also equals N. s. mv is kg. m/s, which Since the equation of impulse and momentum is a vector equation, in addition to the equalit of the magnitudes of and G, the direction of the resultant force coincides with the direction of the rate of change in linear momentum, which is the direction of the rate of change of velocit. Linear impulse momentum equation is one of the most useful and important relationships in dnamics, and it is valid as long as mass m of the particle is not changing with time. F

6 We now write the three scalar components of linear momentum equation as Fx G x F G Fz G z These equations ma be applied independentl of one another.

7 The Linear Impulse-Momentum Principle All that we have done so far is to rewrite Newton s second law in an alternative form in terms of momentum. But we ma describe the effect of the resultant force on the linear momentum of the particle over a finite period of time simpl b integrating the linear momentum equation with respect to time t. Multipling the equation b dt gives Fdt,, which we integrate from time t to time t 2 to obtain dg t t 2 Fdt G 2 dg G G G 2 G F

8 Here the linear momentum at time t 2 is G 2 =mv 2 and the linear momentum at time t is G =mv. The product of force and time is defined as the linear impulse of the force, and this equation states that the total linear impulse on m equals the corresponding change in linear momentum of m. Alternativel, we ma write G Fdt G 2 I

9 which sas that the initial linear momentum of the bod plus the linear impulse applied to it equals its final linear momentum. m v G mv + = F dt G m 2 v 2

10 The impulse integral is a vector which, in general, ma involve changes in both magnitude and direction during the time interval. Under these conditions, it will be necessar to G express F and in component form and then combine the integrated components. The components become the scalar equations, which are independent of one another. t2 Fxdt 2 2 t mvx mvx Gx Gx Gx t2 Fdt 2 2 t mv mv G G G t2 Fzdt 2 2 t mvz mvz Gz Gz Gz

11 In some cases, certain forces are ver large and of short duration. Such forces are called impulsive forces. An example is a force of sharp impact. We frequentl assume that impulsive forces are constant over their time of duration, so that the can be brought outside of the linear impulse momentum integral. In addition, we frequentl assume that nonimpulsive forces can be neglected in comparison with impulsive forces. An example of a nonimpulsive force is the weight of a baseball during its collision with a bat the weight of the ball, about.425 N, is small compared with the force exerted on the ball b the bat, which is about several thousand Newtons in magnitude.

12 There are cases where a force acting on a particle changes with the time in a manner determined b experimental measurements or b other approximate means. In this case, a graphical or numerical integration must be performed. If, for example, a force F acting on a particle in a given direction changes with the time t as indicated in the figure, the impulse, t 2 Fdt t of this force from t to t 2 is the shaded area under the curve.

13 Conservation of Linear Momentum If the resultant force on a particle is zero during an interval of time, its linear momentum G remains constant. In this case, the linear momentum of the particle is said to be conserved. Linear momentum ma be conserved in one direction, such as x, but not necessaril in the - or z- directions. G 0 G G mv m v 2 This equation expresses the principle of conservation of linear momentum. 2

14 In addition to the equations of linear impulse and linear momentum, there exists a parallel set of equations for angular impulse and angular momentum. First, we define the term angular momentum. The figure shows a particle P of mass m moving along a curve in space. The particle is located b its position vector r respect to a convenient origin O of fixed coordinates x--z. with

15 The velocit of the particle is momentum is momentum vector G mv mv the angular momentum, and its linear. The moment of the linear about the origin O is defined as H O v of P about O and is given b the cross-product relation for the moment of a vector. r H o r mv r G

16 The angular momentum then is a vector perpendicular to the plane A defined b r and v. The sense of O is clearl defined b the right-hand rule for cross products. H

17 The scalar components of angular momentum ma be obtained from the expansion z x o x z x z o v v v z x k j i m H or k v x m v j x v z m v i z v m v mv r H so that z v m v H z ox x v z m v H z x o v x m v H x oz

18 Each of these expressions for angular momentum ma be checked from the figure, which shows the three linear momentum components, b taking the moments of these components about the respective axes. In SI units, angular momentum has the units kg. m 2 /s =N. m. s.

19 If F M o r F r mv represents the resultant of all forces acting on the particle P, the moment about the origin O is the vector cross product M o We now differentiate H o differentiation of a cross product and obtain The term H o v mv d dt r mv with time, using the rule for the r mv r mv r m v r mr 0 ma M o is zero since the cross product of parallel vectors is zero.

20 Substitution into the expression for moment about O gives M o H o This equation states that the moment about the fixed point O of all forces acting on m equals the time rate of change of angular momentum of m about O. This relation, particularl when extended to a sstem of particles, rigid or nonrigid, provides one of the most powerful tools of analsis in dnamics. The scalar components of this equation are M ox H ox M o H o M oz H oz

21 The Angular Impulse-Momentum Principle To obtain the effect of the moment on the angular momentum of the particle over a finite period of time, we M o H o integrate from time t to t 2. or t H M odt dh o Ho Ho H 2 o t 2 H total angularimpulse o o 2 t2 M odt t change in angular momentum r2 mv2 r mv H o

22 The product of moment and time is defined as angular impulse and this equation states that the total angular impulse on m about the fixed point O equals the corresponding change in angular momentum of m about O. Alternativel, we ma write t 2 H M dt H o o o 2 t

23 Plane-Motion Applications Most of the applications can be analzed as plane-motion problems where moments are taken about a single axis normal to the plane motion. In this case, the angular momentum ma change magnitude and sense, but the direction of the vector remains unaltered. t 2 t M o dt H H o 2 o Fr sin dt mv 2 d 2 mv d

24 Conservation of Angular Momentum If the resultant moment about a fixed point O of all forces acting on a particle is zero during an interval of time, its angular momentum H O remains constant. In this case, the angular momentum of the particle is said to be conserved. Angular momentum ma be conserved about one axis but not about another axis. H o 0 H H O O 2 This equation expresses the principle of conservation of angular momentum.