Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc
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1 Physics 170 Summry of Results from Lecture Kinemticl Vribles The position vector ~r(t) cn be resolved into its Crtesin components: ~r(t) =x(t)^i + y(t)^j + z(t)^k. Rtes of Chnge Velocity ~v(t) = d~r(t)= Accelertion ~(t) =d~v(t)= = d 2 ~r(t)= 2 Kinemtics with Constnt Accelertion ~r(t) =~r(0) + ~v(0)t ~t2. The motion is specified by the constnt ccelertion ~ during the intervl nd the initil stte of motion (~r(0);~v(0)) t the beginning of the intervl. This describes (pproximtely) the free fll" motion of n object on the erth where ~ = g^j nd where ^j is unit vector perpendiculr to the surfce. Useful results for Free Fll/Projectile Motion Trjectory: y(x) = x tn( ) gx 2 =(2v 2 o cos( ) 2 ) (relesed from origin with speed v o nd projection ngle. Time of flight: T =2v o sin( )=g Mximum height: h =(v o sin( )) 2 =2g Horizontl Rnge: R =((v 2 o=g)) sin(2 ) Kinemtics in 2D Polr Coordintes In two dimensions it is often useful to specify the position in polr coordintes, giving the distnce of the prticle from the observer nd the direction of the line of sight. ~r(t) =r(t)^r(t) (1) where ^r(t) is unit vector directed outwrd long the line of sight. If ^ (t) is unit vector perpendiculr to ^r(t) directed counterclockwise (in the direction of incresing ) then ~r(t) = r(t)^r(t) ~v(t) = d~r(t) = _r^r + r ^ _ ~(t) = d~v(t) =( r r _ 2 )^r +(2_r _ + r )^ Newton's Second Lw of Motion ~F = m~, i.e. in n inertil frme of reference chnges of motion (mesured by ~) re responses to externl motive forces described by ~F. Weight The mgnitude of the grvittionl force on n object of mss m ner the surfce of the erth is the weight W = mg where g ß 9:8m=sec 2.
2 Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direction of this force is tngent to the string pointing towrds the section. (The string cn pull but cn't push.) In generl the tension cn vry s function of position in the string. An idel string is regrded s mssless nd unstretchble. Contct Forces The norml force N is force perpendiculr to the plne of contct with rigid object. The frictionl force is the force prllel to the plne of contct. We distinguish the cse of sttic friction when there is no slipping nd kinetic friction when there is reltive motion sttic F f» μ s N kinetic F f = μ k N Elstic Forces The force from Hooke's Lw spring F el = ks where s is the spring extension (positive s denotes extension nd negtive s is compression, the negtive sign in the force lw sys tht the elstic force opposes the extension/compression). The eqution of motion of mss m connected to Hooke's Lw spring ttched to fixed support is mẍ = kx (2) with solutions x(t) =A sin(!t)+b cos(!t) nd! = q k=m. A nd B re determined by the initil stte of the motion: A = _x(0)=! nd B = x(0). Center of Mss: For system of prticles with msses m i nd positions ~r i, the center of mss" is the mss-weighted position ~R cm = P N i=1 m i ~r i P N i=1 m i (3) The eqution of motion for ~R cm Letting M = P N i=1 m i we hve is often simpler thn for the msses themselves. M ~R cm = ~F ext (4) For n isolted system, initilly t rest this leds to the center of mss principle ~R cm is constnt. Liner Momentum: The liner momentum of prticle with mss m moving t velocity ~v is ~p = m~v. The liner momentum of n collection of prticles is the
3 (vector) sum of their liner moment: ~P = P i ~p i. The totl liner momentum chnges in response to n externl force _ ~ P = ~ Fext. Impulse Momentum Theorem: The impulse ~ J = R ~ F (t). The impulse is the chnge of the liner momentum: ~P = ~J. Momentum Flow nd Mss Trnsport: The dynmics of system of mss M tht releses mss t rte dm r = (with reltivespeedu r ) nd ccumultes mss t rte dm = (with reltive speed u ), in the presence of n externl force F is described by M dm = F + u r dm r u dm (5) The rocket eqution MdV= = u r dm r = specilizes this result to the cse of n isolted rocket tht ccelertes by relesing mss t stedy rte. Work Energy Theorems: For motion in one dimension K b = 1 2 mv2 b 1 2 mv2 = W (! b) = F (x) dx (6) For motion in higher dimensions the work is computed by projecting the force into direction of the displcement nd integrting over the trjectory of the prticle (this requires evlution of line integrl), thus K b = 1 2 mv2 b 1 2 mv2 = W (! b) = ~F (r) d~r (7) Conservtive Forces: The work done by conservtive force is identified with chnge of the potentil energy. The chnge in potentil energy is the negte of the work done by conservtive force U(! b) =U(b) U() = ~F d~r (8) U is the sme for ny pth tht connects the sme two endpoints. Three Potentil Energy Functions: Potentil energy of Hooke's lw spring with compression/extension s : U(s) =ks 2 =2. Potentil energy of mss m distnce y bove the surfce of the erth: U(y) = mgy with g ß 9:8m=s 2.
4 Grvittionl potentil energy of two point msses m 1 nd m 2 with seprtion s : U(s) = Gm 1 m 2 =s (note the sign) with G ß 6: N m 2 =kg 2. Conservtion of Energy The totl mechnicl energy E = K + U. In terms of the totl mechnicl energy, the work energy theorem sttes E b = E + W nc (! b) (9) when W nc < 0 the totl mechnicl energy decreses. Two Body Collisions In two body elstic collision, in one dimension the finl stte is completely constrined by the conservtion of liner momentum nd conservtion of energy. For n elstic collision of two msses m 1 nd m 2 with initil velocities v 1i nd v 2i the finl velocities re determined by mss rtios only (they don't depend on ny detils of the force lws if the kinetic energy is unchnged). v 1f = m 1 m 2 v 1i + 2m 2 v 2i m 1 + m 2 m 1 + m 2 2m 1 v 2f = v 1i + m 2 m 1 v 2i m 1 + m 2 m 1 + m 2 (10) For n inelstic collision, liner momentum is conserved but the kinetic energy chnges. The chnge of the kinetic energy is Q where 1 2 m 1v 2 1i m 2v 2 2i = 1 2 m 1v 2 1f m 2v 2 2f + Q (11) Positive Q mens tht energy is lost; both positive Q (inelstic) nd negtive Q (superelstic) re possible. A completely inelstic collision hs the lrgest possible Q consistent with the conservtion of momentum. For two body collision, this requires sitution where the two objects stick together" in the finl stte. Collisions in More thn One Dimension Here the outcome of the collision is only prtilly constrined by the conservtion lws: m 1 ~v 1i + m 2 ~v 2i = m 1 ~v 1f + m 2 ~v 2f (12) 1 2 m 1v 2 1i m 2v 2 2i = 1 2 m 1v 2 1f m 2v 2 2f elstic collision (13)
5 In two(three) dimensions, the conservtion lws provide three(four) independent constrints for the four(six) unknown components of the finl velocities ~v 1f nd ~v 2f. Thus the outcome of the collision is not constrined by the conservtion lws. An nlysis of the two body collision in more thn one dimension cn be ccomplished by dditionl mesurement of e.g. the scttering ngle, which provides one(two) dditionl constrints on 2D(3D) collisions.
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