Mechanics 1: Work, Power and Kinetic Energy

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1 Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss). The tem wok was fist used in 1826 b the Fench mathematician/enginee Gaspad-Gustave oiolis (note that this was, oughl, 150 eas afte Newton). Wok. Suppose a foce F acting on a paticle gives it a displacement d. Then the wok done b the foce on the paticle is defined as: d F d. (1) (It should be clea that wok is a scala.) The total wok done b a foce field (o, vecto field) F in moving a paticle fom point to point along a path is given b the line integal: P2 whee 1 is the position vecto of and 2 is the position vecto of, see Fig F d, (2) z d i k j + d Figue 1: It is woth consideing the F moe caefull in the epession fo wok. In ou discussion of Newton s second law, F = ma, F was the vecto sum of all foces acting on the paticle of mass m, i.e. the net foce acting on m. This will almost alwas be the case in ou epessions fo wok, and it is definitel the case when Newton s second law is used in cetain manipulations involving the epession fo wok, e.g., in Theoem 1. Howeve, thee ma be cicumstances when one is inteested in the wok done on a paticle b a specific foce in the vecto sum of all foces. This should be clea fom the contet. Unless eplicitl stated, the F in the epession fo wok is the vecto sum of all foces acting on the paticle of constant mass m. Eample. Find the wok done in moving an object along a vecto = 3i + 2j 5k if the applied foce is F = 2i j k, see Fig. 2 (this means along a distance of the constant length of the vecto in the constant diection of the vecto ). 1

2 F θ Figue 2: Solution: Wok done = magnitude of foce in diection of motion distance moved, = (F cos θ)() = F, = (2i j k) (3i + 2j 5k), = = 9. Eample. Find the wok done in moving a paticle once aound a cicle in the plane, if the cicle has cente at the oigin and adius 3, and if the foce field is given b: see Fig. 3. F = (2 + z)i + ( + z 2 )j + ( z)k, O t Figue 3: Solution: In the plane z = 0, F = (2 )i + ( + )j + (3 2)k and d = di + dj. Then the wok done is: = ((2 )i + ( + )j + (3 2)k) (di + dj), (2 )d + ( + )d. 2

3 hoose the paametic equations of the cicle as = 3cos t, = 3 sin t, whee t vaies fom 0 to 2π. Then the line integal becomes: 2π t=0 2π = 0 (2(3 cos t) 3sin t) ( 3sin t) dt + (3cos t + 3 sin t)(3cos t)dt, (9 9sin t cos t)dt = 9t 9 «2π 2 sin2 t = 18π. 0 In tavesing we have chosen the counteclockwise diection indicated in Fig. 3. We efe to this as the positive diection and sa that has been tavesed in the positive sense. If we had tavesed in the clockwise (negative) diection the value of the integal would have been 18π. Powe. The time ate of doing wok on a paticle is called the instantaneous powe, o, biefl, the powe applied to the paticle. Denoting the instantaneous powe b P, we have: P = dw dt. If F denotes the foce acting on a paticle and v is the velocit of the paticle then we also have: P = F v. Eample. Suppose a paticle of mass m moves unde the influence of a foce field along a space cuve whose position vecto is given b: = (2t 3 + t)i + (3t 4 t 2 + 8)j 12t 2 k. Find the instantaneous powe applied to the paticle b the foce field. Solution: We need to compute the foce and the velocit, and then take thei dot poduct. The velocit of the paticle is given b: v = (6t 2 + 1)i + (12t 3 2t)j 24tk. The acceleation is given b: a = 12ti + (36t 2 2)j 24k. Theefoe the foce is given b: F = ma = m `12ti + (36t 2 2)j 24k. Now we have enough infomation to compute the instantaneous wok: F v = m `12ti + (36t 2 2)j 24k `(6t 2 + 1)i + (12t 3 2t)j 24tk, = m `72t t + 432t 5 72t 3 24t 3 + 4t + 576t = m `432t 5 24t t. Ke point: It should be clea (quoting fom Sommefeld [1952]) that wok does not equal foce times distance as often stated, but component of foce along path times path length o foce times component of path length along foce 3

4 . Kinetic Eneg. We now show that the idea of wok leads natuall to the notion of the idea of the kinetic eneg of a paticle. If the foce is consevative, we show that this enables us to intoduce the idea of potential eneg. In this wa wok can be chaacteized eithe though kinetic eneg, o though the potential eneg. Refe back to Fig. 1 which we epoduce again hee. z d i k j + d Figue 4: We assume that the mass m of the paticle is constant, and that at times and t 2 it is located at points and, espectivel, and moving with velocities v 1 = d d (t1) and v2 = (t2) at points P1 and P2, dt dt espectivel. Then we can pove the following theoem. Theoem 1 The total wok done b the net foces F in moving the paticle along the cuve fom to is given b: 1 2 m(v2 2 v 2 1). (3) Poof: Wok done = F d dt dt = F vdt, = m dv dt vdt = m v dv, Note that Newton s second law is used hee. = 1 2 m d(v v) = 1 t 2 2 mv2 = 1 2 mv mv2 1, which completes the poof. If we call the quantit: T = 1 2 mv2, the kinetic eneg, this theoem states that: The total wok done b the net foces F in moving the paticle of constant mass m fom to along = the kinetic eneg at - the kinetic eneg at. o, smbolicall, 4

5 T 2 T 1, whee T 1 = 1 2 mv2 1 and T 2 = 1 2 mv2 2. It should be clea (but ou should convince ouself of this) that is a solution of Newton s equations. Now we conside the impotant case when the foce is consevative. onsevative Foce Fields. Suppose thee eists a scala function V such that F = V. Then we can pove the following: Theoem 2 The total wok done b the foce F = V. in moving the paticle along fom to is: Poof: Fist, note that: P2 V () V (). dv (,,z) = V V V (,,z)d + (,,z)d + (,, z)dz = V d. z Then the poof follows fom a diect calculation using the specific fom of F = V : P2 P2 V d = dv = V = V () V (), P2 which completes the poof. Now we state (without poof) two theoems that give computable conditions fo detemining if a foce field is consevative. Theoem 3 A foce field F is consevative if and onl if thee eists a continuousl diffeentiable scala field V such that F = V o, equivalentl, if and onl if: F = culf = 0 identicall. Theoem 4 A continuousl diffeentiable foce field F is consevative if and onl if fo an closed nonintesecting cuve (i.e., simple closed cuve) we have: I 0, i.e., the total wok done in moving a paticle aound an simple closed cuve is zeo. Potential Eneg o Potential. The scala function V, such that F = V is called the potential eneg (o also the scala potential o just potential) of the paticle in the consevative foce field F. It should be noted that if ou add an abita constant to the potential, the associated foce does not change. We can epess the potential as: V = F d, 0 whee 0 is chosen abitail. The point 0 is sometimes called the efeence point. Even though it is abita, a wise choice can simplif a poblem. We will see eamples of this late on. 5

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