13.10 Worked Examples

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Charlene Wiggins
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1 13.10 Woked Examples Example Wok Done in a Constant Gavitation Field The wok done in a unifom gavitation field is a faily staightfowad calculation when the body moves in the diection of the field. Suppose the body is moving unde the influence of gavity, F = mg ĵ along a paabolic cuve. The body begins at the point (x 0, y 0 ) and ends at the point (x f, y f ). What is the wok done by the gavitation foce on the body? Solution: The infinitesimal line element d is theefoe d = dx î + dy ĵ. (13.9.1) 13-1

The scala poduct that appeas in the line integal can now be calculated, F d = mg ĵ [dx î + dy ĵ] = mgdy. (13.9.2) This esult is not supising since the foce is only in the y -diection.

2 The scala poduct that appeas in the line integal can now be calculated, F d = mg ĵ [dx î + dy ĵ] = mgdy. (13.9.2) This esult is not supising since the foce is only in the y -diection. Theefoe the only non-zeo contibution to the wok integal is in the y -diection, with the esult that y= y y= y f f f d = F dy = mgdy = ( y ). (13.9.3) W = F y mg y f 0 y= y y= y In this case of a constant foce, the wok integal is independent of path. Example Hooke s Law Sping-Body System Conside a sping-body system lying on a fictionless hoizontal suface with one end of the sping fixed to a wall and the othe end attached to a body of mass m (Figue 13.19). Calculate the wok done by the sping foce on body as the body moves fom some initial position to some final position. Figue A sping-body system. Solution: Choose the oigin at the position of the cente of the body when the sping is elaxed (the equilibium position). Let x be the displacement of the body fom the oigin. We choose the +î unit vecto to point in the diection the body moves when the sping is being stetched (to the ight of x = 0 in the figue). The sping foce on the body is then given by F = F ˆ ˆ x i = kx i. (13.9.4) The wok done by the sping foce on the mass is x = x f W sping = ( kx) dx = 1 k(x 2 2 f x 2 0 ). (13.9.5) x = x

3 Example Wok done by the Invese Squae Gavitation Foce Conside a body of mass m in moving in a fixed obital plane about the sun. The mass of the sun is m s. How much wok does the gavitation inteaction between the sun and the body done on the body duing this motion? Solution: Let s assume that the sun is fixed and choose a pola coodinate system with the oigin at the cente of the sun. Initially the body is at a distance 0 fom the cente of the sun. In the final configuation the body has moved to a distance f < 0 fom the cente of the sun. The infinitesimal displacement of the body is given by d = d ˆ + dθ θˆ. The gavitation foce between the sun and the body is given by Gm sm F gav = Fgav ˆ = ˆ. (13.9.6) 2 The infinitesimal wok done wok done by this gavitation foce on the body is given by dw = F d = (F ˆ) (d ˆ + dθ θˆ) = F d. (13.9.7) gav gav, Theefoe the wok done on the object as the object moves fom i to f is given by the integal f f f gav, Gm sun m d. (13.9.8) W = F gav d = F gav, d = 2 i i i Upon evaluation of this integal, we have fo the wok f f Gm m Gm m 1 1 sun sun W = 2 d = = Gm m. (13.9.9) sun i f i i Because the body has moved close to the sun, f < i, hence 1 / f > 1 / i. Thus the wok done by gavitation foce between the sun and the body, on the body is positive, 1 1 W = Gm m > 0 ( ) sun f i We expect this esult because the gavitation foce points along the inwad adial diection, so the scala poduct and hence wok of the foce and the displacement is 13-3

4 positive when the body moves close to the sun. Also we expect that the sign of the wok is the same fo a body moving close to the sun as a body falling towads the eath in a constant gavitation field, as seen in Example above. Example Wok Done by the Invese Squae Electical Foce Let s conside two point-like bodies, body 1 and body 2, with chages q 1 and q 2 espectively inteacting via the electic foce alone. Body 1 is fixed in place while body 2 is fee to move in an obital plane. How much wok does the electic foce do on the body 2 duing this motion? Solution: The calculation in nealy identical to the calculation of wok done by the gavitational invese squae foce in Example The most significant diffeence is that the electic foce can be eithe attactive o epulsive while the gavitation foce is always attactive. Once again we choose pola coodinates centeed on body 2 in the plane of the obit. Initially a distance 0 sepaates the bodies and in the final state a distance f sepaates the bodies. The electic foce between the bodies is given by 1 q 1 q ˆ ˆ = 2 Felec = F elec = F elec, 2 ˆ. ( ) 4πε0 The wok done by this electic foce on the body 2 is given by the integal f f f 1 q q 1 W = F d = F d = 2 d. ( ) elec elec, 2 4πε 0 i i i Evaluating this integal, we have fo the wok done by the electic foce f f 1 q1 q 2 1 q 1 q W = d = = q 2 1 q 2. ( ) 2 4πε 4πε 4πε i i f i If the chages have opposite signs, q 1 q 2 < 0, we expect that the body 2 will move close to body 1 so f < i, and 1 / f > 1 / i. Fom ou esult fo the wok, the wok done by electical foce in moving body 2 is positive, 1 1 W = q 1 q 2 ( 1 ) > 0. ( ) 4πε0 f i Once again we see that bodies unde the influence of electic foces only will natually move in the diections in which the foce does positive wok. If the chages have the 13-4

5 same sign, then q 1 q 2 > 0. They will epel with f > i and 1 / f < 1 / i. Thus the wok is once again positive: W = q 1 q 2 > 0. ( ) 4πε 0 f i 13-5

6 MIT OpenCouseWae Classical Mechanics Fall 2016 Fo Infomation about citing these mateials o ou Tems of Use, visit:

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