Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate the acceleation equied to keep an object moving in a cicula path. Following the lead of Halliday et al. (21) 1, we begin with the geneic situation shown in Figue 1A: a paticle moving at constant speed s aound a cicle of adius : This gue epesents a snapshot of motion at a time when the paticle lies at angle elative to the x-axis of an x-y coodinate system in which the oigin coincides with the cente of cicula motion. Ou deivation depends on two sets of geometical aguments. Fist, we note that the x-y position of the paticle is: fom which we can deduce that x = cos ; (1) y = sin ; (2) cos = x ; (3) sin = y : (4) Next, we examine the motion of the paticle in geate detail (Figue 1B). The velocity of the paticle is tangential to the cicle in which it tavels, so the aow designating the paticle s velocity lies at a ight angle to a adius connecting the paticle to the cente of the cicle. This means that the angle between the velocity aow and a line paallel to the y-axis is. (To see this, mentally otate the adius shown in Figue 1B so that the angle between the adius and the x-axis is zeo. In that case, the velocity aow is paallel to the 1 Halliday, D, R. Resnick, and J. Walke. 21. Fundamentals of Physics (Extended) (6th Edition). John Wiley & Sons, N.Y. 1
y-axis (pointing up ), so the angle between it and the y-axis is likewise zeo.) With this angle established, we see that the x and y components of speed ae s x = s sin ; (5) s y = s cos : (6) Substituting Eqs. 3 and 4 fo the sine and cosine tems in Eqs. 5 and 6, we nd that s s x = y; (7) s y = s x: (8) These ae the x and y components of speed, wheeas we desie to know the x and y components of acceleation. Acceleation is the time deivative of speed, so (given that s and ae constant) a x = ds x dt = a y = ds y dt = s s dy dt ; (9) dx dt : (1) Now dy=dt is velocity in the y diection, s y, and dx=dt, is velocity in the x diection, s x, so: s a x = s y; (11) a y = s s x: (12) 2
Substituting Eqs. 5 and 6 into these expessions bings us nea ou goal: = s2 a x = a y = cos ; (13) sin : (14) These ae the x and y components of acceleation. To aive at the oveall acceleation, we add components using the Pythagoean theoem (Figue 1C) q a = a 2 x + a 2 y; (15) q ( cos ) 2 + ( sin ) 2 ; (16) = s2 p co + sin 2 : (17) Noting that co + sin 2 = 1, we aive at ou destination. The magnitude of centipetal acceleation is a = s2 : (18) It emains only to detemine the diection in which this acceleation is diected. As shown in Figue 1C, tan = a y a x = sin = tan : (19) cos In othe wods, =, so the diection of acceleation is paallel to the oientation of the adius. This is anothe way of saying that acceleation is diected at the cente of the cicle. 2 Supplement 2.2: The Rotational Moment of Inetia fo a Cylinde This supplement expands on a topic addessed on page 23 of the textbook. Conside a cylinde of adius and height z, otating about its axis (Figue 2). The cylinde s mateial has density. We can think of the cylinde as a nested set of in nitesimally thin tubula shells, each with height z, adius `, and thickness d`. The in nitesimal volume at distance ` fom the axis is thus and this volume has in nitesimal mass: dv = 2`zd`; (2) dm = 2`zd`: (21) 3
Theefoe, fom the de nition of J m (Eq. 2.3 in the textbook), the otational moment of inetia fo ou cylinde is J m = = Z Z = 2z `2dm; (22) `2 (2`zd`) ; (23) Z `3d`; (24) = z4 : (25) 4 Thus, fo a cylinde, otational moment of inetia inceases as the fouth powe of adius. 3 Supplement 2.3: Calculating Distance Using Calculus This supplement expands on a topic addessed on page 29 of the textbook. As a efeshe fo you calculus skills, we calculate distance x in a fashion that can be moe easily genealized. The integal of a function is, by de nition, the aea unde that function s cuve; thus, the integal of velocity as a function 4
of time is the aea shown in Figue 3. This aea has units of velocity times time, that is, of distance. So, to calculate distance taveled, we integate velocity u(t) ove time t f Z tf x = u(t)dt: (26) In tun, because a x is constant, velocity is simply the poduct of acceleation and time, u(t) = a x t: (27) Thus, x = Z tf a x tdt; (28) whee t f is the time at which we emove the foce fom ou acceleating object. Again noting that a x is constant, we bing it outside the integal and solve: Z tf t 2 x = a x tdt = a x 2 : (29) But fom Figue 3 and Eq. 2.46 in the text we know that a x = u t f, so the same answe we aived at in Eq. 2.49. x = u t f t 2 f 2 = ut f 2 ; (3) 5
4 Supplement 2.4: Rest Mass This supplement expands on a topic addessed on page 31 of the textbook. The connection between Einstein s equation and the concepts of enegy as discussed in the text is moe diect than one might think (Shu 1982, pg 59) 2. Accoding to the special theoy of elativity, the mass of an object changes with its speed, u: m m = q : (31) u 1 2 c 2 Hee m is the object s est mass, its mass when stationay. Because c is so lage (3 1 8 ms 1 ), u 2 =c 2 is negligibly small fo speeds commonly encounteed on Eath, and m is so nealy equal to m that we commonly neglect the di eence. But let s be diligent and take the speed-dependent vaiation in mass into account. Inseting Eq. 31 fo m into Einstein s equation (E = mc 2, o using ou symbols, W = mc 2 ), we see that Now, when u 2 =c 2 is small, W = m q 1 u 2 c 2 c 2 : (32) 1 u 2 q 1 + u 1 2 2c 2 ; (33) c 2 and the total enegy of an object is thus appoximately W m c 2 + m u 2 : (34) 2 In othe wods, an object contains enegy by dint of simply having mass (m c 2 ) and additional enegy (kinetic enegy) if that mass moves (m u 2 =2). 2 Shu, F.H. 1982. The Physical Univese. Univesity Science Books, Bekeley, Califonia. 6