Lab M4: The Torsional Pendulum and Moment of Inertia


 Jeffery Lloyd
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1 M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the axis of the wie, the wie exets a toque on the mass, tending to otate it back to its oiginal position. f twisted and eleased, the mass will oscillate back and foth, executing simple hamonic motion. This is the angula vesion of the bouncing mass hanging fom a sping. This lab will give you a bette gasp of the meaning of moment of inetia. n Pats 13 you will investigate the moment of inetia of disks and ings and the tosional sping constant of a od. n Pat 4, you will use you data to discove how the sping constant depends upon the diamete of the od. Conside a thin od with one end fixed in position and the othe end twisted though an angle about the od s axis. untwisted twisted This end fixed. τ f the angle is sufficiently small that the od is not plastically defomed, the od exets a toque τ popotional to the angle, (1) τ = κ (like F = k x fo a sping) whee κ (Geek lette kappa) is called the tosion constant. The minus sign indicates that the diection of the toque vecto τ is opposite to the angle vecto, so the toque tends to undo the twist. This is just like Hooke s Law fo spings. f a mass with moment of inetia is attached to the od, the toque will give the mass an angula acceleation α accoding to Fall 004
2 M4. () τ = α = d (like F = ma = m d x ) Combining (1) and () yields the equation of motion fo the tosional pendulum, d = κ (like m d x kx = ) (4) d κ = The solution to this diffeential equation is (5) () t = m cos( ωt + φ) (like xt () = xm cos( ωt + φ )) (6) ω = κ (like ω= k m ) whee m and ϕ ae constants which depend on the initial position and angula velocity of the mass. (The equation of motion is a second ode diffeential equation so its solution must have two constants of integation.) m is the maximum angle; oscillates between + m and  m. The constant ω is elated to the fequency f and the peiod T of the simple hamonic motion by (7) ω = πf = π T. So, fom (6) and (7), the peiod T is given by (8) T = π κ. t s always good to stae at a fomula like this until it makes physical sense. f the wie is vey stiff (lage κ) then the mass oscillates apidly and peiod T is shot. f the mass is lage so that is lage, then its oscillates slowly and T is long. The tosion constant can be detemined fom measuements of T if is known Fall 004
3 M4.3 (9) κ = 4π T. Convesely, if κ is known, the moment of inetia can be detemined fom measuements of T, T (10) =κ 4 π. Expeiment n this fist pat of this expeiment, you will compute the moments of inetia of a disk and an annulus fom measuements of the masses and dimensions. n the second pat, you will detemine the tosion constant of a od fom measuements of T, using a solid disk as a tosion mass. Fom the definition of moment of inetia, (11) = mi i it can be shown (you will show it!) that fo a solid disk of mass M disk and adius R with the axis of otation along the symmety axis, i, (1) M disk = 1 disk R. Thus disk can be computed fom measuements of M and R. Notice that the thickness of the disk does not ente into eq n (1). od annulus n the thid pat, you will add to the disk an annula ing with inne adius R 1 and oute adius R. f the mass of the annulus is M ing, its moment of inetia is (13) ing = 1 Ming ( R + ) R. 1 disk [This fomula can be undestood qualitatively as follows: Fom (11), the of a ing of negligible adial thickness is MR. The fomula (13) is the aveage of two ings, one of adius R 1 and one of adius R.] Fall 004
4 M4.4 Finally, in the last pat of the expeiment, you will detemine the tosion constant κ of seveal wies of diffeent adius and expeimentally discove the functional dependence of κ on. Pocedue Thee ae 4 tosion ods used in this expeiment. Thee of them ae made of steel and one is made of bass. To detemine which is the bass od, use a magnet it will attact the steel ods, but not the bass od. All the ods have the same length, but the thee steel ods have diffeent diametes. The thickest steel od and the bass od have the same diamete. Pat 1. Calculation of disk, ing Begin by weighing the solid disk and the annula ing. They ae quite heavy (a few kilogams) so use the 6000gamcapacity scales only. [Do not attempt to weigh them on the smalle capacity scales! They ae heavy enough to damage those scales.] With a mete stick, measue the adius R of the disk and R 1, R of the annulus. (To get the adius, measue the diamete and divide by.) Then compute disk and ing. Fo this lab, it is pefeable to use cgs units(centimetegamsecond), athe than mks units, because the values of in this lab ae vey small in mks units. As usual, ecod you computed values fo with the coect numbe of significant figues. The disk has a small cylindical potusion at its hub which is needed to attach the tosion od. n one of the pelab questions, you will compute the moment of inetia of this little hub. s its contibution to the total moment of inetia of the disk significant o negligible? Pat. Detemination of κ of thinnest od Choose the thinnest od and use it to suspend the solid disk fom the backet on the wall as shown on the diagam below. Note that the ends of the od have indentations whee the set scews on the backet and the disk should seat. Tighten the set scews caefully so you can feel them seat popely. To pevent the disk fom wobbling when it is oscillating, the axis of the disk can be kept fixed with a centeing post, consisting of a pointed od that spins feely on a mount. Place the centeing post exactly below the cente of the disk and aise the post so that its pointed end is seated in the hole in the cente of the disk. Thee ae two ways to measue the peiod of the oscillation. You can use a photogate time in PEROD mode and tape a piece of pape to the disk so that it inteupts the photogate. Use the eset switch to quickly make seveal measuements and estimate the best value of T. An altenative way to measue the peiod is to use a stop watch and time the inteval fo seveal complete oscillations. f the time fo N complete peiods (N 0 o moe) is T total, then the peiod is T = T total. [DO NOT measue one peiod twenty N times! Measue the time fo twenty peiods once.] By measuing the time fo seveal peiods, the uncetainty in T, due to you eaction time, is educed by a facto of N. Fall 004
5 M4.5 δttotal = human eaction time 0.1 sec. δt = δttotal N Fo this pat of the lab, measue the peiod both with the photogate time and with the stopwatch and compae the two esults. Fo subsequent measuements of T, you can choose one o the othe. Now, using you computed disk and you measued T, compute the tosion constant κ of the thin od. wall backet set scew od pointed set scew seated in indentation solid disk centeing post set scew pape slip lowfiction pivot photogate time Pat 3. Measuement of ing With both the disk and the annulus suspended fom the thin od, measue the new peiod T. Fom you measuements of the peiod with and without the annulus and you computed disk, detemine ing, as shown below, and compae with you ealie computed value of ing. Fall 004
6 M4.6 Td = peiod with disk only T d = peiod with disk + ing d d + T κ = 4π = 4 d Td π ( ) Td Td = d + d = 1 + d T d = d Td 1 Pat 4. Dependence of κ on adius of od. Using the pecision calipes available (the metal ones in the gay plastic case), detemine the adius of each of the fou ods. Using the disk alone (no annulus), measue the peiod T of the tosion oscillato with each of the thee thicke ods. Fom you measued T s and computed disk, compute the tosion constant κ fo each of the fou ods. (κ fo the thinnest od was aleady detemined in pat.) Fo a fixed length od and a fixed composition, the tosion constant κ vaies as some powe of the adius ; that is, κ=a n, whee A and n ae constants. You job is to detemine the exponent n. With the data fom the thee steel ods make a plot of κ vs. n whee n is some intege. Vay n until the gaph is a staight line though the oigin. Veify that you have the coect powe n by computing the atio κ fo the thee steel n ods. [f the gaph is eally a staight line though the oigin, then the atio should be a constant fo all thee ods.] On the same gaph, plot the point fo the bass od. You will have to ceate a second tace fo that single point, and, in the plot fomat window, set the tace type to points and symbol to box o diamond, etc. s bass stiffe than steel o vicevesa? Fall 004
7 M4.7 PeLab Questions: 1. What is the definition of moment of inetia? (Wite down a geneal equation fo and explain what the symbols in the equation mean.) What ae the units of in the cgs system(cgs = centimetegamsecond)? What ae the units of in the MKS system? (MKS = metekilogamsecond, also called the S system.). Use the definition of to show that the moment of inetia of a thin hoop of mass M, adius R, and negligible thickness is given by hoop = M R. 3. Use the definition of moment of inetia to show that MR integal fom of eq n (11), = ρ disk = 1. [Hint: Use the dv, whee ρ = M / V is the density of the disk.] 4. A solid disk of adius R and mass M is glued to a thin hoop of adius R and mass m (m M). The symmety axes of the disk and hoop ae aligned. What is total, the total moment of inetia of the disk + hoop about the axis of symmety? 5. The disk has a little hub at its cente (in the shape of an annulus) with a mass m = 70±1 gams and with inne and oute adii of 1 = 6.4 mm and = 16.8 mm. Use eq n (13) to compute the moment of inetia of the hub. (Caeful of you units! Ae you going to use kilogams o gams?, metes o centimetes o...?) 6. What ae the units of the tosion constant κ in the MKS system? n the cgs system? 7. What is the definition of toque? (Don t just give a vague, qualitative definition; give the pecise, technical definition, as given in physics text books. As usual, if you give an equation, explain what the tems in the equation mean.) What ae the units of toque in the MKS system? 8. Conside a tosion pendulum consisting of a solid disk of mass M and adius R, suspended fom a wie with tosion constant κ Wite an equation which shows how the peiod T depends on M, R, and κ What happens to the peiod T, if m, the amplitude of the angula motion, is deceased by a facto of two? 9. Conside two disks, labeled A and B, with disk A having two times the mass and thee times the adius of disk B (M A = M B, R A = 3R B ). Each disk is suspended fom a wie with same tosion constant κ, and the peiod of each (T A, T B ) is measued. What is the atio T A T? B 10. Conside a set of ods all of the same mateial and the same length, but with diffeing adii,. Assuming that you have the coect value of the exponent n, show with a sketch what a gaph of κ vs. n should look like. Fall 004
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