Chapter 8. Rotational Equilibrium and Rotational Dynamics. ! = Fd. ! = Fr sin" !F x = 0 and!f y = 0. Wrench Demo. Torque is vector quantity.

Size: px

Start display at page:

Download "Chapter 8. Rotational Equilibrium and Rotational Dynamics. ! = Fd. ! = Fr sin" !F x = 0 and!f y = 0. Wrench Demo. Torque is vector quantity."

Tyrone Scott Pitts
7 years ago
Views:

1 Wrench emo hapter 8 Rotatonal Equlbrum and Rotatonal ynamcs Torque Torque,!, s tendency of a force to rotate object about some axs! = Fd F s the force d s the lever arm (or moment arm) Torque s vector quantty recton determned by axs of twst Perpendcular to both r and F lockwse torques pont nto paper. efned as negatve ounter-clockwse torques pont out of paper. efned as postve Unts are Newton-m Non-perpendcular forces! = Fr sn" Torque and Equlbrum!F x = 0 and!f y = 0 Forces sum to zero (no lnear moton)! s the angle between F and r Torques sum to zero (no rotaton)!" = 0

2 Meter Stck emo xs of Rotaton Torques requre pont of reference Pont can be anywhere Use same pont for all torques Pck the pont to make problem least dffcult Example 8.1 Gven M = 120 kg. Neglect the mass of the beam. nother Example Gven: W=50 N, L=0.35 m, x=0.03 m Fnd the tenson n the muscle W a) Fnd the tenson n the cable x L b) What s the force between the beam and the wall a) T=824 N b) f=353 N F = 583 N enter of Gravty Example 8.2 Gravtatonal force acts on all ponts of an extended object However, t can be consdered as one net force actng on one pont, the center-of-gravty, X. Gven: x = 1.5 m, L = 5.0 m, w beam = 300 N, Fnd: T w man = 600 N!(m g)x =! m! m x! g m = MgX, where X =!! m x m Weghted verage T = 413 N x L

3 Example 8.3 Example 8.4a onsder the 400-kg beam shown below. Fnd T R Gven: W beam =300 W box =200 T R = N Fnd: T left What pont should I use for torque orgn? Example 8.4b Example 8.4c Gven: T left =300 Gven: W beam =300 T rght =500 W box =200 Fnd: W beam What pont should I use for torque orgn? Fnd: T rght What pont should I use for torque orgn? Example 8.4d Example 8.4e Gven: T left =250 Gven: T left =250 T rght =400 W beam =250 Fnd: W box What pont should I use for torque orgn? Fnd: W box What pont should I use for torque orgn?

4 Example 8.5 (skp) 80-kg beam of length L = 100 cm has a 40-kg mass hangng from one end. t what poston x can one balance them beam at a pont? L = 100 cm 80 kg aton emo Moment-of-Inerta emo x 40 kg x = cm Torque and ngular cceleraton nalogous to relaton between F and a F = ma,! = I" Moment of Inerta Mass analog s moment of nerta, I! I = m r 2 r defned relatve to rotaton axs Moment of Inerta SI unts are kg m 2 More bout Moment of Inerta Moment of Inerta of a Unform Rng I depends on both the mass and ts dstrbuton. If mass s dstrbuted further from axs of rotaton, moment of nerta wll be larger. vde rng nto segments The radus of each segment s R I =!m r 2 = MR 2

5 Example 8.6 Other Moments of Inerta What s the moment of nerta of the followng pont masses arranged n a square? a) about the x-axs? b) about the y-axs? c) about the z-axs? a) 0.72 kg"m 2 b) 1.08 kg"m 2 c) 1.8 kg"m 2 Other Moments of Inerta cylndrcal shell : I = MR 2 sold cylnder : I = 1 2 MR2 rod about center : I = 1 12 ML2 rod about end : I = 1 3 ML2 sphercal shell : I = 2 3 MR2 soldsphere : I = 2 5 MR2 bcycle rm flled can of coke baton baseball bat basketball boulder Example 8.7 Treat the spndle as a sold cylnder. a) What s the moment of Inerta of the spndle? (M=5.0 kg, R=0.6 m) b) If the tenson n the rope s 10 N, what s the angular acceleraton of the wheel? c) What s the acceleraton of the bucket? d) What s the mass of the bucket? a) 0.9 kg"m 2 b) 6.67 rad/s 2 c) 4 m/s 2 d) 1.72 kg M Example 8.8(skp) cylndrcal space staton of (R=12, M=3400 kg) has moment of nerta 0.75 MR 2. Retrorockets are fred tangentally at the surface of space staton and provde mpulse of 2.9x10 4 N s. Example kg sold cylnder of radus 0.6 m whch can rotate freely about ts axs s accelerated by hangng a 240 kg mass from the end by a strng whch s wrapped about the cylnder. a) Fnd the lnear acceleraton of the mass m/s 2 a) What s the angular velocty of the space staton after the rockets have fnshed frng? b) What s the centrpetal acceleraton at the edge of the space staton? a) #= rad/s b) a=10.8 m/s 2 b) What s the speed of the mass after t has dropped 2.5 m? 4.67 m/s

6 Rotatonal Knetc Energy Each pont of a rgd body rotates wth angular velocty #. KE = 1 2! m v = 1! m r 2 " Example 8.10 What s the knetc energy of the Earth due to the daly rotaton? Gven: M earth =5.98 x10 24 kg, R earth = 6.36 x10 6 m. KE = 1 2 I! 2 Includng the lnear moton KE = 1 2 mv I! 2 KE of center-of-mass moton KE due to rotaton 2.56 x10 29 J Example 8.11 sold sphere rolls down a hll of heght 40 m. What s the velocty of the ball when t reaches the bottom? (Note: We don t know R or M!) emo: Moment of Inerta Olympcs v = 23.7 m/s Example 8.12a The wnner s: ) Hollow ylnder ) Sold ylnder Example 8.12b The wnner s: ) Hollow ylnder ) Sphere

7 Example 8.12c The wnner s: ) Sphere ) Sold ylnder Example 8.12d The wnner s: ) Sold ylnder ) Monster Example 8.12e The wnner s: ) Sphere ) Mountan ew ngular Momentum Rgd body L = I! L = mvr = m!r 2 Pont partcle nalogy between L and p ngular Momentum L = I# Lnear momentum p = mv! = $L/$t F = $p/$t onserved f no net outsde torques onserved f no net outsde forces Rotatng har emo ngular Momentum and Kepler s 2nd Law For central forces, e.g. gravty,! = 0 and L s conserved. hange n area n $t s:! = 1 2 r(v "!t) L = mrv "!!t = 1 2m L

8 Example kg student sprnts at 8.0 m/s and leaps onto a 110-kg merry-go-round of radus 1.6 m. Treatng the merry-go-round as a unform cylnder, fnd the resultng angular velocty. ssume the student lands on the merrygo-round whle movng tangentally. = 2.71 rad/s Example 8.14 Two twn ce skaters separated by 10 meters skate wthout frcton n a crcle by holdng onto opposte ends of a rope. They move around a crcle once every fve seconds. y reelng n the rope, they approach each other untl they are separated by 2 meters. a) What s the perod of the new moton? T F = T 0 /25 = 0.2 s b) If each skater had a mass of 75 kg, what s the work done by the skaters n pullng closer? W = 7.11x10 5 J

Rotation Kinematics, Moment of Inertia, and Torque

Rotation Kinematics, Moment of Inertia, and Torque Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute