Chapter 12 Equilibrium & Elasticity

Transcription

1 Chapter 12 Equlbrum & Elastcty If there s a net force, an object wll experence a lnear acceleraton. (perod, end of story!) If there s a net torque, an object wll experence an angular acceleraton. (perod, end of story!) How can we keep thngs from mong, then? r r dp r d r Recall, net = and τ = dt dt Chapter 12 Equlbrum & Elastcty Equlbrum: P = constant and = constant Statc equlbrum: Objects that are not mong ether n translaton or rotaton. P = 0 = 0 Requrements of equlbrum r r dp r net = (balance of forces) dt net = 0 r d r τr = 0 τ = net (balance of torques) dt

2 or the stuatons n whch the forces that act on the body le n the xy plane. Then the only torque that can act on the body s τ z. We hae net,x = 0 net,y = 0 τ net,z = 0 τ net,z s the net torque that the external forces produce ether about the z axs or about any axs parallel to t Center of graty Center of graty: gratatonal force on a body effectely act at a sngle pont. or eeryday objects, the center of graty concdent wth ts center of mass.

3 Sample Problem 12-1 A beam of length and mass m = 1.8 kg, s at rest wth ts ends on two scales. A block of M = 2.7 kg, s at rest on the beam, wth ts center a dstance /4 from the beam s left end. What do the scales read ( l and r )? Sample Problem 12-1 A beam of length and mass m = 1.8 kg, s at rest wth ts ends on two scales. A block of M = 2.7 kg, s at rest on the beam, wth ts center a dstance /4 from the beam s left end. What do the scales read? Sum of all the forces on the beam must equal zero. ŷ : = 0 l Mg mg + r = 0

4 Sample Problem 12-1 A beam of length and mass m = 1.8 kg, s at rest wth ts ends on two scales. A block of M = 2.7 kg, s at rest on the beam, wth ts center a dstance /4 from the beam s left end. What do the scales read? Sum of all the torques about any pont on the beam must equal zero. Choose ths pont to be the left end of the beam. ẑ : τ = 0 (0)l Mg mg + ()r = Sample Problem 12-1 A beam of length and mass m = 1.8 kg, s at rest wth ts ends on two scales. A block of M = 2.7 kg, s at rest on the beam, wth ts center a dstance /4 from the beam s left end. What do the scales read? Whch yelds, r = Mg + mg 4 2 and 3 l = Mg + mg r = Mg + mg 4 2

5 Sample problem 12-2: a ladder = 12 m and mass m = 45 kg leans aganst a frctonless wall. h = 9.3 m. The ladder s c.o.m. s /3 from the lower end. A frefghter of M = 72 kg clmbs the ladder untl her c.o.m. s /2 from the lower end. What are the magntudes of the forces on the ladder from the wall and the paement? Queston = 65 N What should 1 be n order to keep the unform rod n statc equlbrum?

6 Queston ŷ : = 0 20 N 10 N 1 30 N + 65 N = 0 1 = 45 N ẑ : τ = 0 (8d)20 N + (4d)10 N + (2d) + (d)30 N (0d)65 N = 0 = 45 N 1 = 65 N 1 What should 1 be n order to keep the unform rod n statc equlbrum? 1) 45 N 2) 0 N 3) 40 N 4) t s ponted n the wrong drecton 5) none of the aboe Daly Quz, October 20, 2004 Suppose that the (unform) horzontal rod s held just barely n statc equlbrum by the rope and the pot pont. The grey arrow s the weght of the rod at the center of mass and the green arrow s a moable appled downward force. What happens when s moed toward A? 1) nothng 2) the net force ncreases 3) the net torque ncreases 4) the net torque decreases 5) none of the aboe mg pot pont

7 Daly Quz, October 20, 2004 Suppose that the (unform) horzontal rod s held just barely n statc equlbrum by the rope and the pot pont. The grey arrow s the weght of the rod at the center of mass and the green arrow s a moable appled downward force. What happens when s moed toward A? 1) nothng 2) the net force ncreases 3) the net torque ncreases 4) the net torque decreases 5) none of the aboe mg pot pont ŷ : xˆ : τ = l + Daly Quz, October 20, 2004 = = 2 pot,y pot,x mg + Tsn θ + T cosθ mg Tsn θ > 2 snce l > /2, the torque ncreases + 2 mg Tsn θ 1) nothng 2) the net force ncreases 3) the net torque ncreases 4) the net torque decreases 5) none of the aboe mg /2 l pot pont

8 Stress and Stran Mcroscopc ew of materals: Materals are made of atoms held n place by electrostatc nteractons wth neghborng atoms. These nteractons are such that the atoms are constantly n harmonc moton about ther equlbrum postons. Stress and Stran External forces can be exerted on these atoms. The atoms wll react to these forces dependng on ther mcroscopc enronment.

9 Deformaton Types elongaton (tensle strength) shear hydraulc compresson modulus s a constant depended on how much the materals react (deform) to the appled forces. stress = modulus x stran Elongaton (tensle strength) and compresson Stress s defned as perpendcular force per unt area. Stress = /A Stran s defned as the fractonal change n the length of the object. Stran = / Young s modulus E s the proportonalty constant. A = E

10 Shear stress Stress s defned as force n the plane per unt area. Stress = /A Stran s defned as the fractonal change n the moement of the object. Stran = x/ Shear modulus G s the proportonalty constant. A = G x Hydraulc Stress or three dmensons we use pressure, whch s also defned as force per unt area. Pressure p = /A Stran s defned as the fractonal change n the olume of the object. Stran = V/V Bulk modulus B s the proportonalty constant. p = A V = B V

11 Materals scence studes cause of specfc shape of ths cure. Elastc mt Regon we ll be dscussng ths semester. Addtonal regon of nterest to engneers. World Trade Center Study Suggests Desgn laws Ddn't Doom Towers By ERIC IPTON NY Tmes Artcle Publshed: October 20, Ths artcle dscusses the stresses, strans exerted on the supportng columns.