Equilibrium. Today: Center of Gravity Static Equilibrium. Sign convention for torques: (-) CW torque (+) CCW torque

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Sydney Brendan Green
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1 Tue Sept 29 Assign 5 Friday Exam Mon Oct 5 Morton 235 7:15-9:15 PM by Wednesday afternoon at the latest if conflict Q/A Sunday 1-3,4-6,7-9 Today: Center of Gravity Static Equilibrium Sign convention for torques: (-) CW torque (+) CCW torque Equilibrium ΣF = 0 and Στ = 0 In 2-d: ΣF X = 0 and ΣF Y = 0 Goal: Write expression for Στ and ΣF Choose axis where don t know force and don t care

2 At which point and at which direction would the least amount of force be required to hold the lever stationary? Longest lever arm - perpendicular force

3 Torque More Formal Pick axis. Can calculate torque due to any force. Torque (τ) = (mag of Force) x (perpendicular lever arm) τ = F r Axis of Rotation τ = F r sinθ τ = F r τ = 0 Perpendicular Lever arm line drawn through hinge and perpendicular to line of action.

4 Torque Perpendicular Lever Arm axis May have to calculate using trig Be careful to make sure you use right trig function May be given to you Perpendicular Lever arm line drawn through hinge and perpendicular to line of action.

5 Pushups Varying Force How to make pushups easier (On homework presume horizontal) θ Tilted Pushups: Torque from weight smaller because of angle Arms still pushing perpendicular Less force required

6 Two Different Approaches Find lever arm through trig Look at Components of force r θ F θ F θ F d d, r τ = F r τ = F (d cos θ) τ = F τ = F d τ = (F cosθ) d

7 A cook holds a 2.00 kg carton of milk at arm's length. The bicep muscle exerts a force F b on the forearm. What is the expression for the torque of the bicep muscle on the forearm? Use the elbow as the axis of rotation. (1) (F b cosθ) 8cm (2) (F b sinθ) 8cm (3) F b 8cm (4) -(F b cosθ) 8cm (5) -(F b sinθ) 8cm (6) -F b 8cm (7) None of the above

8 A cook holds a 2.00kg carton of milk at arm's length. What force F b must be exerted by the bicep muscles? (ignore the weight of the forearm for now).

9 A cook holds a 2.00 kg carton of milk at arm's length. The bicep muscle exerts a force F b on the forearm. What direction is the force of the elbow on the forearm? (Choose 1,3,5,7 only if exactly on axis. Choose zero for no force) F E To balance torque, need to push down. To balance forces horizontally, need to push to left.

10 A cook holds a 2.00 kg carton of milk at arm's length. What force F b must be exerted by the bicep muscles? (ignore the weight of the forearm for now). "Write the expression for the sum of the torques." Write out each torque in terms of variables or quantities given. Choose axis at elbow since don't know those forces yet. Στ = 0 F g (25cm+8cm) F b cos(75º)(8cm) + F ELBOW (0cm) = 0 F g = 2.00kg*9.8 = 19.6N F b = 312N

11 What is the magnitude of the horizontal component of the force that the elbow exerts on the forearm? (1) 61.2N (2) 292N (3) 301N (4) 307N (5) 312N (6) 332N ΣF x = 0 F b sin(75) F ELBOW,X = 0 F ELBOW,X = 312N*sin(75) = 301N

12 What are the magnitude and direction of the force exerted on the forearm by the elbow? If we can find the components of the force, we can reconstruct the full force vector. F ELBOW,X = 301N ΣF y = 0 -F g + F b cos(75) F ELBOW,Y = 0 F ELBOW,Y = 312N*cos(75) 19.6N F ELBOW,Y = 61N 301N θ 61N

13 Center of Gravity (Center of Mass) If extended object, can choose one place to apply Force due to Gravity (weight) This is called the Center of Gravity (CG) Finding CG: If symmetrical, along line of symmetry "real center If irregular, can balance or hang

14 Stability If CG over or under point(s) of support then stable. Stable Equilibrium: displace from equilibrium and it returns Unstable equilibrium: displaced but does not return to equilibrium

15 Stability Unstable equilibrium Neutral equilibrium: displaced stable in new state

16 Stability - Stance Lower CG, more stable in this situation Hit sideways at CG, easier to knock higher CG off balance

17 A child is sitting on an adult s leg. Given the measurements shown to the right, what is the torque due to the child if we use the point shown as the axis of rotation? axis (1) Nm (2) Nm (3) Nm (4) 3.80 Nm (5) Nm (6) 3.72 Nm (7) Nm (8) 37.2 Nm τ = -F*r = -mg*r = -(10 kg) (9.8 m/s 2 ) (0.38 m) τ = Nm

18 A child is sitting on an adult s leg. Given the measurements shown to the right, what is the torque due to weight of the leg if we use the point shown as the axis of rotation? axis (1) +0.8 Nm (2) 0.8 Nm (3) Nm (4) 3.20 Nm (5) Nm (6) 7.84Nm (7) Nm (8) 15.8 Nm Use location of CG for point of application τ = -F*r = -mg*r = -(4 kg) (9.8 m/s 2 ) (0.20 m) τ = Nm

19 A child is sitting on an adult s leg. What is the tension in upper leg muscle? axis Στ(about knee) = 0 -τ Child τ Leg + τ Muscle = Nm 7.84 Nm + F * 0.02m= 0 F = 45.0Nm / 0.02m F = 2252N

20 A 800 N painter stands on a 400 N scaffold (a uniform rectangular board) with one rope supporting each end. The painter stands onequarter of the length of the scaffold from the left side. Compare the tension in the two ropes. (1) T L > T R (2) T L < T R (3) T L = T R What is the tension in each rope? Στ(about left rope) = 0-800*(0.25L) -400*(0.5L)+T R *L = 0 Στ(about right rope) = 0 -T L *L + 800*(0.75L)+ 400*(0.5L) = 0 L/2 from left or right end

21 Draw, Draw, Draw!!!! Look for similarities If don t know L, take leap of faith and stick it in as a variable. See if it cancels. Break forces into components perpendicular and parallel to the board, bridge, arm, tray, fishing pole, wrench,. Στ = 0, ΣF X = 0, ΣF Y = 0

22 Look for similarities 35 Στ = 0, ΣF X = 0, ΣF Y = 0

23 A 1500N beam is attached to a wall by a hinge. The beam is 4.0m long, and the mass of the beam is distributed evenly along the beam. The beam is supported by a cable that is attached 1.0m to the beam from the wall and makes an angle of 35 from the vertical. The goal is to find the tension in the cable. 35 1m At what point does it make the most sense to place the axis of rotation? (1) Hinge (2) Point Cable attaches (3) Center of beam (4) Right End 4m

24 A 1500N beam is attached to a wall by a hinge. The beam is 4.0m long, and the mass of the beam is distributed evenly along the beam. The beam is supported by a cable that is attached 1.0m to the beam from the wall and makes an angle of 35 from the vertical. The goal is to find the tension in the cable. 35 What is the lever arm for the force due to gravity? F Use the hinge as the axis of rotation. g (1) 0m (2) 1m (3) 2m (4) 3m (5) 4m 1m 4m Center of gravity applied at center of board. Center of board is 2m from the hinge.

25 A 1500N beam is attached to a wall by a hinge. The beam is 4.00m long, and the mass of the beam is distributed evenly along the beam. The beam is supported by a cable that is attached 1.00m to the beam from the wall and makes an angle of 35 from the vertical. The goal is to find the tension in the cable. 35 What is the tension in the cable? (1) 1500N (2) 2460N (3) 3000 N (4) 3660 N (5) 5230 N (6) 6000N (7) 7320N 1m 4m Στ(about hinge) = 0 T(cos35 )*(1m) -1500*(2m)+F HINGE * 0 = 0 T = (3000N*m)/(cos35 *1m) T = 3660N

26 A 1500N beam is attached to a wall by a hinge. The beam is 4.00m long, and the mass of the beam is distributed evenly along the beam. The beam is supported by a cable that is attached 1.00m to the beam from the wall and makes an angle of 35 from the vertical. 35 What direction does the hinge push on the beam? (1) up and to the right (2) directly to the right (3) down and to the right ΣF X = *sin35 + F xhinge = 0 F xhinge = N ΣF Y = cos N + F yhinge = 0 F yhinge = -2998N N F yhinge = -1498N Or put axis at 1m: CG applies CW torque Hinge must apply CCW torque.

27 A cable is attached to an elliptical rim and attached via a pulley to a weight. The black dot represents the axis. In which configuration is the torque on the ellipse the greatest? (1) (2) Exercise machines or bikes may find elliptical gears or cams adjusts force at different points in motion You have fixed lever arm (black handle) but lever arm for load is adjusting throughout cycle Longer lever arm for 2

28 A picnic basket weighing 100N is hanging from the end of a 3.0mlong beam. The beam weighs 250 N and is supported by a cable which attaches to the middle of the beam and makes an angle of 30 relative to the perpendicular to the beam. The beam is attached to the wall by a hinge. Suppose a bear weighing 1250N walks out onto the beam. The cable can only withstand a tension of 1800N. How close can the bear get to the basket before the cable snaps?

ΣF = 0 and Στ = 0 In 2-d: ΣF X = 0 and ΣF Y = 0 Goal: Write expression for Στ and ΣF

ΣF = 0 and Στ = 0 In 2-d: ΣF X = 0 and ΣF Y = 0 Goal: Write expression for Στ and ΣF Thur Sept 24 Assign 5 Friday Exam Mon Oct 5 Morton 235 7:15-9:15 PM Email if conflict Today: Rotation and Torques Static Equilibrium Sign convention for torques: (-) CW torque (+) CCW torque Equilibrium