Week7_Torque_Written 2

1 Physics 131 Written Homework Problems: Week 7 Problem 1: Merry-go-round shenanigans Tom and Jerry both push on the 2 m radius merry-go-round as shown in the figure on the right. Tom pushes with a force of magnitude 30 N . The pivot is the black dot shown at the center. A. What force should Jerry push with so there is no net torque on the merry-go-round about its center? B. In order for the net force to be zero, there must be a force ࠵హ࠵? ⃗ ࠵ౝ࠵?࠵ౝ࠵?࠵ౝ࠵?࠵ౝ࠵?࠵ౝ࠵? exerted on the merry-go- round at the pivot. Calculate the ࠵౥࠵? - and ࠵౦࠵? -components of this force (using the usual coordinates). Hint: You only need to worry about forces here – not torques! Problem 2: Tension in the Deltoid Muscle When we hold a weight at arm’s length at the height of our shoulder, the deltoid muscle is largely responsible for keeping our arm and the object at their fixed height. To estimate the tension in the Deltoid, we model the arm as in the drawing below, with all forces acting on the arm at the indicated distances from the origin ( O ). The arm’s mass is 10 kg and its weight ࠵హ࠵? ⃗ ࠵఺࠵? ࠵స࠵?→࠵ఴ࠵? acts on the center of gravity (CG), while the small object’s mass is 6 kg , and it exerts a force ࠵హ࠵? ⃗ ࠵ు࠵? ࠵ొ࠵?→࠵ఴ࠵? all the way at the end of the arm. The deltoid exerts a force at an angle of ࠵༃࠵? = 15° . CG θ 18 cm ࠵హ࠵? ⃗ ࠵ే࠵? ࠵ష࠵?→࠵ఴ࠵? O ࠵హ࠵? ⃗ ࠵఺࠵? ࠵స࠵?→࠵ఴ࠵? 36 cm 72 cm ࠵హ࠵? ⃗ ࠵ు࠵? ࠵ొ࠵?→࠵ఴ࠵? ࠵హ࠵? ⃗ ࠵ే࠵? ࠵ష࠵?→࠵ఴ࠵? ࠵హ࠵? ⃗ ࠵఺࠵? ࠵స࠵?→࠵ఴ࠵? ࠵హ࠵? ⃗ ࠵ు࠵? ࠵ొ࠵?→࠵ఴ࠵? I fxt FOP 5 5124,0 T 24109 3 Fg 15 0 15 5 Tt 30 2523 3053 IFg 26 sn F FI Es oF C3ON Ccos6OE sin6005 26.38N C cos 80 sin 80 1 0 10.425 51.965 lx wmpe 10 VN 1ywmp 51.96Nf

2 For all questions below, torques are measured relative to the origin ( O ) and counterclockwise rotations are positive. A. What is the torque (in N ∙ m ) exerted by ࠵హ࠵? ⃗ ࠵఺࠵? ࠵స࠵?→࠵ఴ࠵? ? B. What is the torque (in N ∙ m ) exerted by ࠵హ࠵? ⃗ ࠵ు࠵? ࠵ొ࠵?→࠵ఴ࠵? ? C. For the arm+object system to be static, what is the torque exerted by the deltoid muscle? Express your answer in N ∙ m . D. If the system is in equilibrium, what is the magnitude of the tension, �࠵హ࠵? ⃗ ࠵ే࠵? � , in Newtons? E. If you look carefully at the free body diagram, the system cannot truly be in static equilibrium because the sum of the drawn forces is not zero. There must be a force on the arm acting at the origin (O) exerted by the scapula. What are the ࠵౥࠵? - and ࠵౦࠵? - components of this force? FEE magda to kg 9.8m 1st o 36m Fgs a ÉÉÉ dI kgx a.am s xo 72m 42.336NTyTdeltoid TFE AtTNWDA 35.28 N m 42 336 N m 477.616Mt FI TEE.in f ftfnnFo 41666.03T R component Fx FI cos 0 Fy Ft s in Q 41609.26mi 1666.03N Ws 150 166.03 in 150 10kg6kg 9 AM D1 274 conL y

4 Problem 4: A Hanging Beam A beam of length 5 . 0 m is hung from the ceiling with two strings. The mass of the beam is small enough that you can neglect it. The beam has two masses attached, as shown in the figure. The system is in equilibrium such that ࠵హ࠵? ⃗ ࠵౛࠵?࠵౛࠵?࠵ౝ࠵? = 0 and ࠵༏࠵? ࠵౛࠵?࠵౛࠵?࠵ౝ࠵? = 0 . A. Draw an extended free-body diagram of the beam by itself. B. Use the torque equation for static equilibrium to solve for the magnitude of the tension force � F �⃗ T1 � . Hint: choose the pivot at the right end of the beam. With that choice of the pivot, the torque from ࠵హ࠵? ⃗ ࠵ే࠵?2 would be zero . C. Use the force equation for static equilibrium with your answer above to solve for the magnitude of the tension force �࠵హ࠵? ⃗ ࠵ే࠵?2 � . ࠵హ࠵? ⃗ ࠵ే࠵?1 ࠵హ࠵? ⃗ ࠵ే࠵?2 Wi X I Wz 5 1 IFT I X 5 0 29 4 19 6 X 4 I Ft I x 5 0 117,1 21.571 I Ft I I Ftz l Wp Wz o 21.56 Ftz 19 6 29 44 0 11721 27442