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LECTURE 21 TORQUE Instructor: Kazumi Tolich Lecture 21 2 ¨ Reading chapter 11-1 to 11-2 ¤ Torque ¤ Newton’s 2nd law for rotation Torque about an axis 3 ¨ Torque (𝜏) is a measure of twisting, and the magnitude of torque is defined as 𝐅⃗ 𝜏 = 𝑟𝐹 sin 𝜃 = 𝐹) 𝑟 = 𝐹𝑟* ¨ ¨ 𝑟* is called moment arm, or lever arm of 𝐅⃗. The SI unit for torque is N·∙m, the same as the unit of work. But torque and work are different physical quantities. 𝜃 𝐅⃗) 𝐫⃗ Axis 𝑟* Line of action Newton’s 2nd law for rotation 4 ¨ Newton’s 2nd law for rotation is given by . 𝜏 = 𝐼𝛼 ¨ ¨ ¨ The sign of torque is the same as the sign of angular acceleration it causes if it were the only torque acting in the system. If two or more torques act on a rigid object, the net torque is the sum of the torques with correct sign assigned to each torque. This is analogous to Newton’s 2nd law for linear motion: ∑ 𝐅⃗ = 𝑚𝐚 Quiz: 1 5 ¨ Two forces produce the same torque. Does it follow that the magnitudes of the forces are the same? A. B. Yes No Quiz: 21-1 answer 6 ¨ ¨ ¨ No Because torque is 𝜏 = 𝑟𝐹 sin 𝜃 = 𝐹) 𝑟 = 𝐹𝑟* , two different forces that act at different distances or angles could still give the same torque. For example, the diagrams below show the applications of various forces to various perpendicular level arms. The magnitudes of the torques about the dark blue dots are the same in all cases. L a 2L F b 0.5F F1 F2 0.5L c 2F Applications/Demo: 1 7 ¨ ¨ ¨ ¨ Door knobs: why are all door knobs located farthest from the door hinges? Bicycle pedals: why is it hard to get going if you try to start your bike with the pedal at the highest point? Wheelchairs: why are the handrims for wheelchairs for racing and basketball different? Demo: torque bar Quiz: 2 8 ¨ In case 1, one end of a horizontal massless rod of length 𝐿 is attached to a vertical wall by a hinge, and the other end holds a ball of mass 𝑀. In case 2 the massless rod is twice as long and makes an angle of 30° with the wall as shown. In which case is the total torque about an axis through the hinge bigger? A. B. C. Case 1 Case 2 same Quiz: 21-2 answer 9 ¨ ¨ ¨ ¨ Same The weight of the balls are the same in both cases. The perpendicular lever arms are the same in both cases. 𝜏 = 𝐹𝑟* 𝐹 = 𝑀𝑔 𝐹 = 𝑀𝑔 𝑟* = 𝐿 𝑟* = 2𝐿 sin 30° = 𝐿 Quiz: 3 10 ¨ Suppose a ladder on a rough floor is leaning against a frictionless wall as shown, and you are trying to calculate the net torque on the ladder about the axis through where the ladder touches the floor. What is the lever arm of the normal force by the wall? A. B. C. 2𝑟 𝑟 ℎ 𝐍=D h 𝐖=@ D. ℎF E. ℎF + 𝑟 F + 2𝑟 F 𝐍=> 𝐟⃗B,=> r r Quiz: 21-3 answer 11 ℎ ¨ The lever arm is the shortest distance between the line of action and the axis. ¨ Line of action 𝐍=D 𝑟* = ℎ h 𝐖=@ 𝐍=> 𝐟⃗B,=> r r Quiz: 4 12 ¨ Forces 𝐹⃗H and 𝐹⃗F have the same magnitude and are applied to identical disks on a frictionless horizontal table as shown. Which of the following statements is/are correct? Choose all that apply. A. Disk 1 has angular acceleration. B. Disk 2 has angular acceleration. Disk 1 has linear acceleration. Disk 2 has linear acceleration. C. D. Disk 1 Disk 2 Quiz: 21-4 answer 13 ¨ ¨ ¨ Disk 2 has angular acceleration. The net force on both disks are zero as magnitudes of 𝐹⃗H and 𝐹⃗F are the same, and they point in the opposite directions. The net torque on Disk 1 is zero since the lever arms of 𝐹⃗H and 𝐹⃗F on Disk 1 about its center of mass is zero. ¨ On the other hand, the lever arms of 𝐹⃗H and 𝐹⃗F on Disk 2 about its center of mass are non-zero. ¨ 𝐹⃗H and 𝐹⃗F both would cause Disk 2 to rotate counterclockwise, having a positive angular acceleration. 𝑟* 𝑟* Disk 1 Disk 2 Demo: 2 14 ¨ Hinged Stick and Ball ¤ When the bar is just about to become horizontal, the acceleration of the free-end of the bar is greater than 𝑔. ¤ 𝛼 ¤ 𝑎 I J M KLN = =O P KQN = 𝐿𝛼 = RL F = RL FQ >𝑔 Example: 1 15 ¨ Figure shows the massive shield door at a neutron facility at Lawrence Livermore National Laboratory; this is the world’s heaviest hinged door. The door has a mass of 𝑀 = 44,000 kg, a rotational inertia about an axis through its hinges of 𝐼 = 8.7 × 104 kgm2, and a front face width of 𝑤 = 2.4 m. Neglecting friction, what steady force, applied at its outer edge and perpendicular to the plane of the door, can move it from rest through an angle of 𝜃 = 90º in 𝑡 = 30 s? Example: 2 16 ¨ A wheel on a game show is given an initial angular speed of 𝜔X = 1.22 rad/s. It comes to rest after rotating through ∆𝜃 = 0.75 of a turn. a) b) Find the average torque exerted on the wheel given that it is a disk of radius 𝑟 = 0.71 m and mass 𝑚 = 6.4 kg. If the mass of the wheel is doubled, and its radius is halved, will the angle through which it rotates before coming to rest increase, decrease, or stay the same assuming that the average torque exerted on the wheel is unchanged?