Torque • Tutorial homework due Thursday/Friday. • Covering torque today. 1 Torque Force causes linear acceleration while torque causes angular acceleration Force applies Force applies small torque large torque Force applies no torque Use Greek letter tau for torque: τ Axis of rotation τ = rF sin φ where r is the distance between the force application and the axis, F is the force magnitude, and φ is the angle between r and F. 2 Calculating torque 3 ways τ = rF sin φ distance r times force times sine of angle between them. τ = r ( F sin φ ) = rF⊥ τ = rF sin φ F φ F φ r F sin φ distance r times the perpendicular force τ = F ( r sin φ ) = Fl F φ Force times the perpendicular distance. Also called lever arm or moment arm. r l = r sin φ r 3 Clicker question 1 Set frequency to BA Three forces labeled A, B, C are applied to a rod which pivots on an axis through its center. Which force causes the largest magnitude torque? sin 45° = cos 45° = 1/ 2 = 1/ 1.414 2F A. A L B. B C L/2 C. C 45° D. Two are tied for largest A B L/4 F F E. All three are the same L LF LF L LF F sin 90° = LF sin 45° = 2F sin 90° = C: B: A: 2 2 1.4 4 2 A generates the largest magnitude torque 4 Newton’s 2nd law for angular motion Consider a force pushing on mass m perpendicular to the rod R. At the moment it is into the screen but in general, the force is always tangent to the motion. Newton’s 2nd law for this is Ftan = matan R m Ftan But we know that atan = Rα so Ftan = mRα 2 Multiplying both sides by R: RFtan = mR α 2 But RFtan is the torque applied and mR is the moment of inertia for this situation so τ = Iα This is Newton’s 2nd law for angular motion A force causes linear acceleration; a torque causes angular acceleration Moment of inertia resists angular acceleration 5 like mass resists linear acceleration Summary of torque so far A force F applied a distance r from an axis provides a torque of ! τ = rF sin φ , where φ is the angle between the r! and F vectors. SI units of torque are newton·meter (N·m). Not joules. ! ! Newton’s 2nd law Fnet = ma can be rewritten as ! ! τ net = Iα for rotational motion. α is the angular acceleration. It is also related to the tangential acceleration: atan = rα The addition of torques requires a modification to the free body diagram. We need to show where the force acts on the object. Special note: Gravity always acts at the center-of-mass of the object (also called the center-of-gravity). 6 Generally use M for a rotating mass and m for a point-like mass Clicker question 2 Set frequency to BA A mass is hanging from the end of a horizontal bar which pivots about an axis through its center and is initially stationary. The bar is released and begins rotating. As the bar rotates from horizontal to vertical, the magnitude of the torque on the bar… A. increases B. decreases C. remains constant τ = rF sin φ r and F stay the same but φ decreases from 90° when horizontal to 0° when vertical 7 Massive pulleys A m=2 kg weight is attached to the end of a rope coiled around a pulley with mass of M=6 kg and radius of 0.1 m. T What is the acceleration of the weight? Weight free body diagram: T − mg = ma Fg = mg For the pulley, rope (and therefore the tension force) comes off at 90° relative to R so τ = RT (the other forces produce no torque). Disk moment of inertia is 1 2 T R 2 kg MR 2 so τ = Iα becomes RT = 12 MR 2α Acceleration on outside of pulley equals acceleration of 2 kg weight because connected by rope. However, signs need to match so −a = Rα 1 2 a RT = − MR 2 R 1 becomes T = − 2 Ma 8 Massive pulleys A 2 kg weight is attached to the end of a rope coiled around a pulley with mass of 6 kg and radius of 0.1 m. What is the acceleration of the weight? 1 T = − Ma We have two equations T = mg + ma and 2 2 kg mg + ma = − 12 Ma rearranges to −2mg = Ma + 2ma 2 −2mg −2 ⋅ 2 kg ⋅10 m/s −40 N Solving a = = = = −4 m/s2 M + 2m 6 kg + 2 ⋅ 2 kg 10 kg gives Since we defined up as positive, the acceleration is 4 m/s2 downward. 9 Clicker question 3 Set frequency to BA For the 2 kg weight attached to 6 kg pulley with radius of 0.10 m, what is the kinetic energy of the weight after dropping for 1 second. Remember a=4 m/s2 and K=½mv2, K=½Iω2,and Idisk=½MR2. A. 2 J B. 4 J C. 8 J D. 16 J E. 32 J After 1 second v = v0 + at = 0 + 4 m/s2 ⋅1 s = 4 m/s 2 kg Kinetic energy for the weight is 2 2 1 1 K = 2 mv = 2 2 kg ⋅ ( 4 m/s) = 16 J 10
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