Chapter 9 Rigid Bodies and Rotational Motion Angular velocity an object which rotates about a fixed axis has an average angular velocity av av 2 1 usually rad s but sometime rpm rps t 2 t1 instantaneous angular velocity is given by d lim t 0 t dt since s r ds d r or v r dt dt Phys 250 Ch9 p1 t r s Angular Acceleration the rate of change of angular speed d d 2 av lim 2 t 0 t t dt dt related to linear acceleration in circular motion dv d atan r at r dt dt v2 ac 2 r r ac 2r aT r total linear acceleration 2 a ac aT 2 Example In a hammer throw a 7 25 kg shot is swung in a circle 5 times and then released The shot moves with an average radius of 2 1 m and an average angular acceleration of 2 3 rad s2 What is the average tangential force and what is the maximum centripetal force on the hammer Phys 250 Ch9 p2 Rotation with constant angular acceleration just like linear 1 d Angular Linear 1 2 0 0t t 2 0 0 t 2 1 2 x x0 v0t at 2 v0 v x x0 t 2 0 t v v0 at 2 2 0 2 0 watch units consistency Phys 250 Ch9 p3 2 v 2 v0 2a x x0 Example The wheel on a moving car slows uniformly from 70m rad s to 42 rad s in 4 2 s What is the angular acceleration of the wheel What angle does the wheel rotate in those 42 s How far does the car go if the radius of the wheel is 0 32 m Phys 250 Ch9 p4 Torque the rotational analogue of force Torque force x moment arm FL F r sin moment arm perpendicular distance through which the force acts L L F F L F Example The bolts holding a head gasket are to be torqued down to 90 N m If a 45 cm wrench is used what force should be applied perpendicular to the wrench handle Phys 250 Ch9 p5 Example The crank arm of a bicycle pedal is 16 5 cm long If a 52 0 kg woman puts all her weicht on one pedal how much torque is developed when the crank is horizontal How much torque is developed when the pedal is 15 from the top Phys 250 Ch9 p6 Equilibrium stability steadiness balance etc Mechanical Equilibrium absence of change in motion Net Force 0 F 0 usually no motion sum of x force components sum of y force components F F x y 0 0 With Rotational Equilibrium Rotational Equilibrium absence of change in rotation usually no rotation net torque is zero i 0 about any axis i 0 for all torques lying in the same plane Watch signs for torque F L Positive torque for counterclockwise rotation F L Phys 250 Ch9 p7 L F Negative torque for clockwise rotation F L Center of Gravity CG aka Center of Mass the point of an object from which it could be suspended without tending to rotate The point where all the mass of an object can be considered to be located CG does not need to be located within the physical object Horseshoe for example usually easily identified from symmetry Example A 5 kg mass hangs from the 5 cm mark on a 1 meter long rod An unknown mass hangs from the 85 cm mark The rod has a mass of 2 0 kg and is balanced at the 35 cm mark What is the unknown mass Phys 250 Ch9 p8 Example A sign weighing 400 N is suspended from the end of a 350 N horizontal uniform beam What is the tension in the cable 35o w Phys 250 Ch9 p9 Elasticity stretchiness springiness how materials respond to stress compression tension shear Stretch ability amount of stress applied force produces a strain elongation compression shear Hooke s Law the amount of stretching is proportional to the applied force F kx The details of such springiness depends upon the size and shape of the material as well as how the forces are applied 1 Ton x x 2 Tons Phys 250 Ch9 p10 Elastic Limit the maximum stress force which can be applied to an object without resulting in permanent deformation Plastic Deformation the permanent deformation which results when a materials elastic limit has been exceeded Ultimate strength greatest tension or compression or shear the material can withstand snap A malleable or ductile material has a large range of plastic deformation Fatigue small defects reduce materials strength well below original strength Phys 250 Ch9 p11 Young s Modulus how things stretch elastically stress force per area F A A L0 L0 compression L A tension A L strain fractional change in length change in length per original length L Lo Elastic modulus stress strain Young s modulus for stretching in one direction Y Phys 250 Ch9 p12 F A L L0 A Example A steel elevator cable supports a load of 900 kg The cable has a diameter of 2 0 cm and an initial length of 24 m Find the stress and the strain on the cable and the amount that it stretches under this load Phys 250 Ch9 p13 Torque and Moment of Inertia For a single mass FT maT FTr maTr m r r mr2 moment of inertia I mr2 I looks like F ma for a system of objects a rigid object I miri2 Phys 250 Ch9 p14 ac 2r L R2 L R2 R 1 I ML2 3 Thin Rod axis at end 1 I ML2 12 Thin Rod a 1 I MR 2 2 Solid Disk 1 2 2 I M R1 R2 2 Hollow Cylinder a R b b 1 I M a 2 b 2 12 Rectangular Plate through center 1 I M a 2 b 2 3 Thin Rectangula r Plate about edge R Phys 250 Ch9 p15 2 I MR 2 5 Solid Sphere I MR 2 Thin Walle d Hollow Cylinder R 2 I MR 2 3 Thin Walle d Hollow Sphere Example A cylindrical winch of radius R and moment of inertia I is free to rotate without friction A cord of negligible mass is wrapped about the shaft and attached to a bucket of mass m What is the acceleration of the bucket when it is released Phys 250 Ch9 p16 Angular Momentum L I like p m v Angular momentum is conserved in the absence of external torques Lstart Lend for a point mass moving in a circle L mvr mr2 conservation of angular momentum implies Kepler s 3rd law Example Ann ice skater starts spinning at a rate of 1 5 rev s with arms extended He then pulls his arms close to his body decreasing his moment of inertia to of its initial value What is the skater s final angular velocity Phys 250 Ch9 p17 Rotational Kinetic Energy for a single point particle 1 1 KE mv 2 mr 2 2 2 2 …