Lecture 18
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The document discusses internal and external torques, the three laws of angular motion, and key concepts related to rotational dynamics including: 1) Internal torque is applied within a system while external torque is applied across the system boundary. 2) The three laws of angular motion describe how torques cause changes in angular velocity and acceleration according to an object's moment of inertia. 3) Key concepts like angular impulse, work, power, and kinetic energy can be analyzed similarly to linear motion but involve torque, angular velocity/acceleration, and moment of inertia rather than force and linear velocity/acceleration.
Lecture 18
- 1. Internal vs. External Torque • Internal Torque : is applied to a system by a force Laws of Angular Motion acting within the system • External Torque : is applied to a system by a force or torque acting across the boundary of the system Objectives: Fquads • Define internal & external torques, rigid bodies • Understand and apply the 3 laws of angular motion Tflexor • Define angular impulse and understand the System relationship between angular impulse and angular Fgastroc momentum • Introduce the concepts of angular work, power, & rotational kinetic energy Wleg Wfoot Rigid Body 1st Law (Law of Inertia) • An object whose change in shape is negligible. • Objects made up of multiple parts can be • A rigid body in rotation will maintain a constant considered a rigid body if the parts don’t move angular velocity unless acted upon by an external relative to one another. torque. • Example: the leg + foot is a rigid body if no motion • If there is no net external torque acting on a rigid (or very little motion) occurs at the ankle body: – if the body is not rotating, it will continue not to • The laws of angular kinetics that follow apply only rotate. to rigid bodies – if the body is rotating, it will continue to rotate • In non-rigid bodies, each rigid part making up the at a constant velocity body must be analyzed separately (i.e. at the same speed in the same direction) 1
- 2. 2nd Law (Law of Acceleration) Effects of Torque • For rotation of a rigid body about its center of mass • Net torque and angular velocity ω in same direction: (or a fixed axis): magnitude of angular velocity increases T= Iα • Net torque and angular velocity ω in opposite direction: where: magnitude of angular velocity decreases (deceleration) – T : net external torque about the COM (or axis) Velocity Torque Change in Velocity – I : body’s moment of inertia about the COM (or axis) – α : angular accel. of the body about the COM (or axis) (+) (+) Increase in + dir. • If there is a net external torque acting on a body, the (+) (–) Decrease in + dir. angular acceleration is: – directly proportional to the net torque (–) (–) Increase in – dir. – inversely proportional to the moment of inertia – in the direction of the net torque (–) (+) Decrease in – dir. 3rd Law (Law of Reaction) Example Problem #1 During a squat lift, a person is holding a 450 N weight • For every action, there is an equal and opposite as shown below. What resultant hip moment is reaction. required for the lifter to remain motionless? • If the forces acting across a joint between two If the hip extensors have an average moment arm of 5 bodies causes body 1 to experience a torque, body cm, what total force do they need to generate? 2 will experience a torque: What = 430 N – of the same magnitude femur 15 cm – in the opposite direction HIP Mextension Mextension W = 450 N 40 cm tibia 2
- 3. Example Problem #2 Radial & Tangental Acceleration During a sit-up, the hip flexors generate a torque of • The acceleration of a body in angular motion can 85 Nm on the head-arms-torso. What torque do be resolved into two components: they generate on the lower limbs? – Tangental: along at Given the body position and inertial properties shown path of motion below, what are the accelerations of the head- v – Radial: perpendicular arms-torso and lower limbs? to path of motion a 15 cm 40 cm Ihat = 11.0 kg m2 Ilower = 6.0 kg m2 at = r α ar v2 α r ar = = r ω2 ω r W = 465 N W = 220 N Fgrf Torques & Tangental Acceleration Centripetal Force • Centripetal force produces radial acceleration • Torques produce tangental acceleration only • Magnitude of centripetal force: at = r α at m v2 I F c = m ar = = m r ω2 v at = T r r m T =Iα • Force required increases with: ar • Radial acceleration must α – object mass (m) Fc – velocity (v or ω) come from some other T r r – distance (r) from axis of source! ω rotation • F c always directed inward towards the axis of rotation axis of rotation 3
- 4. Example Problem #3 Angular Impulse A 3500 lb. race car is attempting to go through a flat • The linear motion of a body depends both on the turn of radius 500 ft. at 100 mi/hr. force and the duration that the force is applied What total friction force between the road and tires is • The angular motion of a body depends both on the required? torque and the duration that the torque is applied If the coefficient of friction between the road and tires • Angular Impulse : a measure related to the net is 1.0, will the car be able to negotiate the turn? effect of applying a torque (T) for a time (t): Angular Impulse =Tt • Angular impulse increases with: – Increased torque magnitude – Increased duration of application Angular Impulse & Momentum Example Problem #4 • The angular impulse due to the net external A hammer thrower is able to apply an average torque torque acting on a system equals the change in of 100 Nm to the hammer while spinning about his the angular momentum of the system over the longitudinal axis. same period of time The ball of the hammer has a mass of 7.25 kg and angular momentum at time t1 spins at a distance of 1.5 m from the axis of angular momentum at time t2 rotation If the hammer ball starts from rest, what is its angular velocity after 3 s, just prior to release? Ang. Impulse = Texternal (t2 – t1) = I2 ω2 – I1 ω1 What is the magnitude of its linear velocity upon release? ang. impulse when Texternal is constant between t1 and t2 4
- 5. Work, Power, & Energy • The concepts of work, power and kinetic energy also apply to objects in rotation: Linear Angular Work W = Fx∆px + Fy∆py Wa = T θ Power P = W / ∆t Pa = Wa / ∆t Kinetic Energy KE = ½ m v2 KEa = ½ I ω2 • General relationship between work and energy: W + Wa = ∆KE + ∆KEa + ∆PE + ∆TE 5