Physics Week 9 Notes
Physics Week 9 Notes PHYS2001
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This 12 page Class Notes was uploaded by Grace Lillie on Saturday March 12, 2016. The Class Notes belongs to PHYS2001 at University of Cincinnati taught by Alexandru Maries in Fall 2016. Since its upload, it has received 17 views. For similar materials see College Physics 1 (Calculus-based) in Physics 2 at University of Cincinnati.
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Date Created: 03/12/16
Chapter 10 – Rotation of a Rigid Object About a Fixed Axis 10.1 – Angular Position, Velocity, and Acceleration kinematics are similar to translational motion - instead of x or y for position, we use θ for angular position s=rθ arc length s is related to angular position: π *The unit used for θ is the radian: θ rad)= θ (deg) 180° θf−θ i ∆θ - average angular speed: ω avg t −t = ∆t f i ∆θ dθ - instantaneous angular speed: ω= lim = ∆t→ 0t dt ω fω i ∆ω - average angular acceleration: α avg t −t = ∆ t f i ∆ω dω α= lim = - instantaneous angular acceleration: ∆t→ 0∆t dt *right-hand rule: when you wrap your four fingers around the axis of rotation and extend your thumb, it points in the direction of angular velocity. If ω is increasing, angular velocity points in the same direction; if ω is decreasing, angular velocity points in the opposite direction. 10.2 – Analysis Model: Rigid Object Under Constant Angular Acceleration - an example of a rigid object under constant acceleration is a CD - the kinematic equations are the same! Just switch x,v,and a with θ, ω, and α: - There are two things to keep in mind that are different in rotational motion: 1) you have to specify a rotation axis 2) the object keeps returning to its original orientation (can make multiple revolutions) 10.3 – Angular And Translational Quantities - There are some helpful relationships between angular and translational quantities: v= ds=r dθ =rω dt dt tangential velocity related to angular velocity. *every point on a rigid object has the same angular speed, but different tangential speeds (because r is different) dv dω at= =r =rα dt dt tangential acceleration related to angular acceleration 2 v 2 ac= r =rω centripetal acceleration, equals the magnitude of radial acceleration, a r total acceleration is the vector sum of tangential and radial acceleration, and has a magnitude: a= √ +t = r √ +r ω =r α +ω √ 2 4 10.4 – Torque - Force causes changes in translational motion, what causes changes in rotational motion? - torque τ is the tendency of a force to rotate an object around an axis - In the wrench example, the force F is being applied. Fsinφ The component is what causes the rotation. The quantity d is the moment arm or lever arm - τ=rFsinφ=Fd - d=rsinφ ∑ τ=τ +τ +… - 1 2 If two or more forces act on an object *Torque depends on your choice of axis! In the wrench example, the axis is at O pointing out of the page. Sometimes the choice will be obvious, but once you pick it, stick to it! *Torque is NOT the same as force. Forces cause change in motion, but the effectiveness of forces causing changes in rotational forces depends on the magnitude and the moment arm, which is torque - The unit or Torque is force times length (N·m) *Torque is also not the same as work, even though they have the same units! 10.5 – Analysis Model: Rigid Object Under a Net Torque - the angular acceleration of a rigid object rotating about a fixed axis is proportional ∑ τ= ∑ Ftr= ( at)= (mrα )r=(mr 2α=Iα to the net torque acting about that axis: I= ∑ m r2 - I is the moment of inertia i ii - the moment of inertia is the resistance to changes in rotational motion. It depends on the mass of the object and how the mass is distributed around the rotation axis * The figure below is from Physics for Scientists and Engineers, p.304 * 10.6 – Calculation of Moments of Inertia 2 2 2 - moment of inertia of a rigid object: I= lim ∑ ri∆m i r∫dm= ρr ∫V ∆ i →0i (ρ is density and V is volume) - There are multiple ways of expressing density: ρ=m/V volumetric mass density (mass per unit volume) σ=ρt surface mass density (mass per unit area) M λ= =ρ/A linear mass density (mass per unit length) L - parallel-axis theorem – simplifies the calculation of the moments of inertia of an object: I=I CM+M D 2 10.7 – Rotational Kinetic Energy 1 2 2 1 2 KR= ∑ m iiω = Iω rotational kinetic energy 2 (i ) 2 10.8 – Energy Considerations in Rotational Motion dW P= dt=τω Power delivered to a rotating rigid object ωf W= ∫ Iωdω= Iω1 2 1Iω 2 ωi 2 f 2 i work-kinetic energy theorem for rotational motion. - Net work done by external forces is the change in total kinetic energy (the sum of translational and rotational kinetic energy). Ex: a baseball moves through space and spins after it’s pitched. - Conservation of energy can also be applied to rotational situations. Just remember to include both translational and rotational kinetic energy 10.9 – Rolling Motion of a Rigid Object - When a cylinder rolls, a point on the rim moves in a cycloid path, but the center of mass moves in a straight line. In such pure rolling motion (an object with uniform radius rolls without slipping), there is a simple relationship between rotational and translational motion: ds dθ vCM =R =Rω condition for pure rolling motion dt dt dvCM dω aCM =R =Rα dt dt (Figure 10.25 from page 317 of Physics for Scientists and Engineers) 1 2 - The total kinetic energy at point P is:K= I ω P 2 K= 1 I ω + 1 M v2 2 CM 2 CM Total Kinetic Energy of a rolling object is the sum of rotational energy about its center of mass and the translational kinetic energy of its center of mass. - For an object to undergo accelerating rolling motion without slipping, there must be friction present to produce a net torque. **Even though there’s friction, mechanical energy is still conserved because the contact point is at rest relative to the surface (See Figure 10.25C)** - rolling friction will transform mechanical energy into internal energy, but is mostly ignored for now (From page 314 of Serway Jewett Physics for Scientists and Engineers)