Rolling Motion We can find Ip using the parallel axis theorem CM CM Ip=ICM+ MR2 Ko=1/2(cM+MR2)2 Kor=12lcM02+12M(R202)=1/2lcM02+1/2MvM2! TOT CM0+M∥ Physics 121: Lecture 19, Pg 6
Physics 121: Lecture 19, Pg 6 Rolling Motion We can find IP using the parallel axis theorem KTOT = CM + MVCM 1 2 1 2 2 2 I VCM P Q CM IP = ICM + MR2 KTOT = 1/2 (ICM + MR2 ) 2 KTOT = 1/2 ICM 2 + 1/2 M (R22 ) = 1/2 ICM 2 + 1/2 M vCM 2 !
Rolling Motion Cylinders of different /rolling down an inclined plane 0 △K=-AU=Mgh R 0 M K=O TOT CM +-M CM V=0 Physics 121: Lecture 19, Pg7
Physics 121: Lecture 19, Pg 7 Rolling Motion Cylinders of different I rolling down an inclined plane: h v = 0 = 0 K = 0 R K = - U = Mgh v = R M KTOT = CM + MVCM 1 2 1 2 2 2 I
Rolling If there is no slipping(due to friction 2V Where= OR In the lab reference frame In the cm reference frame Physics 121: Lecture 19, Pg 8
Physics 121: Lecture 19, Pg 8 Rolling... If there is no slipping (due to friction): v 2v In the lab reference frame v In the CM reference frame v Where v = R
Rolling Kot =lIcmo2+lMvM use y= or and /=cMR2, hoop: C disk C=1/2 sphere: C=2/5 C etc So C+1Mv2= Mgh V=√2qh C+1 The rolling speed is always lower than in the case of simple Sliding since the kinetic energy is shared between CM motion and rotation Physics 121: Lecture 19, Pg 9
Physics 121: Lecture 19, Pg 9 Rolling... Use v= R and I = cMR2 . So: ( 1)Mv Mgh 2 1 2 + = 1 1 v 2gh + = The rolling speed is always lower than in the case of simple Sliding since the kinetic energy is shared between CM motion and rotation. hoop: c=1 disk: c=1/2 sphere: c=2/5 etc... c c c c KTOT = CM + MVCM 1 2 1 2 2 2 I
Example: Rolling Motion A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane Ball has radius r 0 M Physics 121: Lecture 19, Pg 10
Physics 121: Lecture 19, Pg 10 Example : Rolling Motion A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ? M h M v ? Ball has radius R