Physics 3210In central-force motion, the areal velocity is a constant What is the period t of an elliptical orbit with area A?Kepler’s First Law states that planets move in elliptical orbits with the sun at one focus. Why is the sun at one focus of the orbit? Because...A system of n particles is described by the masses and positions of each particle: What is the position of the center of mass R of the system?Consider particles a and b, and the internal force fab=the force on particle a due to particle b. How does Newton’s third law relate the internal forces?Suppose a system of particles experiences only internal forces (no external forces). What can you say about the momentum of the system?Slide 7A chain with length b and mass density r is initially attached at both ends. One end of the chain is cut. After this falling end has moved down a distance x, what is the mass of the right (falling) side of the chain?A chain with length b and mass density r is initially attached at both ends. One end of the chain is cut; the other end of the chain experiences a tension T. As the chain falls, what is the total external force on the chain (in the x direction)?A chain with length b and mass density r is initially attached at both ends. The height is measured by x (increasing down) and the potential energy is zero at the attachment point. What is the total potential energy of the chain?A system of n particles is described by the masses and positions of each particle, relative to the center of mass: What can you say about the quantity ?Slide 12Consider particles a and b; the internal force fab=the force on particle a due to particle b; the position vector rab= the vector from particle a to particle b. If the force between the two particles is central, what can you say about rabfab?A system of n particles is described by the masses and positions of each particle, relative to the center of mass: The squared velocity of each particle is therefore What is the total kinetic energy of the system?The work done in unwrapping a rope from a cylinder is given by What is the total work done?What is the total kinetic energy of the rope-cylinder system?Physics 3210Week 6 clicker questionsIn central-force motion, the areal velocity is a constant What is the period  of an elliptical orbit with area A?A.B. C.D. 2=mlAmt =l2Amt =lA2t =mlAt =mlArea=AKepler’s First Law states that planets move in elliptical orbits with the sun at one focus. Why is the sun at one focus of the orbit? Because...A. Otherwise the planets wouldn’t all be in the same orbital plane.B. In two-body central-force motion one mass is always at the focus on the orbit.C. In two-body central-force motion the center of mass is always at the focus of the orbit, and the center of mass position is approximately given by the position of the sun. D. The moon is at the other focus.SunA system of n particles is described by the masses and positions of each particle: What is the position of the center of mass R of the system?A.B. C.D. E. 1Maa=�R rm , a arM maa=�1mMaa=�Rma aa=�R r1mMa aa=�R rMaa=�R rConsider particles  and , and the internal force f=the force on particle  due to particle . How does Newton’s third law relate the internal forces? A.B. C.D. E. ab ba=f f0ab ba= =f fab ba=-f f1ab ba+ =f fab ab=-f ffSuppose a system of particles experiences only internal forces (no external forces). What can you say about the momentum of the system? A. Nothing – having no external forces doesn’t determine the momentum of the system.B. The momentum of each particle is constant.C. The momentum of some (but not all) of the particles is constant.D. The momentum of the center of mass of the system is constant.Physics 3210Wednesday clicker questionsA chain with length b and mass density  is initially attached at both ends. One end of the chain is cut. After this falling end has moved down a distance x, what is the mass of the right (falling) side of the chain?A.B. C.D. E. ( )rightm b x2r= -( )rightm b x2r= +( )rightm b x=r +( )rightm b x=r -rightm b=rxA chain with length b and mass density  is initially attached at both ends. One end of the chain is cut; the other end of the chain experiences a tension T. As the chain falls, what is the total external force on the chain (in the x direction)?A.B. C.D. F T=F Mg=xF Mg T= +F Mg T= -A chain with length b and mass density  is initially attached at both ends. The height is measured by x (increasing down) and the potential energy is zero at the attachment point. What is the total potential energy of the chain?A.B. C.D.E. U gb=- rxU gb=r2gbU2- r=2gbU2r=2gbU4- r=A system of n particles is described by the masses and positions of each particle, relative to the center of mass: What can you say about the quantity ?A.B. C.D. m , a a�rma aa��rm 0a aa�=�rm 0a aa�>�rm 0a aa�<�rm the position of the CMa aa�=�rPhysics 3210Friday clicker questionsConsider particles  and ; the internal force f=the force on particle  due to particle ; the position vector r= the vector from particle  to particle If the force between the two particles is central, what can you say about rf? A. Nothing.B. C.D. 0ba ab� >r fr0ba ab� =r f0ba ab� <r fA system of n particles is described by the masses and positions of each particle, relative to the center of mass: The squared velocity of each particle is thereforeWhat is the total kinetic energy of the system?A.B. C.D. m , a a�r2 2 2 2 2v 2 v 2 Va a a a a� � � �= + � + = + � +r r R R r R& & && & &2 21 1 1T m v m v V MV2 2 2a a a aa a� �= + +� �2 21 1T m v m v V MV2 2a a a aa a� �= + +� �2 21 1T m v MV2 2a aa�= +�( )21T m v V2a aa�= +�The work done in unwrapping a rope from a cylinder is given byWhat is the total work done?A.B. C.D. R0xW g x R sin dxRq� �� �= r -� �� �� �� ��22W gR cos 12�q �=r + q-� �� �22cosW gR 12 R�q q �=r + -� �� �22W gR cos2�q �=r + q� �� �2W gR cos 12�q �=r + q-� �� �What is the total kinetic energy of the rope-cylinder system?A.B. C.D. ( )2 21T m M R2= + q&( )21T m M R2= + q&( )2 2 2 21 1T R 2 R MR2 2= r p- q q + q& &( )2 21 1T R 2 R MR2 2= r p- q q + q&