2013 Semifinal Exam Part A 4 Question A2 A solid round object of radius R can roll down an incline that makes an angle 6 with the horizontal. Assume that the rotational inertia about an axis through the center of mass is given by I=BmR2 The coefficient of kinetic and static friction between the object and the incline is u.The object moves from rest through a vertical distance h. a.If the angle of the incline is sufficiently large,then the object will slip and roll;if the angle of the incline is sufficiently small,then the object with roll without slipping.Determine the angle 0c that separates the two types of motion. b.Derive expressions for the linear acceleration of the object down the ramp in the case of i.Rolling without slipping,and ii.Rolling and slipping. Question A3 A beam of muons is maintained in a circular orbit by a uniform magnetic field.Neglect energy loss due to electromagnetic radiation. The mass of the muon is 1.88 x 10-28 kg,its charge is-1.602 x 10-19 C,and its half-life is 1.5234s. a.The speed of the muons is much less than the speed of light.It is found that half of the muons decay during each full orbit.What is the magnitude of the magnetic field? b.The experiment is repeated with the same magnetic field,but the speed of the muons is increased;it is no longer much less than the speed of light.Does the fraction of muons which decay during each full orbit increase,decrease,or stay the same? The following facts about special relativity may be useful: The Lorentz factor for a particle moving at speed v is 1 Y三 V1-u2/c2 The Lorentz factor gives the magnitude of time dilation;that is,a clock moving at speed v in a given reference frame runs slow by a factor in that frame. The momentum of a particle is given by p=Ymi where m does not depend on v. The Lorentz force law in the form d师 =9(E+元×月) continues to hold. Copyright C2013 American Association of Physics Teachers2013 Semifinal Exam Part A 4 Question A2 A solid round object of radius R can roll down an incline that makes an angle θ with the horizontal. Assume that the rotational inertia about an axis through the center of mass is given by I = βmR2 . The coefficient of kinetic and static friction between the object and the incline is µ. The object moves from rest through a vertical distance h. a. If the angle of the incline is sufficiently large, then the object will slip and roll; if the angle of the incline is sufficiently small, then the object with roll without slipping. Determine the angle θc that separates the two types of motion. b. Derive expressions for the linear acceleration of the object down the ramp in the case of i. Rolling without slipping, and ii. Rolling and slipping. Question A3 A beam of muons is maintained in a circular orbit by a uniform magnetic field. Neglect energy loss due to electromagnetic radiation. The mass of the muon is 1.88 × 10−28 kg, its charge is −1.602 × 10−19 C, and its half-life is 1.523 µs. a. The speed of the muons is much less than the speed of light. It is found that half of the muons decay during each full orbit. What is the magnitude of the magnetic field? b. The experiment is repeated with the same magnetic field, but the speed of the muons is increased; it is no longer much less than the speed of light. Does the fraction of muons which decay during each full orbit increase, decrease, or stay the same? The following facts about special relativity may be useful: • The Lorentz factor for a particle moving at speed v is γ = 1 p 1 − v 2/c2 • The Lorentz factor gives the magnitude of time dilation; that is, a clock moving at speed v in a given reference frame runs slow by a factor γ in that frame. • The momentum of a particle is given by ~p = γm~v where m does not depend on v. • The Lorentz force law in the form d~p dt = q(E~ + ~v × B~ ) continues to hold. Copyright c 2013 American Association of Physics Teachers