A uniform marble rolls without slipping down the path shown in Figure, starting from rest. (a) Find the minimum height ?h? required for the marble not to fall into the pit. (b) The moment of inertia of the Figure marble depends on its radius. Explain why the answer to part (a) does not depend on the radius of the marble. (c) Solve part (a) for a block that slides without friction instead of the rolling marble. How does the minimum ?h? in this case compare to the answer in part (a)? Figure:

PHYS 1010 Notes Week 14 April 1822 Heisenberg uncertainty principle: the more we know about the position of a particle, the less we know about its momentum (and vice versa) Before we measure a particle, it doesn't have a position, just a probability of a position; by measuring it, we force it to choose properties Tunneling Particles can pass through walls Energytime uncertainty Photons borrow energy from the universe and turn into an electron or a positron They can only borrow the energy temporarily Positron = an electron moving backward in time Antimatter is matter moving backward in time Indistinguishability: we can't tell two electrons apart EPR Paradox: a photon turns into an electron and a positron (energy > mass) one goes to the right and one goes to the left; both have an equal chance of being a positron or an electron, but we don't know which, because we haven't measured them Even though these particles are far away, measuring one immediately describes the properties of the other Strange connection between particles If this is true, information travels faster than the speed of light QuantumXeno Paradox Unstable situation like balancing a ball on a sharp point Describing the situation as a factor of time and probability Beginning of time 100% As time goes on, probability goes down Not watching it for a brief amount of time measures it > probability goes away/starts over By continually watching it, it is a stable situation Hidden variables interpretation: there must be more information that we don't know Copenhagen interpretation: assuming that measuring something collapses a wave function is problematic for some reason we don't know Many worlds interpretation: measuring something makes it choose both possibilities, which creates an infinite number of universes PHYS 1010 Notes Week 14 April 1822 Heisenberg uncertainty principle: the more we know about the position of a particle, the less we know about its momentum (and vice versa) Before we measure a particle, it doesn't have a position, just a probability of a position; by measuring it, we force it to choose properties Tunneling Particles can pass through walls Energytime uncertainty Photons borrow energy from the universe and turn into an electron or a positron They can only borrow the energy temporarily Positron = an electron moving backward in time Antimatter is matter moving backward in time Indistinguishability: we can't tell two electrons apart EPR Paradox: a photon turns into an electron and a positron (energy > mass) one goes to the right and one goes to the left; both have an equal chance of being a positron or an electron, but we don't know which, because we haven't measured them Even though these particles are far away, measuring one immediately describes the properties of the other Strange connection between particles If this is true, information travels faster than the speed of light QuantumXeno Paradox Unstable situation like balancing a ball on a sharp point Describing the situation as a factor of time and probability Beginning of time 100% As time goes on, probability goes down Not watching it for a brief amount of time measures it > probability goes away/starts over By continually watching it, it is a stable situation Hidden variables interpretation: there must be more information that we don't know Copenhagen interpretation: assuming that measuring something collapses a wave function is problematic for some reason we don't know Many worlds interpretation: measuring something makes it choose both possibilities, which creates an infinite number of universes