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1. The position of a particle of mass m moving in a circle of radius R about the z-axis with angular speed w is given by the vector function r(t) = a(t)), +y(t)ed 20, where .(2) = R cosut, y(t) = Rsinwt and 2 =constant. Answer the following: (a) What is the angular velocity w about the z-axis? (b) Find the velocity v of the particle in terms of w. 2, y and 2. (c) Find the angular momentum of the particle in terms of w. .2. y and 2. (d) Confirm L = mr2w m(r w)r in this case. 2. For the system in Problem 1, show that L = Iw where is the moment of inertia about the axis of rotation. 3. The total angular momentum of a system of particles is thesum of the angular moments of the individual particles. If the system in Problem 1 is replaced by system of two particles of mass m with positions r1 = ((t)s + (f)ey + 2e-and find the total angular momentum L L1 + Lyand show that L = Iw where 1 is the total moment of inertia about the axis of rotation. 4. Ifr. vand a denote the position, velocity and acceleration of a particle, respectively, prove that d a.(vxr)(=A. (vxr) dt 5. Suppose that all the atoms in a monatomic gas move with the same speed v. but that the directions of motion are randomly distributed. Compute the average number of atoms striking an element of wall area dA per second, and the average momentum imparted per second to dA in terms of N. V, v and d.A. Note J is the total number of atoms and V is the volume of the box. (Hint: Average over all directions of incidence.) 1 6. New parameters are often born out of the need to explain inconsistencies between experimental results and prevailing theory. In physics, mass is the property that determines the magnitude of the gravitational force, and charge is the property that determines the magnitude of the electromagnetic force. Particle physicists have ascribed a property they call color, which determines the magnitude of the strong force that binds protons and neutrons together to form atomic nuclei. What is it about the structure of the atomic nucleus that would lead physicist to believe a new property other than mass and charge was required to explain observations? Estimate the energy of this force in 12c using E = mc² with the mass determined from Avogadro's number and the definition of the mole (i.e., 12 grams of 1 mole III 12C). 2

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1. a) Ans . The angular velocity is the vector quantity while angular speed is a scalar quantity.         

ω =ω ez i.e. a vector quantity with magnitude ω in the z-direction...

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