*Time: 30 Minutes*

A rigid flywheel rotates about its spin axis parallel to the z-axis at an angular velocity equal to w, as shown in the diagram above. Let r denote the distance of the axis of symmetry of the flywheel to any other point in the flywheel.

1)What is the

*curl of the magnitude*of the*velocity field*of the flywheel at a given point in the flywheel? (The velocity field of the flywheel is the velocity of the points of the flywheel considered as a function of their position.)A) 0, except at r = 0

B) 2w

C) 2w/ r

D) 2w/ r + 4 pw

E) 4 pw

^{2}r + w/ r2) In what direction is the curl of the velocity field of the flywheel at a given point in the flywheel?

A) ^x cos wt ^y sin wt

B) ^x cos wt - ^y sin wt

C) ^r, where r is directed from the axis of symmetry to the given point.

D) +z – direction

E) –z direction

3)Find the curl of

**ix**^{2}+ jy^{2}+ k (x^{2}– y^{2})4)A string of length L and uniform density r is fastened at both ends and maintained at a tension T. At what frequencies can this string vibrate? (Note that n is a positive integer)

The number of degrees of freedom of a system is the number of quantities that must be specified in order to completely specify the velocity of all the particles of the system for any motion consistent with the constraints of the system.

5)How many degrees of freedom does a rigid body have:

A) 1…..B) 2……C) .3…….D) 5 …….E) 6

6) If a rigid body is constrained to rotate about a given axis, how many degrees of freedom does it have?

A)1…..B) 2……C) 3…….D) 5 …….E) 6

7)A disk is constrained to roll without slipping on a plane such that the plane of the disk always remains perpendicular to the plane. How many degrees of freedom does the disk have?

A)1…..B) 2……C) .3…….D) 4 …….E) 6

__Problems (8 – 9)__

In a certain rectangular coordinate system, the directions of whose axes are given by the unit vectors i, j and k, the inertia tensor of an object is given by:

I = K x

(1….0…..0)

(0….1…..1)

(0….1… ..1)

8) What are the

*principal moments of inertia*of the object (the moments of inertia along the principal axis)*relative to the origin of the above coordinate system*?A) 0, ½K, K

B) 0, K, 2K

C) 0, K, 3K

D) K, 2K, 4K

E) K, 2K, 5K

9) What is the direction of the principal axis corresponding to the principal moment of inertia and equal to K?

A) i

B) j

C) (1/ Ö2)j – (1/Ö2)k

D) 1/ Ö3)j + (1/3 Ö6)k

E) 1/ Ö3)i + (1/Ö3)j + (1/Ö3)k

10) If the origin of the above rectangular coordinate system is at the center of mass of the object and the total mass of the object is M, what is the change in the inertial tensor of the object if the rectangular coordinate system is displaced parallel to itself a distance r

_{o}in the direction**(1/****Ö****2)j + (1/****Ö****2)k**?11) The Earth’s magnetic field at a certain point is 0.70 Gauss. This field is to be annulled (neutralized) by the magnetic field at the center of a circular conducting loop 5.0 cm in radius. Find the required current.

12) How much work is required to bring a point charge q from infinity to the center of a dielectric shell of dielectric constant k, inner radius b and outer radius a? (Note – use cgs, Gaussian units)

13)One mole of an ideal monatomic gas is heated at a constant pressure of 1 atm from 0C to 100C. What is the change in the internal energy of the gas?

14) What is the

*change in the entropy*of the gas as a result of being heated?15) A two-electron atom for which the orbital angular momentum quantum numbers are ℓ1 =3 and ℓ2 = 2 can have the following values for the total orbital angular momentum number L:

A) 0, 1, 2, 1, 3

B) 0, 1, 2, 3, 4

C) 1, 2, 3, 4, 5

D) 2, 3, 4, 5, 6

E) 2, 3, 4

Determine the possible values of the total angular momentum quantum number of a single f electron.

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