# October 12, 2004: 9.40 am - 10.55 am

Instructions:

·      Do not turn this page until you are instructed to do so.

·      Each answer must be well motivated.  You will need to show your work in order to get full credit.

·      If we can not read what you wrote, we can only assume it is wrong.

·      Useful equations are attached to the exam and a copy of Appendices D, E, and F is available as a separate handout.

·      The total number of points you can get on this exam is 105 (you get 5 bonus point if you answer question 4e correctly).

Problem 1 (30 points)

Consider a uniform cylinder of length L and radius R.  The mass density of the cylinder is r.  Consider a point P on the axis of the cylinder at a distance z from its center (z > L/2).  Calculate the gravitational field at P.  Make sure you specify both the magnitude and the direction of the gravitational field.

Note: The gravitational potential on the axis of an infinitesimal thin disk of mass M and radius R a distance z from the center of the disk is equal to

Problem 2 (20 points)

Consider the following two phase diagrams that describe the one-dimensional motion of an object of mass m.

1.

2.

For each phase diagram answer the following questions:

a.     What is the location of the equilibrium position?  You need to motivate your answer!

b.     Is the equilibrium a stable or an unstable equilibrium?  You need to motivate your answer!

c.     Sketch the potential associated with the phase diagram.  In particular, pay attention to the symmetry of the potential around the equilibrium position.

Problem 3 (30 points)

Consider the plane pendulum shown in the Figure below.  The mass m is constraint by a weightless, extensionless string that can move frictionless in a vertical circle of radius l.  The mass is at rest at a position that corresponds to a polar angle q0.

a.     Derive the equation of motion for the polar angle q.  Do not assume that q is small.

b.     Is this a linear or a non-linear system?  If the system is non-linear, under what conditions will the system behave like a linear system?

Now consider a more complicated system where the pendulum is suspended from the cusp of a cycloid cut into a rigid support (see Figure below, and useful equations on page 7).

The path of the mass in this case is not circular, but has a cycloidal dependence:

where the length of the pendulum is l = 4a, and where f is the angle of rotation of the circle generating the cycloid.

c.     Show that the oscillations are exactly isochronous with a frequency w0 = √g/l, independent of the amplitude.

Problem 4 (25 points)

a.     A system with a spherically symmetric mass distribution generates the gravitational potential and the gravitational field shown in the figure below.  Sketch the mass density of this system as function of R.

b.     Which of the curves in the following figure shows the time dependence of the amplitude of an over-damped oscillator?

c.     Consider two systems, each consisting of a particle of mass m.  The forces acting on these particles are shown in the following figure.  The forces deviate from a linear dependence on x at large |x|.  Which of the two forces will produce a soft system?

d.     For each of the two forces shown in the figure above, sketch the potential energy as function of x.  In the same figures, also sketch expected dependence of the potential energy on x for a linear system.

e.     Match the pictures to the names listed below.

 A B C D E

___   Orlando Hernandez

___   Derek Jeter

___   Bernie Williams

___   Hideki Matsui

Useful Relations

The term cycloidal motion was introduced to describe the motion of a point on a wheel that is rolling, without slipping, on a surface (see Figure below).

In Problem 3, the cycloidal motion is described by the following equations:

The angle f is the angle of rotation of the circle that generates the cycloid, and this is NOT equal to the angle a between the vertical and the tangent to the cycloidal path at the position of mass m.  The angle a is related to the change in the vertical position dy and the displacement along the cycloid ds:

This relation can also be expressed in terms of the angle f:

Here we have used the expression for dy obtained from the description of the path, and the following expression for ds, which can also be directly obtained from the description of the path:

## Practice Exam # 1.

This exam was used in Phy 235 in Fall 2004.

You are encouraged to use this practice exam to determine how well prepared you are for exam # 1.

Sit down for 1 hour and 15 minutes and complete the practice exam. Then compare it with official solutions to determine how well you did.

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