**UIUC
Physics 211 James Scholars Problem Set 4**

**Consider a flexible string of length L and mass M that
is held vertically so that its bottom just touches the floor. The
string is then dropped. Let the position of the top of the string be
y and the position of the floor be y =
0.**

1.1 Is every piece (for * y > 0*) of the string
moving at the same speed when it falls (you might want to consider
whether the string remains straight)?

1.2 Write the mass of the string above the floor in terms of
*y*, *L*, * M* and *g*.

1.3 Write the momentum of the string at time * t* in
terms of *y*, *dy/dt*, *L*, * M* and *g*.

1.4 Write the momentum of the string at a time * t + dt*
using the variables above. Note that both the length of the string and
its velocity are changed from what they are at time *t*.

1.5 Write external force on the string at time * t* in
terms of *y*, *L*, * M* and *g*.

1.6 Using the time rate of change of momentum determined
above in Newton's 2nd law, write the equation for the second derivative
of * y* with respect to time, *d ^{2}y/dt^{2}*.

1.7 What is the normal force exerted by the floor on the
rope (note that it
depends on time)?

1.8 What is the acceleration of the string when it is first
released? What is the acceleration of the string (portion above the
floor) just prior to the last part hitting the floor? What is the
velocity of the string at this point?

**A spool rests on a horizontal surface on which it rolls without
slipping. The middle section of the spool has a radius r and
is very light compared with the ends of the cylinder which have radius
R and together have mass M. A string is wrapped around the
middle section so you can pull horizontally (from the middle section's
top side) with a force T.**

2.1 Determine the total frictional force on the spool in
terms of *T*, * r* and *R*.

2.2 What is the condition for the linear acceleration of the
spool to exceed *T/M*?

2.3 Which direction does the friction force point when the
acceleration is less than *T/M*? Which direction does the
friction force point when the acceleration is greater than *T/M*?

**In this problem we want to learn a little bit about what is
sometimes called dynamical loading. Our simple system consists of a
uniform stick of length L and mass M hinged at one
end. We would like to calculate the forces on the (frictionless) hinge
when the stick is released from rest at an angle theta_0 with
respect to the vertical. You may find it useful to combine work and
energy equations with torque (N II) equations. **

3.1 Show that the radial force exerted on the stick by the
hinge is * F_r = Mg/2*(5 cos(theta) - 3 cos(theta_0))*, where
theta is the angle of the stick with respect to the vertical after it
is released.

3.2 Show that the tangential (tangent to the direction of
motion, perpendicular to the stick) force exerted on the stick by the
hinge is * F_t = Mg/4*sin(theta)*.

3.3 Make a graph of the total force measured in units of *
Mg* (i.e. plot the ratio of the total force to * Mg* using the
same ideas as in Assignment 3) as a function of theta for * theta_0
= Pi/4*. Make a second graph as a function of * theta* for *
theta_0 = 0*. Attach your graph to your e-mail.

3.4 Calculate theta corresponding to the maximum value of
the total force.

3.5 For this value of theta, calculate * theta_0*
corresponding to the maximum value of the force.

3.6 What are these values for the ratio of the total force
to *Mg*? Make sure they agree with your graph.

3.7 What is the value of the total force for t*heta =
theta_0* for, say, * theta = Pi/4*? Why is the total force not
* Mg* for this situation?