UIUC Physics 211 James Scholars Problem Set 4

## Problem 1

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, d2y/dt2.

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?

## Problem 2

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?

## Problem 3

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 theta = theta_0 for, say, theta = Pi/4? Why is the total force not Mg for this situation?