Mechanics I Final Exam

Problem #1.)

Consider a force F(t) acting on a massless sliding block as shown below.

Through experimentation, the coefficient of static friction is found to vary parabolically with respect to time:

To simplify the analysis, assume that the coefficients of static and kinetic friction are equal. Define F(t) as follows:

Part A.)
Generate an expression that states at which time t the block will begin to move. When will it stop?
Part B.)
At which time t does the block move for the sixth time?
Part C.)
What values for the polynomial coefficients in the coefficient of static friction cause the block to stay in static equilibrium until t = 1000?
Part D.)
Discuss the ramifications of t < 0. What occurs as t approaches the initial singularity (t = -12 billion years, the so-called Big Bang)?

Problem #2.)

A standard 30" wide flange beam of density equal to that of a neutron star and length = 800 miles floats through space at some constant velocity Vo < c (the speed of light). Draw the shear and bending moment diagrams as the beam passes a bilinear star system at distance R = Ro. Assume uniform gravitational loading.

Problem #3.)

Generate an expression for the centroid of the vector sum of the arbitrary placement of the two-torus embedded in zero curvature three-dimensional space with a hyperboloid. Discuss the ramifications of the hyperboloid being of one sheet or two. Hint: Solve by method of direct integration.

Problem #4.)

After years of planning, you construct your Doomsday Computer with one of the original Pentium Floating Point Units. Awkwardly, your DC (as you affectionately call it) predicts that the moon will come to a sudden halt in the near future. You take it upon yourself to save your home planet (Earth) by designing the truss shown below.

Part A.)
Calculate the force of mutual attraction between the Earth and its single satellite using Newton's Law of Gravitation. Leave the universal gravitation constant, G, an unknown in this calculation.
Part B.)
Assume all truss members to be of uniform length. What tension or compression is necessary in member BC to hold the moon in static equilibrium? Note: take into account the mutual gravitational attraction between the beams of the truss by applying these forces as point loads at the joints.

Problem #5.)

Assume Newton to be wildly incorrect. In fact:

Newton's Second Law is replaced by:

Rederive a few of the fundamental equations of vector mechanics to prove a rudimentary understanding of this material.

Good Luck! See you in Mechanics II next semester!