TEST 4 TAKE HOME

ON ALL TEST QUESTIONS PLEASE WRITE LARGE ENOUGH FOR THE YOUR WORK TO BE LEGIBLE.   DO NOT SQUEEZE WRITING BETWEEN THE QUESTION STATEMENTS.   USE WHITE SPACES ONLY AND USE EXTRA SPACES TO MAKE YOUR SOLUTION LARGE ENOUGH TO READ WITHOUT ME GUESSING WHAT YOU ARE  TRYING TO DO.

1. (40 POINTS) CHAPTER 10.  This problem “maps” into #16, 17, 70, 73, 68, at the very least!  See online lecture notes related to these problems ! A thin,  light horizontal string is wrapped  around the rim of a 4.00-kg solid uniform disk that is 30.0 cm in diameter.  A 2.00-kg box is connected  to the right end of the string and moves to the right along the ground horizontally with no friction. The box is subjected to a horizontal force of magnitude F =100. 0 N parallel to the ground.  The disk is rotates clockwise about a fixed axis attached to a steel structure bolted to the ground:

(a) (15 points) What is a, the linear acceleration magnitude the box?

(b) (10 points) What is the tension T in the string? 
(c) (10 points) Assuming the system starts from rest, what is the  direction and magnitude of the pulley’s angular momentum after a time period of 2.0 seconds?

(d) (5 points) If the disk is replaced with a hollow thin walled cylinder of the same mass, and diameter, what will be the acceleration in part (a)?  Explain the difference in your answer with this replacement.

2. (40 POINTS) During  SPRING, a well known Chabot professor performs a classroom demonstration using two identical wooden turntables in the shape of flat uniform disks. Each turntable has radius R = 1.00 m and mass M = 4.00 kg. CHECK OUT DIAGRAM BELOW BEFORE READING ON.

      The bottom turntable is initially rotating at angular velocity ω_i = 10.00 rad/s about a vertical axis through its center.  Suddenly the professor vertically drops,  from directly above,  a spinning disk on the rotating bottom turntable as shown. The top identical disk has initial angular speed 2.00 rad/s spinning in the opposite direction along the common axis looking down.  The disks have aligned circumferences as shown after impact and turn together as one unit with common angular speed.

      In other words, the top disk lands on the bottom one along the turntables’ common central axis of rotation,   sticks to bottom turntable’s surface and turns with bottom one without slipping.    Below is a schematic of the system just before and after top disk lands on bottom one.
      (a) (30  points) What is the common final angular velocity ω_f of the system after the top disk lands on bottom one?   SHOW ALL WORK!    
       (b) (10 points) Assuming the axis shown runs vertically up and down the page, what is the direction of  angular momentum of the system, Up or Down? Circle one and explain.

3. (40 points)

One end of a uniform beam of mass 50-kg is attached to a wall with a hinge. A cable supports the other end. The relevant angles are shown.  In particular, the cable makes  an angle of 60-degrees  with the vertical wall and the beam makes an angle of 60-degrees with the horizontal. Other needed angles must be derived. SHOW ALL WORK!    

(a) (32 points) Find the tension T in the cable. SHOW ALL WORK!    

(b) (4 points)   Find the horizontal component F_h of the force of the hinge on the beam.

(c) (4 points) Find the vertical component F_v of the force of the hinge on the beam.

4.   (40 points) Three uniform spheres are fixed at the three corners of a square of side a = 0.50 m. A 4^th sphere, of mass m ‘ represented by the dot, is placed at the lower RIGHT corner of square—at point P. Note: The sphere in the upper LEFT corner has mass M = 2.00 kg and the  spheres in the LOWER left (AT ORIGIN 0) and UPPER right corners have equal masses m = 1.00 kg.  The sphere in the lower RIGHT corner has mass m’ = 0.0150 kg.  See left diagram in the figure below.

(a) (8 points) What is the magnitude of the force on the sphere of mass m’ due to the sphere directly above it in the upper RIGHT corner.

(b) (8 points) What is the magnitude of the force on the sphere of mass m’ due to the sphere across the diagonal in the upper LEFT corner?    
(c) (8 point) What is the magnitude of the force on the sphere of mass m’ due to the sphere directly to the  LEFT  of it in the lower LEFT corner?
(d) (8) What is the magnitude of the resultant force on the sphere of mass m’ ?
(e) (8) What is the direction of the resultant force on the sphere of mass m’? Show this angle by drawing the resultant on the blank x-y axes in the correct quadrant. Label the angle the vector makes with the x-axis.
    

5. (34 points) CH. 14--PHYSICAL PENDULUM. THE DERIVATION OF THE PERIOD INVOLVES AN UNDERSTANDING OF GRAVITATIONAL TORQUE IN CH. 10.
1.80-kg uniform rod is pivoted about a horizontal axis as shown.  The rod length is L = 2.00 m.  The rod is set into small amplitude oscillations in a vertical plane parallel with the Earth’s  downward gravitational field.
(a) (15) What is the period  T  of oscillations? The momentum of inertia of a rod about an  axis though an end is ML²/3, where L is the length and M is the mass.
(b) (19) If the rod is initially displaced 0.400 rad from its vertical equilibrium position, use conservation of energy to find the angular speed of  wrench about the shown axis when wrench passes through vertical equilibrium position. Assume rod was released from rest.