PHYSICS 1500 FS 2018

Homework #9

(due Monday, April 9, 2018 at the beginning of class)

Exercises (easy ones on new material; graded on the following scale: 0 nothing useful, 1 effort was made, 2 correct, but notational problems or incorrect, but good notation, 3 excellent)

  1. a. Find the net force (vectors!) exerted on the Earth at full moon and first quarter moon by the Sun and the Moon.

b. How much weaker is the force of Jupiter on the Earth if Jupiter, Earth are at the same position angle in their orbits around the Sun, i.e. form a straight line? You can neglect eccentricities of orbits.

  1. The displacement of a SHM is given as a function of time: x(t) = 1.8m cos(5πt/(4s) -π/6).
  1. Determine period, amplitude, angular frequency, phase angle and frequency of the SHM.
  2. Find velocity and acceleration as a function of time and evaluate them at t=0 and t =2s.
  3. Plot the kinetic, potential and total energy as a function of time.

Examish Questions (usually from previous material, medium difficulty, you should be able to do one of these problems in about 15 minutes without help or textbook, just usinga formula sheet; graded on the Exercise Scale)

  1. Two boxes on a frictionless, horizontal surface are connected by a string of mass 100g. A force of 50N pointing to the right is applied to the right box of mass 0.5kg. The left box has a mass of 0.4kg. Find the acceleration of the three objects.Qualitatively compare the tension to a setup with a massless string.

Problems (New, take several passes: reading textbook, thinking, trying out a couple of things, asking questions; 10 points each)

  1. Using Kepler’s laws …
  2. … prove that the ratio of distances of the planets from the sun at the closest and farthest points of their orbits is the same as the inverse ratio of their velocities at these points.
  3. … compute the maximal and minimal velocities of the Earth in a year, given that the semi-major axis and the eccentricity of Earth’s orbit are 149.6 million km and 0.0167, respectively.
  4. Compare these to the escape speed of the Sun at the position of the Earth.
  5. How much work needs to be done on the Earth to remove it from the solar system?

(over)

  1. Consider a ladybug (m=0.5g) at the edge of a turntable of radius 15cm going at 33rpm around the z axis. Assume the ladybug starts its counter-clockwise rotational motion on the negative y axis.
  1. Write down the position vector in two dimensions as a function of time. You want to express its components x(t) and y(t) as sinusoidal functions. Determine period, amplitude, angular frequency, phase angle and frequency of the SHM.
  2. Take the derivatives of x(t) and y(t) to determine velocities and accelerations as a function of time.
  3. Draw the six motion diagrams associated with this rotation in two dimensions.
  4. What is the connection to SHM?
  5. Confirm that the centripetal acceleration is |a| = |v|2/|r| by explicitly calculating the lengths of the vectors a(t),v(t), and r(t). Is there a time when this equation is not satisfied?
  1. Suppose there is a 350g mass at the end of a spring oscillating 2½ times each second with an amplitude of 15cm.
  1. What is the velocity of the mass as it passes the equilibrium point, and at 10cm away from equilibrium?
  2. Determine the total energy of the system.
  3. Find the position of the mass as a function of timeassuming the mass was initially at maximum elongation. (You might have done this already in a.)
  1. Consider a block (835g) attached to the end of a spring (k=41N/m) moving horizontally in a fluid providing a resistive force F = – bv, with b=0.662Ns/m.
  1. Find the period of the motion.
  2. Determine the fractional decrease in amplitude per cycle.
  3. Write down x(t) under the assumption that the mass is at the origin at t=0 and at +12cm at t = 1s.
  1. Comment on this homework (2 points)