COMPLETING THE CIRCLE

Problems from the section titled ``Energy Redux''

• 4.1:
  Show explicitly that energy is conserved for a freely-falling mass by explicitly working with the equations for motion and the definitions of kinetic and potential energy. Start by considering a ball of mass m thrown straight upward with initial velocity v₀ from a height y₀ above the ground.
  • a.) In Maple, define the vertical position, y, as a function of time which obeys the appropriate equation for motion, y0 + v0*t - 0.5*g*t^2. Now define v, the velocity, as a function of time using the equation for motion,

    

    Define the kinetic, potential, and total energies as functions of time using your functions v(t) and y(t). For the values of m=1 kg, g=9.8 m/s² , y0=2 meters, and v0=49 m/s, plot the kinetic energy, potential energy, and total energy on the same graph for time between 0 and 10 seconds. What do you observe?

  • b.) Using your kinetic energy and potential energy functions, find the values of time for which the two energies are equal. Does the time at which the kinetic energy is zero (or the potential energy is maximum) correspond to a time when the ball is at its greatest height?

• 4.2:
  
  • a.) You have just learned that the potential energy for a horizontally-moving mass on the end of a spring is U = (1/2)kx^2, where x is the distance of the mass from the equilibrium point of the spring. Plot, using Maple, the potential energy as a function of x for x from -3 to 3 and a given value of k (you can choose your own value of k).
  • b.) Since you know that energy is conserved for springs, sketch what the kinetic energy for the mass at the end of the spring should look like as a function of x. From your sketch, at what value or values of x should the velocity of the mass be greatest in magnitude? At what value or values should the velocity be smallest in magnitude? Verify your sketch by plotting the actual kinetic energy as a function of x using Maple.
  • c.) If, instead of moving horizontally, the mass were suspended from a spring attached to the ceiling, how would the potential energy change? How would your sketch of the kinetic energy change?
  • d.) Now, since you know the force acting on the mass at the end of the spring you know everything you need to completely specify its motion. Instead of plotting the potential energy, kinetic energy, and total energy as functions of x, plot them as functions of time assuming that the mass starts at time t=0 at a distance 3 units from the origin and an initial velocity of zero. You can assume any values of k and m that you wish. What do you observe?

• 4.3:
  Suppose that you have a mass of 10 kg attached to a non-ideal spring that does not quite obey Hooke's Law. There is no friction acting and you know that the potential energy as a function of energy for the non-ideal spring follows the formula

  

  where k = 10 N/m and q = 0.5 N/m^3

  • a.) Use Maple to plot both the potential energy for a Hooke's Law spring with spring constant k = 10 N/m, and the non-ideal spring. Choose your own range of values for plotting, but make them reasonable so that you can see clearly the behavior of both the ideal and non-ideal spring.
  • b.) Find the expression for the force exerted on the mass by the non-ideal spring. Plot this force along with the two potential energy curves you plotted earlier. How does the non-ideal spring force differ from the ideal case?

  Problems from the section titled ``Motion in Two Dimensions''

• 4.4:
  A thief is creeping around a jewelry store persistently looking for a hidden safe with valuable diamonds. He suddenly hears the call of police sirens racing down the street in his direction. The thief quickly abandons his search and runs outside around the corner into an alley. The police jump out of their cars and chase the thief up a ladder to the roof of the building. The thief decides to throw the sack of stolen goods over to the roof of the next building where it will be safe until he is able to return. There is a considerable distance between the two buildings.
  • a.) At what angle does the thief need to direct the bag in the air to get the furthest distance for his throw? Think fast, the police are coming up that ladder quickly!! Sack size and air resistance can be neglected.
  • b.) Suppose the thief has never had physics and doesn't know what angle to use. If he heaves the sack with an initial velocity of 12 m/s and it travels a distance of 14 meters horizontally, at what angle did the thief throw the sack?

• 4.5:
  The space shuttle orbiter is in a circular orbit 220 km above the earth's surface. It completes a revolution around the earth every 89 minutes. The equatorial radius of the earth is 6.374 x 10⁶ meters.
  • a.) What is the orbiter's speed?
  • b.) What is its acceleration?

• 4.6:
  Before the year 200 B.C., the ancient astronomer Hipparchus of Greece had already determined an accurate distance from the Earth to the moon (approximately 3.84 x 10⁵ km). From this and a few easily obtained observations, a great number of other parameters about the moon can be determined. For example,
  • a.) The period of the moon's orbit is approximately 28 days. What is the magnitude of the moon's acceleration, in units of g, as the moon orbits the earth?
  • b.) The angular size of the full moon, as observed from earth, is about 0.52 degrees. What is the diameter of the moon?

  Problems from the section titled ``Enzyme Kinetics II''

• 4.7:
  A certain enzyme-substrate system is studied kinetically. Ten reaction mixtures, each containing the same concentration of enzyme, i.e. the same [E]_T, were made up to various substrate concentrations ([S]) and the initial velocities (v₀) were determined as shown in the data table below.

  [S] (in M)	v0 (in M/min.)
  1.00 x 10-3	65.0 x 10-6
  5.00 x 10-4	63.0 x 10-6
  1.00 x 10-4	51.0 x 10-6
  5.00 x 10-5	42.0 x 10-6
  3.00 x 10-3	33.0 x 10-6
  2.00 x 10-5	27.0 x 10-6
  1.00 x 10-5	17.0 x 10-6
  5.00 x 10-6	9.50 x 10-6
  1.00 x 10-6	2.20 x 10-6
  5.00 x 10-7	1.10 x 10-6

  1. Perform a Lineweaver-Burk plot of 1/v₀ versus 1/[S] for this data. Determine the best-fit line through the plotted data by use of Maple's stats package. As usual, be sure to display your data points along with the best-fit line on the same graph.
  2. According to your analysis in (1), determine the slope, the 1/v₀ ("y") intercept, and the 1/[S] ("x") intercept. Express your results along with the appropriate units.
  3. According to your results from (2), determine K_M and V_max for this enzyme-substrate system. Express your results along with the appropriate units.
  4. Referring to the table of representative enzyme-substrate systems shown in this section - and your results above as needed - which of the enzyme-substrate systems does this data most nearly represent? How do you know?