Phil 221                                                     Questions                                                             Weeks 8 and 9

 

  1. In "De Gravitatione," (H 107-15) what is Newton's main dynamical objection (i.e., an objection based on physical forces) to Descartes's relational theory of space and motion? (The key passage is on H 108.) Note: Huggett gives versions of this objection on H 103-4 and H 127-8.
  2. Newton also criticizes Descartes’s theory of motion on conceptual grounds, arguing that it leads to a number of absurdities. (See H 108-10). Explain Newton’s criticism.
  3. In class, we worked through a derivation of the formula a= v2/r for the centripetal acceleration, a, of a body moving with constant speed, v, in a circle of radius, r. This particular derivation, based on Galileo’s formula s = ½at2 for the distance traveled by a body moving with constant acceleration, a, was not published in Newton’s Principia but does appear in Newton’s Waste Book (the name for one of Newton’s notebooks). Explain the derivation.
  4. The derivation of a= v2/r that Newton does give in the Principia is based on an argument that he invented as early as 1665. Newton considers a ball colliding with the inside wall of a rigid cylinder of radius r. Suppose, as Newton first did, that the ball moves with constant velocity v in a square path, thus rebounding from the wall exactly four times before retracing its path. What is the component of velocity of the ball perpendicular to the wall prior to a collision? What is the component of velocity of the ball perpendicular to the wall immediately after a collision? (Hint: the ball rebounds from the inside wall of the cylinder just as if it had hit a rigid wall that is tangential to the cylinder at the point of impact.) Thus, what is the total change in velocity? (Remember that velocity, unlike speed, is a vector quantity: it has a direction.) Given that the cylinder has radius r and the ball moves with constant speed v, what is the time taken from one collision to the next? So, what is the rate of change of velocity (i.e., the centripetal acceleration) of the ball? Note: Newton then generalizes this result for any regular N-sided polygon inscribed in a circle, arguing that the same formula for the centripetal acceleration must hold as the number of sides of the polygon increases and the path of the ball becomes ever closer to the circumference of the circle.
  5. In class we discussed the Newtonian significance of Kepler’s second law, K2, that the line joining a planet to the sun sweeps out equal areas in equal times. In Book I of the Principia, Newton proves first (Proposition 1) that if a body moves under the action of a centripetal force (that is, under the action of a central force that is always directed towards the same unmoving body, such as the sun) the moving body must obey K2. Then, in Proposition 2, Newton proves the converse, that if a body obeys K2, then the force acting on the body must be centripetal. Explain each derivation in turn, making clear the role played by Newton’s first law (N1) and the law of composition of velocities. Which Proposition, 1 or 2, does Newton need for his eventual derivation of the law of universal gravitation?

    6. [Note: the centripetal or central force in Props. 1 and 2 does not have to be an inverse-square law force. It can vary with distance or time in any fashion. All that matters is that it is always directed towards the same central body about which K2 holds.]
  6. In class, we showed how, for the simplified case of a body moving with constant speed v in a circle, the formula for centripetal acceleration, a= v2/r, together with Kepler’s third law (K3) that T2 µ r3, implies that the constant force acting on the body towards the center of the circle varies as the inverse square of the distance. Explain the derivation.

    7. [Note: in Book 1 of the Principia (Proposition 11) Newton also proves the much harder and more relevant case, namely that if a body moves in an ellipse (or, indeed, any conic section, whether circle, ellipse, hyperbola, or parabola), the centripetal force towards the focus is as the inverse square of the distance.]
  7. The satellites (moons) around a planet also obey K3. If a planet (such as the earth) has only one satellite, can we show that it obeys K3? What problem does this present in showing that the earth’s moon obeys an inverse-square law force directed towards the earth?
  8. Another result that Newton obtained in 1666 (his annus mirabilis) was Newton’s famous moon test. The point of this calculation was to show that the earth's gravity (assumed to vary inversely with the square of the distance from the earth's center) is sufficient, all by itself (without the need for any Cartesian vortices) to hold the moon in its orbit. Work through a version of Newton's calculation as follows. (1) Calculate the centripetal acceleration of the moon in its orbit, assuming that moon moves at constant speed in a perfect circle about the earth, the radius of the moon's orbit is 60 times the earth's radius, the earth's radius is 4000 miles, and the moon orbits the earth once every 28 days. (2) Calculate the same centripetal acceleration of the moon in its orbit assuming that g, the acceleration due to gravity at the earth's surface is 32 feet per second squared, and that the force causing this acceleration falls off as the inverse square of the distance as it extends to the moon. How closely do the two separate calculations of the moon's centripetal acceleration agree?

    9. [When Newton did his own moon test in 1666, he found that the numbers were only roughly or "pretty nearly" in agreement. Mainly this was because he used an inaccurate value of the earth's radius. That value was corrected in the Principia. Also, in 1666, Newton was unable to justify his assumption that a homogeneous sphere gravitationally attracts a particle external to the sphere as if all its mass were concentrated at its center. Newton proved this result in the Principia, Book 1, Proposition 71.]
  9. In Book 1 of the Principia, (in the Scholium following Corollary 6 to the Axioms or Laws of Motion) Newton gives a fascinating argument that, for bodies (A and B) that are in contact and which attract one another, N1 logically implies N3 (action and reaction are equal and opposite). For, if the force that A exerts on B were not exactly equal and opposite to the force that B exerts on A, the combined body, consisting of A and B together, would accelerate even though no external force is acting on it. Is this argument logically valid? Does it suffice to show that gravitational action at a distance also must obey N3 if N1 is true?
  10. Just like Euclid's Elements, Newton's Principia begins with Definitions and Axioms (the laws of motion). The very first of these definitions defines mass ("quantity of matter") as density times volume. Since density is, in turn, mass per unit volume, Newton's definition of mass (inertial mass) is circular. An alternative approach would be to regard mass as the constant, m, that appears in (the modern version) of Newton’s second law, N2, F =ma. Does taking this second approach to defining mass (inertial mass) entail that N2 is no longer an empirical law capable of being refuted by experiment? Explain.
  11. Newton’s laws of motion hold only in inertial frames of reference. An inertial frame of reference is either at rest in absolute space or moving with constant velocity in a straight line in absolute space. But we cannot directly observe absolute space (or absolute time). So we need another way of determining which frames are inertial if Newton’s laws are to be empirical laws capable of being tested against experience. What would be the problem with simply defining inertial frames as ones in which Newton’s laws hold? How would you go about determining whether a frame is inertial prior to testing to see whether Newton’s laws hold in that frame?