Solved Problems in Classical Mechanics potx

[...]... the uncertainty ∆x the motion as deterministic (in the sense mentioned in Question 3.1) However, if we wait long enough the particle can be found anywhere between the walls: determinism has changed into complete indeterminism (iii) It is only in the ideal (and unattainable) case ∆v0 = 0 (i.e the initial velocity is known exactly) that deterministic motion persists indefinitely (iv) In non-linear systems... Furthermore, this constant can be [1] [2] M Born, Physics in my generation, pp 78–82 New York: Springer, 1969 F Waismann, in Turning points in physics Amsterdam: North-Holland, 1959 Chap 5 ½ Solved Problems in Classical Mechanics incorporated in the universal constant G, which is already present in (1) and (2) That is, we can set mP = mA There is no need to distinguish between active and passive gravitational... to certain problems is facilitated by choosing a suitable non-inertial frame Thus the trajectory of a particle at rest on a rotating turntable is simplest in the frame of the turntable, where the particle is in static equilibrium under the ½¼ Solved Problems in Classical Mechanics action of four forces (weight, normal reaction, friction and centrifugal force) Similarly, for a charged particle in a uniform... x-axis, bouncing between two perfectly reflecting walls at x = 0 and x = In between collisions with the walls no forces act on the particle Suppose there is an uncertainty ∆v0 in the initial velocity v0 Determine the corresponding uncertainty ∆x in the position of the particle after a time t Solution In between the instants of reflection, the particle moves with constant velocity equal to the initial value... According to (5), the mass of a system is an extensive variable ½ Solved Problems in Classical Mechanics Question 2.8 Consider a ‘mass dipole’ consisting of two particles having opposite masses‡ m (> 0) and −m Describe its motion in the following cases: (a) The dipole is initially at rest in empty inertial space (b) The constituents of the dipole in (a) have electric charge q1 and q2 (c) The charged mass... are moving relative to each other – see Chapter 15 We can interpret the equation of motion (24) in the following way: if we wish to write Newton’s second law in a non-inertial frame S in the same way as in an inertial frame S (i.e as force = mass × acceleration), then the force F due to physical interactions (such as electromagnetic interactions) must be replaced by an effective force Fe that includes... systems the uncertainty can increase much faster with time (exponentially rather than linearly) due to chaotic motion (see Chapter 13) ½¾ Solved Problems in Classical Mechanics Question 2.2 A ball moves freely on the surface of a round billiard table, and undergoes elastic reflections at the boundary of the table The motion is frictionless, and once started it continues indefinitely The initial conditions... Chapters 14 and 15) In a non-inertial frame these properties do not hold For example, if one stands on a rotating platform it is noticeable that positions on and off the axis of rotation are not equivalent: space is not homogeneous in such a frame Notwithstanding the fact that, in general, Newtonian dynamics is most simply formulated in inertial space, one should keep in mind the following proviso Namely,... Thus, we obtain the desired expression for the period T = 2π mI mG g (3) When mI = mG this reduces to the result in Question 4.3 [3] C M Will, “Relativity at the centenary,” Physics World, vol 18, pp 27–32, January 2005 ½ Solved Problems in Classical Mechanics Comments (i) Newton used the result (3) in conjunction with experiments on pendulums to test the equality, in modern terminology, of inertial... and time In addition to the fact that the laws of motion assume their simplest forms in inertial frames, these frames also possess unique properties with respect to space and time For a free particle in an inertial frame these are: First, all positions in inertial space are equivalent with regard to mechanics This is known as the homogeneity of space in inertial frames Secondly, all directions in space . class="bi x0 y0 w0 h0" alt="" Solved Problems in Classical Mechanics This page intentionally left blank Solved Problems in Classical Mechanics Analytical and. rotating turntable is simplest in the frame of the turntable, where the particle is in static equilibrium under the Solved Problems in Classical Mechanics action