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Review of Classical Mechanics CHAPTER 1 REVIEW OF CLASSICAL MECHANICS Newton’s laws of motion provide the starting point for all of the physics that is discussed in this text. This brief review of elementary classical mechanics discusses those laws, and the useful conservation theorems that follow from them. 1.1 NEWTON’S LAWS OF MOTION I: A body remains at rest or in uniform motion unless acted upon by a force. In other words, its velocity is constant when the force on that body is F =0. II: Abodyacteduponbyaforcemovessuchthatthetimerateof changeofitsmomentum equals that force, namely, p˙ = F, where p = m˙r is the body’s linear momentum , m its mass, r its position vector, and its velocity ˙r = dr/dt where the derivative is with respect to time t. This is the familiar F = m¨r law. III: If two bodies exert forces on each other, those forces are equal in magnitude and opposite in direction. Thus if F12 is the force on particle 1 that is exerted by particle 2, then F21 = F12. − 1.2 REFERENCE FRAMES AND COORDINATE SYSTEMS A reference frame is the coordinate grid that that is used to measure all particles’ positions and velocities. Newton’s laws are valid in an inertial reference frame, and law I indicates that an inertial reference frame is one that is stationary or moving with a constant velocity. 1 2 REVIEW OF CLASSICAL MECHANICS Figure 1.1 Position vector r for a particle at point P. Note that the unit vectors ˆr and θˆ lie in the ˆx–ˆy plane. Cartesian andcylindricalcoordinatesystems will be used in this text. In those coordinate systems, the position vector for particle at point P is (see Fig. 1.1) r = xˆx + yˆy + zˆz inCartesiancoordinates (1.1) = rˆr + zˆz in cylindrical coordinates. (1.2) In this cylindrical coordinate system, the unit vector ˆr is always confined to the ˆx–ˆy plane. Note also the distinction in the lengths r = x2 + y2 and r = x2 + y2 + z2. In these coordinate systems, the particle’s velocity is | | p p dr ˙r = =x ˙ ˆx +y ˙ˆy +z ˙ˆz =r ˙ˆr + rθ˙θˆ+z ˙ˆz, (1.3) dt and its acceleration is d2r 1 d ¨r = =x ¨ˆx +¨yˆy +¨zˆz = (¨r rθ˙2)ˆr + (r2θ˙)θˆ+¨zˆz. (1.4) dt2 − r dt 1.3 LINEAR AND ANGULAR MOMENTA Law II indicates that a particle’s linear momentum p = m˙r is conserved (i.e., a constant) when the total force on it is zero. That particle’s angular momentum is L = r p = mr ˙r, and rate at which L varies is the the torque on that particle, T = dL×/dt = m(˙r× ˙r + r ¨r) = r F. When the net torque on that particle is zero, its angular momentum× is× conserved.× 1.4 WORK AND ENERGY Suppose force F displaces a particle a small differential distance dr′ during a short time interval dt; see Fig. 1.2. The small amountof work that that force performs on that particle WORK AND ENERGY 3 Figure 1.2 A particle is displaced from r0 to r1 by force F along three possible paths. The work r done on the particle is = 1 F · dr′, and if is independent of the choice of path ( or ), W Rr0 W a,b, c then the force is said to be conservative. is dW = F dr′, so the total work done on that particle as that force drives the particle · from position r0 to r1 is the sum of all the contributions dW along that path, which is r1 W = F dr′. (1.5) r · Z 0 Note also that dr 1 d 1 d(v2) dW = m¨r dr = m¨r dt = m¨r ˙rdt = m (˙r ˙r)= m , (1.6) · · dt · 2 dt · 2 dt where v2 = ˙r ˙r is the square of the particle’s velocity. The total work that F must do to · drive the particle from r0 r1 is then → r1 1 2 1 2 2 W = m d(v )= m(v1 v0 )= T1 T0, (1.7) 2 r 2 − − Z 0 1 2 where Ti = 2 mvi is the particle’s kinetic energy when at position ri. Thus the work done on the particle is simply its change in kinetic energy. Also note that the rate at which force F does work on the particle is dW P = = m¨r ˙r, (1.8) dt · which is also known as the power delivered to that particle by force F. The work done on the particle by force F is also related to changes in its potentialenergy U. This text is largely concerned with conservative forces, and a conservative force is one where the work W performed on a particle is independent of the particular path that takes 4 REVIEW OF CLASSICAL MECHANICS the particle from r0 to r1 ; see Fig. 1.2 When that is the case, then the vector force F can always be written as the gradient of a scalar U(r) that is a function of position r only: F = U, (1.9) −∇ where U is the system’s potential energy. The gradient of U in Cartesian and cylindrical coordinates is ∂U ∂U ∂U U = ˆx + ˆy + ˆz ∂x ∂y ∂z ∇ (1.10) ∂U 1 ∂U ∂U = ˆr + θˆ+ ˆz ∂r r ∂θ ∂z So for example, the component of force along the ˆx axis is Fx = ∂U/∂x, while the azimuthal force is F = (∂U/∂θ)/r. Then the work, Eqn. (1.5), becomes− θ − r1 W = U dr, (1.11) − r ∇ · Z 0 where U(r) is a function of the particle’s trajectory r(t), which is the path traced by the particle over time. Next, use the chain rule to calculate dU/dt in a Cartesian coordinate system: dU ∂U dx ∂U dy ∂U dz = + + = ( U) ˙r. (1.12) dt ∂x dt ∂y dt ∂z dt ∇ · Thus dU = ( U) dr is the small change in the particle’s potential energy that occurs as it advances a∇ small· distance dr along its trajectory during the short time interval dt. The work done on the particle can now be written as r1 W = dU = (U1 U0) (1.13) − r − − Z 0 where Ui = U(ri) is the potential energy of the particle when it is at position ri. Thus the work done on the particle is also 1 times its change in potential energy. And since − W = T1 T0 = (U1 U0), this means that the particle’s energy at the endpoints of the − − − trajectory, E1 = T1 + U1 = T0 + U0 = E0, is a constant, which tells us that the particle’s energy E = T + U is conserved, provided of course that the force acting on the particle is conservative. Conservative systems are frictionless (i.e.. have no velocity–dependent forces), and any external forces, if present, do not have any time dependence. 1.4.1 potential energy, and the potential According to Eqns. (1.5) and (1.13), the system’s potential energy U(r) is 1 the work − × done on the particle as it is moved from the reference position r0 to its present position r, so r U(r)= W = F(r′) dr′, (1.14) − − r · Z 0 where r′ is a dummy variable that runs along the integration path (see Fig. 1.2). Note also that the unimportant constant U0 has been dropped from the above, which means that the system’s energy scale has been calibrated such that U(r0)=0. WORK AND ENERGY 5 Figure 1.3 Two gravitating particles, one of mass m at r, and the other of mass M at rM . The lower figure illustrates how this system’s potential energy U is calculated at particle m is drawn from the reference site r0 to its final position at r. EXAMPLE 1.1 To illustrate the calculation of U, consider a simple gravitating two–particle system. The field particle, which is the particle of interest, has a mass m and a position vector r, and it is free to move about the system, while the source mass M, which is the source of the force that disturbs m, remains at a fixed position rM . According to Newton’s law of gravity, the force on m due to M is GMm(r r ) F = − M , (1.15) − r r 3 | − M | and this force law is written so that it is evident that force F draws the field mass m towards the source mass M. Put the origin on M so that rM = 0, and recall that U is 1 the work done on the particle as M’s force delivers particle m from − × the reference site r0 to its present position r. Thus the force on m when it is at an 2 intermediate distance r′ = r rM away from M is F = (GMm/r′ )ˆr where ˆr is the usual unit radial vector;| − see the| lower part of Fig. 1.3.− The potential energy of this two–particle system is then r r GmM GmM U(r)= F(r′) dr′ = 2 dr′ = , (1.16) − r0 · r′ − r Z Z∞ where the arbitrary reference distance r0 has been set to infinity, as is required since Eqn. (1.14) has set U(r0)=0 at the reference site. 6 REVIEW OF CLASSICAL MECHANICS Figure 1.4 A Gaussian surface S surrounds a volume that contains mass m. The position vector r also points to the small area element da = ˆnda on surface S. Another useful quantity is the potential energy per unit mass, Φ(r)= U/m, also known as the system’s potential.
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