These new ideas attracted some of the best minds in physics back to the study of classical mechanics. Each chapter ends with a summary called "Main Definitions and Equations of Chapter xx." I hope you find this useful as a check on your understanding of the chapter after you've finished reading it, and as a reference later when you're trying to find that formula whose details you've forgotten.
Classical Mechanics
Quantum mechanics requires a whole separate book (or several books),
Space and Time
Similarly, if r(t) is a vector and f(t) is a scalar, then the derivative of the product f(t)r(t) is given by the appropriate version of the product rule. Clearly, the frame 8" is non-inertial, and neither of the first two laws can hold in S".
Newton’s Second Law in Cartesian Coordinates
Therefore, Newton's second law implies that the y component of the net force is zero; so Fy = O. The two forces on the skateboard are its weight w = mg and the wall's normal force N, as shown in figure 1.14.
Space and Time
- Air Resistance
- Linear Air Resistance
Use the result of Exercise 1.18 to show that the area of the triangle is given by. Write down and solve Newton's second law for the overall motion of the puck, including the effects of gravity.
Trajectory and Range in a Linear Medium
In this case, it is often clear that the effects of air resistance should be small. Thus, when evaluating the last term in (2.42), we can replace R with the approximate value of R % Rvac and as our second approximation we find [remember that the first term in (2.42) is just Rm]. To get the second row, I replaced the second Rvac in the previous row with 215601)yo/g and rg with vter.) Note that the drag correction is always Rvac less than Rvac as you would expect. In general, it can be seen (Problem 2.32) that the importance of air resistance is indicated by the ratio v/vt‘,r of projectile velocity to terminal velocity.
If v/vter << 1 throughout the flight, the effect of air resistance is very small;.
Quadratic Air Resistance
Next, I will briefly discuss a general case (motion in horizontal and vertical directions) that can only be solved numerically. Initially, position increases as it would in a vacuum (that is, y = %gt2), but lags as v increases and air resistance becomes more important. Obviously, the effect of air resistance is to lower the trajectory compared to the vacuum trajectory (shown dashed).
Air resistance slows the horizontal motion so that the sphere approaches the vertical asymptote just above x = 100 meters.
Motion of a Charge in a Uniform Magnetic Field
I claimed that the baseball of Example 2.6 approaches a vertical asymptote as t —> 00, and we can now prove that this is always the case. First, it is easy to convince yourself that once the ball starts moving downward, it continues to accelerate downward, with vy approaching —vter as t —> 00. The solution of this equation is of course an exponential function, vx = Ae 'k', and we see that vx approaches zero very quickly (exponentially) as t —> 00.
As is so often the case, the simplest way to solve the equation of motion is to resolve it into components.
Complex Exponentials
And it can be shown (not always so easily) that it has all the other known properties of the exponential function - for example, that ezew = e(z+w). See Problems 2.50 and 2.51.) In particular, the function Aekz (with A and k any constant, real or complex) has the property that. Since it satisfies the same equation regardless of the value of A, it is actually the general solution of the first-order equation df/dz = kf. At the end of the last section, I introduced the complex number n(t) and showed that it satisfied the equation 7'7 2 —iwn.
Noting that i2 = —1, i3 = —i, and so on, you can see that all the even powers in this set are real, while all the odd powers are purely imaginary.
Solution for the Charge in a B Field
2.2 * The origin of the linear drag force on a sphere in a fluid is the viscosity of the fluid. 2.4 ** The origin of the square drag force on any projectile in a fluid is the inertia of the fluid sweeping up the projectile. Show for two spheres of the same material that the larger one will eventually fall faster.
Given the mass and terminal velocity, what should we use for the diameter of the sphere.
Solution for the Charge in a B Field
If you change variables of the form “x = vx — vdr and My = vy, the equations for (ux, uy) will have exactly the form (2.68), of which you know the general solution.] ((1) Integrate the velocity to find the position as a function of t and sketch the trajectory for various values of vxo These three laws are closely related to each other and are perhaps the most important of the small number of conservation laws that are considered cornerstones of all modern physics As a result from the third law, the internal forces all cancel out the rate of change of the total momentum.
3.2) We see that the final velocity is just the weighted average of the original velocity.
3.2) We see that the final velocity is just the weighted average of the original veloc-
Equivalently, it is the sum of the ra, each multiplied by the fraction of the total mass at ra.). For example, the CM of the solar system is very close to the Sun. You can prove that this lies on the line joining ml to m2, as shown, and that the distances from the CM to m1 and m2 are in the ratio m2/m.
Considering the importance of CM, feel comfortable calculating the CM position for different systems.
Angular Momentum for a Single Particle
Equation (3.16) is the rotational analog of the equation p = F for the linear momentum, and (3.16) is often described as the rotational form of Newton's second law. The only force on the planet is the gravitational pull GmM/r2 of the sun, as shown in Figure 3.6. One of the earliest triumphs for Newtonian mechanics was that he was able to explain Kepler's second law as a simple consequence of the conservation of angular momentum.
In contrast, we will see in Chapter 8 that the first and third laws (particularly the first, which says that the orbits are ellipses with the Sun at one focus) depend on the inverse square nature of gravity and are not true. for other force laws.
Angular Momentum for Several Particles
Clockwise rotation of the crank means that the left mass is moving up relative to CM with the gearbox and the total initial speed is Each walker can run to the other end of the car and jump at the same speed (relative to the car). What is the thrust over the same period and how does it compare to the total initial weight of the ship (on land).
Find a way to clearly show that the CM of the two pieces continues to follow the original parabolic path.
Angular Momentum for a Single Particle
- Kinetic Energy and Work
- Potential Energy and Conservative Forces
- Force as the Gradient of Potential Energy
- The Second Condition that F be Conservative
- Time-Dependent Potential Energy
- Energy for Linear One-Dimensional Systems
Evaluate the line integral for the work done by the two-dimensional force F = (y, 2x) going from the origin 0 to the point P = (1., 1) along each of the three paths shown in Figure 4.2. We can now derive a crucial expression for the work of F in terms of the potential energy U(r). Note that the answer depends only on the magnitude r of the position vector r and not on the direction.
However, the spatial dependence of the force is the same as for the time-independent Coulomb force of Example 4.5 (page 119).
W(ABCB) = W(AB) + W(BC) + W(CB)
W(ABCB) = W(AB),
Curvilinear One-Dimensional Systems
The cube cannot slide on the tire of the cylinder, but it can certainly rock from side to side, as shown in Figure 4.14. Because the length of the string is fixed, the position of the entire system is specified by the distance x of m1 below any suitable fixed level. Thus, the position of the entire system can be specified by a single parameter, for example the height x of m1 below the center of the pulley as shown, and the system is.
If the system is also one-dimensional (position specified by only one parameter, as with the Atwood machine), then all the considerations of Section 4.6 apply.
Central Forces
Thus, considering the total energy of the system, we can simply ignore the limiting forces. Second, it is spherically symmetric or rotationally invariant; that is, the magnitude function f(r) in (4.65) is independent of the direction of r and, therefore, has the same value at all points at the same distance from the origin. On the surface of the earth, f' points up, in the direction of the local vertical.).
I will leave the proof of the converse result, that a central force which is spherically symmetric is necessarily conservative, to the problems at the end of this chapter.
Energy of Interaction of Two Particles
An impressive property of the force (4.77) is that it depends on the two positions r1 and r2 only through the special combination r1 — r2. For example, if the force F12 on particle 1 is to be conservative, then it must be satisfied. This means that the force between the particles at r1 and r2 must be the same as that at s1 and $2.
If (4.79) is satisfied, we can define a potential energy U (r1) such that the force on particle l is equal.
The Energy of a Multiparticle System
For N particles, denoted by = l, - - - , N , the total kinetic energy is simply the sum of the N separate kinetic energies. As you probably remember from elementary physics, the total kinetic energy of the N particles rigidly bound together is just the kinetic energy of the motion of the center of mass plus the rotational kinetic energy. That is, the potential energy U int of the internal forces is a constant and can therefore be ignored.
The external forces on the cylinder are the normal and frictional forces of track and gravity.
Kinetic Energy and Work
4.6 * For a system of N particles subjected to a uniform gravitational field g acting vertically downward, prove that the total gravitational potential energy is the same as if all the mass were concentrated at the center of mass of the system; that is,. If the puck is slightly pushed to start sliding down, through what vertical height will it descend before leaving the surface of the sphere. Hint: Use conservation of energy to find the speed of the puck as a function of its height, then use Newton's second law to find the normal force of the ball on the puck.
Show that the total potential energy (spring plus gravity) has the same form %ky2 if we use the coordinate y equal to the displacement measured from the new equilibrium position at x = x0 (and redefine our reference point so that U = 0 at y =O).
Force as the Gradient of Potential Energy
- Simple Harmonic Motion
4_31 * (a) Write the total energy E of the two masses in the Atwood machine of Figure 4.15 in terms of the coordinate x and for the coordinate x by differentiating the equation E = const. One force on the heel is the normal force N of the wire (which keeps the head on the wire). The positions of the two masses can be specified by the one angle 6. a) Write the potential energy U (6).
4.50 ** The formalism of the potential energy of two particles depends on the claim in (4.81) that.
- Two-Dimensional Oscillators
- Damped Oscillations
- Driven Damped Oscillations
To conclude this section on simple harmonic motion, let's briefly look at the energy of the oscillator (the cart on a spring or whatever it is) as it oscillates back and forth. In particular, nothing stops us from finding a solution of the form. This happened because the two solutions of the auxiliary equation (5.30) happen to coincide when )3 = coo.].
Be sure to distinguish the driving frequency a) from the natural frequency we of the oscillator.
