Questions and Answers of Classical Dynamics Of Particles

If a projectile moves such that its distance from the point of projection is always in creasing, find the maximum angle above the horizontal with which the particle could have been projected. (Assume
A gun fires a projectile of mass 10kg of the type to which the curves of Figure 2-3 apply. The muzzle velocity is 140m/s. Through what angle must the barrel be elevated to hit a target on the same
Show directly that the time rate of change of the angular momentum about the origin for a projectile fired from the origin constant g is equal to the moment force or torque about the origin.
The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation for the force on a particle in such a field, if the electric field vector is E and the magnetic
A particle of mass m = 1 kg is subjected to a one dimensional force F(t) – kte–at, where k = 1 N/s and a = 0.5 s–1. If the particle is initially at rest, calculate and plot with the aid of a
A skier weighing 90kg starts from rest down a hill inclined at 17o. He skis 100 m down the hill and then coasts for 70 along level snow until he stops. Find the coefficient of kinetic friction
A block of mass m = 1.62kg slides down a frictionless incline (Figure 2-A). The block is released a height h = 3.91, above the bottom of the loop.(a) What is the force of the inclined track on the
A child slides a block of mass 2 kg along a slick kitchen floor. If the initial speed is 4m/s and the block hits a spring with spring constant 6 N/m, what is the maximum compression of the spring?
A rope having a total mass of 0.4kg and total length 4 m has 0.6 m of the rope hanging vertically down off a work bench. How much work must be done to place all the rope on the bench?
A super ball of mass M and a marble of mass m are dropped from a height h with the marble just on top of the super ball. A super ball has a coefficient of restitution of nearly 1 (i.e., its collision
An automobile driver traveling down and 8% grade slams on his brakes and skids 30 m before hitting a parked car. A lawyer hires an expert who measures the coefficient of kinetic friction between the
A student drops a water-filled balloon from the roof of the tallest building in two trying to hit her roommate on the ground (who is too quick). The first student ducks back but hear the water splash
In Example 2.10, the initial velocity of the incoming charged particle had no component along the x-axis. Show that, even if it had and x component, the subsequent motion of the particle would be the
Two blocks of unequal mass are connected by a string over a smooth pulley (Figure 2-B). If the coefficient of kinetic friction is μv what angle θ of the incline allows the masses to move at
A particle is released from rest (y = 0) and falls under the influence of gravity and air resistance. Find the relationship between v and the distance of falling y when the air resistance is equal
Perform the numerical calculations of Example 2.7 for the values given are Figure 2-8. Plot Figures 2-8 and 2-9. Do not duplicate the solution in Appendix H; compose your own solution.
A gun is located on a bluff of height h overlooking a river valley. If the muzzle velocity is v0 find the expression for the range as a function of the elevation angle of the gun. Solve numerically
A particle of mass m has speed v = a/x, where x is its displacement. Find the force F(x) responsible.
The speed of a particle of mass m varies with the distance x as v(x) = ax –n. Assume v(x = 0) = 0 at t = 0. (a) Find the force F(x) responsible.(b) Determine x (t) and (c) F (t)
A boat with initial speed v0 is launched on a lake. The boat is slowed by the water by a force F = – ae.(a) Find an expression for the speed v (t).(b) Find the time and (c) Distance for the boat to
A particle moves in a two-dimensional orbit defined by(a) Find the tangential acceleration at and normal acceleration an as a function of time where the tangential and normal components are taken
A train moves along the tracks at a constant speed v. A woman on the train throws a ball of mass m straight ahead with a speed v with respect to herself.(a) What is the kinetic energy gain of the
A solid cube of uniform density and sides of b is in equilibrium on top of a cylinder of radius R (Figure 2-C). The planes of four sides of the cube are parallel to the axis of the cylinder. The
A particle is under the influence of a force F – kx + kx3/a2, where k and a are constants and k is positive. Determine U(x) and discuss the motion. What happens when E = (1/4) ka2?
A potato of mass 0.5 kg moves under Earth’s gravity with an air resistive force of –kmv.(a) Find the terminal velocity if the potato is released from rest and k = 0.01 s–1.(b) Find the maximum
A pumpkin of mass 5 kg shot out of a student-made cannon under air pressure at an elevation angle of 45o fell at a distance of 142 m from the cannon. The students used light beams and photocells to
Write the criteria for determining whether an equilibrium is stable or unstable when all derivatives up through order n, (dnU/dxn) 0 = 0.
Consider a particle moving in the region x > 0 under the influence of the potentialWhere U0 = 1 J and a = 2m. Plot the potential, find the equilibrium points, and determine whether they are maxima or
Two gravitationally bound stars with equal masses m, separated by a distance d, revolve about their center of mass in circular orbits. Show that the period r is proportional to d3/2 (Kepler’s Third
Two gravitationally bound stars with unequal masses m1 and m2, separated by a distance d, revolve about their center of mass in circular orbits. Show that the period r is proportional to d3/2
According to special relativity, a particle of rest mass m0 accelerated in one dimension by a force F obeys the equation of motion dp/dt = F. Here p = m0v/(1 – v2/c2)1/2 is the relativistic
Let us make the (unrealistic) assumption that a boat of mass m gliding with initial velocity vo in water is slowed by a viscous retarding force of magnitude bv2, where b is a constant.(a) Find and
A particle of mass m moving in one dimension has potential energy U(x) = U0 [2(x/a) 2 – (x/a) 4], where U0 and a are positive constant.(a) Find the force F(x), which acts on the particle.(b) Sketch
Which of the following forces are conservative? If conservative, find the potential energy U(r).(a) Fx = ayz + bx + c, Fy = axz + bz, Fz = axy + by.(b) Fx = - ze-x, F, = In z, Fx = e–x +y/z.(c) F =
A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is 104 dyne/cm. The mass is displaced 3cm and released from rest. Calculate(a) The natural frequency v0
Allow the motion in the preceding problem to take place in a resisting medium. After oscillating for 10s, the maximum amplitude decreases to half the initial value. Calculate (a) The damping
The oscillator of Problem 3-1 is set into motion by giving it an initial velocity of 1 cm/s at its equilibrium position. Calculate (a) The maximum displacement and (b) The maximum potential energy.
Consider a simple harmonic oscillator. Calculate the time averages of the kinetic and potential energies over one cycle, and show that these quantities are equal. Why is this a reasonable result?
Obtain and expression for the fraction of a complete period that a simple harmonic oscillator spends within a small interval Δx at a position x. Sketch curves of this function versus x for
Two masses m1 = 100 g and m2 = 200g slide freely in a horizontal frictionless track and are connected by a spring whose force constant is k = 0.5 N/m. Find the frequency of oscillatory motion for
Where g is the gravitational field strength, determine the value of T.
A pendulum is suspended from the cusp of a cycloid* cut in a rigid support (Figure 3-A). The path described by the pendulum bob is cycloidal and is given by x = a (Ã˜ - sin
A particle of mass m is at r4est at the end of a spring (force constant = k) hanging from a fixed support. At t = 0, a constant downward force F is applied to the mass and acts for time t0. Show
If the amplitude of a damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 = (8π2n2)–1] times the
Derive the expressions for the energy and energy-loss curves shown in Figure 3-8 for the damped oscillator. For a lightly damped oscillator, calculate the average rate at which the damped oscillator
A simple pendulum consists of a mass m suspended from a fixed point by a weight-less, extension less rod of length l. Obtain the equation of motion and, in the approximation that sin θ ≡
Show that Equation 3.43 is indeed the solution for critical damping by assuming a solution of the form x(t) = y(t) exp( – βt) and determining the function y(t).
Express the displacement x(t) and the velocity x(t) for the over damped oscillator in terms of hyperbolic functions.
Reproduce Figures 3-10b and c for the same values given in Example 3.2, but instead let β = 0.1 s–1 and δ = π rad. How many times does the system cross the x = 0 line before the
Discuss the motion of a particle described by Equation 3.34 in the event that b < 0 (i.e., the damping resistance is negative).
For a damped, driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves* above the kinetic energy resonance as at a frequency of the same number
Show that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately.
For a lightly damped oscillator, show that Q ≡ w0/Δw (Equation 3.65).
Plot a velocity resonance curve for a driven, damped oscillator with Q = 6, and show that the full width of the curve between the points corresponding to xmax/√2 is approximately equal to w0/6.
Use a computer to produce a phase space diagram similar to Figure 3-11 for the case of critical damping. Show analytically that the equation of the line that the phase paths approach asymptotically
Let the initial position and speed of an over damped, non-driven oscillator be x0 and v0, respectively. (a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values A1 =
To better understand under damped motion, use a computer to plot x(t) of Equation 3.40 (with A = 1m) and its two components [e –βt and cos (w1t – δ)] and comparisons (with β = 0)
For β = 0.2 s–1, produce computer plots like those shown in Figure 3-15 for a sinusoidal driven, damped oscillator where xp(t), xc(t), and the sum x(t) are shown . Let k = 1kg/s2 and m = 1kg.
Figure 3-B illustrates a mass m1 driven by a sinusoidal force whose frequency is w. The mass m1 is attached to a rigid support by a spring of force constant k and slides on a second mass m2. The
Show that the Fourier series of Equation 3.89 can be expresses asRelate the coefficients cn to the an and bn of Equation 3.90.
Obtain the Fourier expansion of the functionIn the interval €“ π/w
Obtain the Fourier series representing the function
Obtain the Fourier representation of the output of a full-wave rectifier. Plot the first three terms of the expansion and compare with the exact function.
A damped linear oscillator, originally at rest in its equilibrium position, is subjected to a forcing function given byFind the response function. Allow T → 0 and show that the solution becomes
Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit T → 0) for over damping. Sketch the response functions.
Calculate the maximum values of the amplitudes of the response functions shown in Figures 3-22 and 3-24. Obtain numerical values for β = 0.2w0 when a = 2m/s2, w0 = 1 rad/s, and t0 = 0.
Consider an undamped linear oscillator with a natural frequency w0 = 0.5 rad/s and the step function a = 1 m/s2. Calculate the sketch the response function for an impulse forcing function acting for
Obtain the response of a linear oscillator to the forcing function
Derive and expression for the displacement of a linear oscillator analogous to Equation 3.110 but for the initial conditions x(t0) = x0 and x(t0) = x0.
Derive the Green’s method solution for the response caused by an arbitrary forcing function. Consider the function to consist of a series of step functions-that is, start from Equation 3.105 rather
Use Green€™s method to obtain the response of a damped oscillator to a forcing function of the form
Consider the periodic functionWhich represents the positive portions of a sine function (Such a function represents, for example, the output of a half-wave rectifying circuit)? Find the Fourier
An automobile with a mass of 1000 kg, including passengers, settles 1.0 cm closer to the road for every additional 100kg of passengers. It is driven with a constant horizontal component of speed
(a) Use the general solutions x(t) to the differential equation d2 x/dt2 + 2βdx/dt + w2/0 x = 0 for under damped, critically damped, and over damped motion and choose the constants of
An un-damped driven harmonic oscillator satisfies the equation of motion m (d2x/dt2 + w2/0 = F(t). The driving force F (t) = F- sin (wt) is switched on at t = 0. (a) Find x (t) for t > 0 for the
A point mass m slides without friction on a horizontal table at one end of a mass less spring of natural length a and spring constant k as shown in Figure 3-C. The spring is attached to the table so
Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of the damped oscillator to its
A grandfather clock has a pendulum length of 0.7m and mass bob of 0.4kg. A mass of 2kg falls 0.8m in seven days to keep the amplitude (from equilibrium) of the pendulum oscillation steady at 0.03
Reconsider the problem of two coupled oscillators discussed in Section 12.2 in the event that the three springs all have different force constants. Find the two characteristic frequencies, and
Continue Problem 12-1, and investigate the case of weak coupling: k12
Two identical harmonic oscillators (with masses M and natural frequencies w0) are coupled such that by adding to the system a mass m common to both oscillators the equations of motion becomeSolve
Refer to the problem of the two coupled oscillators discussed in Section 12.2. Show that the total energy of the system is constant. (Calculate the kinetic energy of each of the particles and the
Find the normal coordinates for the problem discussed in Section 12.2 in Example 12.1 if the two masses are different, m1 ≠ m2. You may again assume all the k are equal.
Two identical harmonic oscillators are placed such that the two masses slide against one another, as in Figure 12-A. The frictional force provides a coupling of the motions proportional to the
A particle of mass m is attached to a rigid support by a spring with force constant k. At equilibrium, the spring hangs vertically downward. To this mass-spring combination is attached an identical
A simple pendulum consists of a bob of mass m suspended by an inextensible (and mass less) string of length l. From the bob of this pendulum is suspended a second, identical pendulum. Consider the
The motion of a pair of coupled oscillators may be described by using a method similar to that used in constructing a phase diagram for a single oscillator (Section 3.4). For coupled oscillators, the
Consider two identical, coupled oscillators (as in Figure 12-1). Let each of the oscillators be damped, and let each have the same damping parameter β. A force F0 cos wt is applied to m1. Write
Consider the electrical circuit in Figure 12-B. Use the developments in Section 12.2 to find the characteristic frequencies in terms of the capacitance C, inductance L, and mutual inductance M. The
Show that the equations in Problem 12-11 can be put into the same form as Equation 12.1 by solving the second equation above for I1 in the second equation. The characteristic frequencies may then by
Find the characteristic frequencies of the coupled circuits of Figure 12-C.
Discuss the normal modes of the system shown in Figure 12-D.
In Figure 12-C, replace L12 by a resistor and analyze the oscillations.
A thin hoop of radius R and mass M oscillates in its own plane hanging from a single fixed point. Attached to the hoop is a small mass M constrained to move (in a frictionless manner) along the hoop.
Find the Eigen frequencies and describe the normal modes for a system such as the one discusses in Section 12.2 but with three equal masses m and four springs (all with equal force constant) with the
A mass M moves horizontally along a smooth rail. A pendulum is hung from M with a weightless rod and mass m at its end. Find the Eigen frequencies and describe the normal modes.
In the problem of the three coupled pendula, consider the three coupling constants as distinct, so that the potential energy may be written as with ε12, ε13, ε23 all
Construct the possible Eigenvectors for the degenerate modes in the case of the three coupled pendula by requiring a11 = 2a21. Interpret this situation physically.
Three oscillators of equal mass m are coupled such that the potential energy of the system is given byWhere k3 = √2k1k2. Find the Eigen frequencies by solving the secular equation. What is the
Consider a thin homogeneous plate of mass M that lies in the x1-x2 plane with its center at the origin. Let the length of the plate be 2A (in the x2-direction) and let the width be 2B (in the

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