Questions and Answers of Classical Dynamics Of Particles

Show that Equation 14.31 is valid when a receiver approaches a fixed light source with speed v.
A star is know to be moving away from Earth at a speed of 4 X 104 m/s. This speed is determined by measuring the shift of the Ha line (λ = 656.3nm). By how much and in what direction is the
A photon is emitted at an angle θ by a star (system K’) and then received at an angle θ on Earth (system K). The angles are measured from a line between the star and Earth. The star is
The wavelength of a spectral line measured to be λ on Earth is found to increase by 50% on a far distant galaxy. What is the speed of the galaxy relative to Earth?
Solve Example 14.11 for the case of the observer at rest and the source moving. Show that the results are the same as those given in Example 14.11
Equation 14.34 indicates that a red (blue) shift occurs when a source and observer are receding (approaching) with respect to one another in purely radial motion (i.e., β = β,). Show that,
An astronaut travels to the nearest star system, 4 light years away, and returns at speed 0.3c. How much has the astronaut aged relative to those people remaining on Earth?
The expression for the ordinary force is F = d/dt (mu/√1 €“ β2) takes u to be in the x1-direction and compute the components of the force. Show that F1 = mtu1, F2 = mtu2, F3 =
The average rate at which solar radiant energy reaches Earth is approximately 1.4 X 103 W/m2. Assume that all this energy results from the conversion of mass to energy. Calculate the rate at which
Show that the momentum and the kinetic energy of a particle are related by p2c2 =2Tmc2 + T2.
What is the minimum proton energy needed in an accelerator to produce antiprotons p by the reaction p + p → p + p + (p + p) the mass of a proton and antiproton is mp.
A particle of mass m, kinetic energy T, and charge q is moving perpendicular to a magnetic field B as in a cyclotron. Find the relation for the radius r of the particle’s path in terms of m, T, q,
Show that an isolated photon cannot be converted into an electron-positron pair, y → e + e. (The conservation laws allow this to happen only near another object).
Electrons and positrons collide, form opposite directions head-on with equal energies in a storage ring to produce protons by the reaction e + e → p + p The rest energy of a proton and
Calculate the range of speeds for a particle of mass m in which the classical relation for kinetic energy, ½ mv2, is within one percent of the correct relativistic value. Find the values for an
The 2-mile long Stanford Linear Accelerator accelerates electrons to 50 GeV (50 X 109 eV). What is the speed of the electrons at the end?
A free neutron is unstable and decays into a proton and an electron. How much energy other than the rest energies of the proton and electron is available if a neutron at rest decays? This is an
A neutral pion π0 moving at speed v = 0.98c decays in flight into two photons. If the two photons emerge on each side of the pion’s direction with equal angles θ find the angle θ and
In nuclear and particle physics, momentum is usually quoted MeV/c to facilitate calculations. Calculate the kinetic energy of an electron and proton if each has a momentum of 1000MeV/c.
Show that ∆s2 is invariant in all inertial system moving at relative velocities to each other.
A neutron (mn = 939.6 MeV/c2) at rest decays into a proton (mp = 938.3 MeV/c2), an electron (me = 0.5 MeV/c2), and an antineutrino (m0 ≈ 0). The three particles emerge at symmetrical angles in
A spacecraft passes Saturn with a speed of 0.9c relative to Saturn. A second spacecraft is observed to pass the first one (going in the same direction) at relative speed of 0.2c. What is the speed of
We define the four-vector F (called the Minkowski force) by differentiating the four-vector momentum with respect to proper time. F = dP/dτ Show that the four-vector force transformation is
Show that the relativistic form of Newton€™s Second Law becomes
A common unit of energy used in atomic and nuclear physics is the electron volt (eV), the energy acquired by an electron in falling through a potential difference of one volt; 1 MeV = 106 eV = 1.602
Consider an inertial frame K that contains a number of particles with masses ma, ordinary momentum components paj, and total energies Ea. The center of mass system of such a group of particles is
Show that the relativistic expression for the kinetic energy of a particle scattered through an angle ψ by a target particle of equal mass isThe expression evidently reduces to Equation 9.89a in
The energy of a light quantum (or photon) is expressed by E = hv, where h is Planck€™s constant and v is the frequency of the photon. The momentum of the photon is hv/c. Show that if the
Find the center of mass of a hemispherical shell of constant density and inner radius r1 and outer radius r2.
Find the center of mass of a uniformly solid cone of base diameter 2a and height h.
Find the center of mass of a uniformly solid cone of base diameter 2a and height h and a solid hemisphere of radius a where the two bases are touching.
Find the center of mass of a uniform wire that subtends an arc θ if the radius of the circular arc is a, as shown in Figure 9-A
The center of gravity of a system of particles is the point about which external gravitational forces exert no net torque. For a uniform gravitational force, show that the center of gravity is
Consider two particles of equal mass m. The forces on the particles are F1 = 0 and F2 = F0i. If the particles are initially at rest at the origin, what is the position, velocity, and acceleration of
A model of the water molecule H2O is shown in Figure 9-B. Where is the center of mass?
Where is the center of mass of the isosceles right triangle of uniform a real density shown in Figure 9-C?
A projectile is fired at an angle of 45o with initial kinetic energy E0. At the top of its trajectory, the projectile explodes with additional energy E0 into two fragments. One fragment of mass m1
A cannon in a fort overlooking the ocean fires a shell of mass M at an elevation angle θ and muzzle velocity v0. At the highest point, the shell explodes into two fragments (mass m1 + m2 = M),
Verify that the second term on the right-hand side of Equation 9.9 indeed vanishes for the case n = 3.
Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at 3m/s.
Even though the total force on a system of particles (Equation 9.9) is zero, the net torque may not be zero. Show that the net torque has the same value in any coordinate system.
Consider a system of particles interacting by magnetic forces. Are Equation 9.11 and 9.31 valid? Explain.
A smooth rope is placed above a hole in a table (Figure 9-D). One end of the rope falls through the hole at t = 0, pulling steadily on the remainder of the rope. Find the velocity and acceleration of
For the energy-conserving case of the falling chain in Example 9.2, show that the tension on either side of the bottom bend is equal and has the value px2/4.
Integrate Equation 9.17 in Example 9.2 numerically and make a plot of the speed versus the time using dimensionless parameters, x √2gb vs. t/√2b/g where √2b/g us the free fall time,
Use computer to make a plot of the tension versus time for the falling chain in Example 9.2. Use dimensionless parameters (T/Mg) versus t/t free fall, where t free fall = √ 2b/g. Stop the plot
A chain such as the one in Example 9.2 (with the same parameters) of length b and mass pb is suspended from one end at a point that is a height b above a table so that the free end barely touches the
A uniform rope of total length 2a hangs in equilibrium over a smooth nail. A very small impulse causes the rope to slowly roll off the nail. Find the velocity of the rope as it just clears the nail.
A flexible rope of length 1.0 m slides from a frictionless table top as shown in Figure 9-E the rope is initially released from rest with 30 cm hanging over the edge of the table. Find the time at
A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed 14.9 km/s collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass
A particle of mass m1 and velocity v1 collides with a particle of mass m2 at rest. The two particles stick together. What fraction of the original kinetic energy is lost in the collision?
A particle of mass m at the end of a light string wraps itself about a fixed vertical cylinder of radius a (Figure 9-F). All the motion is in the horizontal plane (disregard gravity). The angular
Slow-moving neutrons have a much larger absorption rate in 235U than fast neutrons produced by 235U* fission in a nuclear reactor. For that reason, reactors consist of moderators to slow down
The force of attraction between two particles is given byWhere k is a constant, v0 is a constant velocity, and r ≡ | r2 €“ r1 |. Calculate the internal torque for the system; why does
Derive Equation 9.90.
A particle of mass m1 elastically collides with a particle of mass m2 at rest. What is the maximum fraction of kinetic energy loss for m1? Describe the reaction.
Derive Equation 9.91.
A tennis player strikes an incoming tennis ball of mass 60 g as shown in Figure 9-G. The incoming tennis ball velocity is vi = 8m/s, and the outgoing velocity is vf = 16 m/s.(a) What impulse was
Derive Equation 9.92.
A particle of mass m and velocity u1 makes a head-on collision with another particle of mass 2m at rest. If the coefficient of restitution is such to make the loss of total kinetic energy a maximum,
Show that T1/T0 can be expressed in terms of m2/m1 ≡ a and cos ψ ≡ y asPlot T1/T0 as a function of ψ for a = 1, 2, 4, and 12. These plots correspond to the energies of protons or
A billiard ball of initial velocity u1 collides with another billiard ball (same mass) initially at rest. The first ball moves off at ψ = 45o. For an elastic collision, what are the velocities
A particle of mass m1 with initial laboratory velocity u1 collides with a particle of mass m2 at rest in the LAB system. The particle m1 is scattered through a LAB angle ψ and has a final
In an elastic collision of two particles with masses m1 and m2, the initial velocities are u1 and u2 = au1. If the initial kinetic energies of the two particles are equal, find the conditions on
When a bullet fires in a gun, the explosion subsides quickly. Suppose the force on the bullet is F = (360 – 107t2s–2) N unit the force becomes zero (and remains zero). The mass of the bullet is 3
A particle of mass m strikes a smooth wall at an angle θ from the normal. The coefficient of restitution is ε. Find the velocity and the rebound angle of the particle after leaving the wall.
The particle of mass m1 and velocity u1 strikes head-on a particle of mass m2 at rest. The coefficient of restitution is ε. Particle m2 is tied to a point a distance a away as shown in Figure
A rubber ball is dropped from rest onto a linoleum floor a distance h1 away. The rubber ball bounces up to a height h2. What is the coefficient of restitution? What fraction of the original kinetic
A steel ball of velocity 5m/s strikes a smooth, heavy steel plate at an angle of 30o from the normal. If the coefficient of restitution is 0.8, at what angle and velocity does the steel ball bounce
A proton (mass m) of kinetic energy T0 collides with a helium nucleus (mass 4m) at rest. Find the recoil angle of the helium if ψ = 45o and the inelastic collision has Q = – T0/6.
A uniformly dense rope of length b and mass density μ is coiled on a smooth table. One end is lifted by hand with a constant velocity v0. Find the force of the rope held by the hand when the
Show that the equivalent of Equation 9.129 expressed in term of θ rather than ψ is
Calculate the differential cross section σ(θ) and the total cross section σt for the elastic scattering of a particle from an impenetrable sphere; the potential is given by
Show that the Rutherford scattering cross section (for the case m1 = m2) can be expressed in terms of the recoil angle as
Consider the case of Rutherford scattering in the event that m1 >> m2. Obtain an approximate expression for the differential cross section in the LAB coordinate system.
Consider the case of Rutherford scattering in the event that m2 >> m1. Obtain an expression of the differential cross section in the CM system that is correct to first order in the quantity m1/m2.
A fixed force center scatters a particle of mass m according to the force law F(r) = k/r3. If the initial velocity of the particle is u0, show that the differential scattering cross section is
It is found experimentally that in the elastic scattering of neutrons by protons (mn ≡ mp) at relatively low energies, the energy distribution of the recoiling protons in the LAB system is
Show that the energy distribution of particles recoiling from an elastic collision is always directly proportional to the differential scattering cross section in the SM system.
The most energies a-particles available to Ernest Rutherford and his colleagues for the famous Rutherford scattering experiment were 7.7 MeV. For the scattering of 7.7 MeV a-particles from 238U
A rocket starts from rest in free space by emitting mass, at what fraction of the initial mass is the momentum a maximum?
An extremely well-constructed rocket has a mass ratio (m0/m) of 10. A new fuel is developed that has an exhaust velocity as high as 4500 m/s. The fuel burns at a constant rate for 300 s. Calculate
A water droplet falling in the atmosphere is spherical. Assume that as the droplet passes through a cloud, it acquires mass at a rate equal to kA where k is a constant (>0) and A its cross-sectional
A rocket in outer space in a negligible gravitational field starts from rest and accelerates uniformly at a until its final speed is v. The initial mass of the rocket is m0. How much work does the
Consider a single0stage rocket taking off from Earth. Show that the height of the rocket at burnout is given by Equation 9.166. How much farther in height will the rocket go after burnout?
A rocket has an initial mass of m and a fuel burn rate of a (Equation 9.161). What is the minimum exhaust velocity that will allow the rocket to lift off immediately after firing?
A rocket has an initial mass of 7 x 104 kg and on firing burns its fuel at a rate of 250 kg/s. The exhaust velocity is 2500m/s. If the rocket has a vertical ascent from resting on the earth, how long
Consider a multistage rocket of n stages, each with exhaust speed u. Each stage of the rocket has the same mass ratio at burnout (k = mi/mf). Show that the final speed of the nth stage is nu in k.
To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of g/6. The exhaust velocity is 2000m/s, but fuel amounting to
A new projectile launcher is developed in the year 2023 that can launch a 104 kg spherical probe with an initial speed of 6000m/s. For testing purposes, objects are launched vertically.(a) Neglect
A new single-stage rocket is developed in the year 2023, having a gas exhaust velocity of 4000m/s. The total mass of the rocket is 105kg, with 90% of its mass being fuel. The fuel burns quickly in
In a typical model rocket (Estes Alpha III) the Estes C6 solid rocket engine provides a total impulse of 8.5 N-s. Assume that total rocket mass at launch is 54 g and that it has a rocker engine of
For the previous problem, take into account the change of rocket mass with time and omit the effect of gravity.(a) Find the rocket’s speed at burn out. (b) How far has the rocket traveled at that
Complete the derivation for the burnout height Hbo in Example 9.13. Use the numbers for the Saturn V rocket in Example 9.12 and use Equations 9.167 and 9.168 to determine the height and speed at
A particle moves in a potential V (r) = −C/(3r3). (a) Given L, find the maximum value of the effective potential. (b) Let the particle come in from infinity with speed v0 and impact
A particle moves in a potential V (r) = −V0e−λ2r2. (a) Given L, find the radius of the stable circular orbit. An implicit equation is fine here. (b) It turns out that if L is too
Given L, find the form of V (r) so that the path of a particle is given by the spiral r = Aeaθ, where A and a are constants.
A particle of mass m moves in a potential given by V (r) = βrk. Let the angular momentum be L. (a) Find the radius, r0, of a circular orbit. (b) If the particle is given a tiny kick so that
A particle moves in a V (r) = βr2 potential. Following the general strategy in Sections 6.4.1 and 6.4.3, show that the particle’s path is an ellipse.
A particle is subject to a V (r) = β/r2 potential. Following the general strategy in Section 6.4.1, find the shape of the particle’s path. You will need to consider various cases for β.

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