Two gravitationally bound stars with equal masses m, separated by a distance d, re- volve about their center of mass in circular orbits. Show that the period 7 is propor- tional to d² (Kepler's Third Law) and find the proportionality constant.
Two gravitationally bound stars with equal masses m, separated by a distance d, re- volve about their center of mass in circular orbits. Show that the period 7 is propor- tional to d² (Kepler's Third Law) and find the proportionality constant.
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![ll Jazz LTE
2:50 PM
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( Classical-Dynamics-of-Particles-and-...
248. Two gravitationally bound stars with equal masses m, separated by a distance d, re-
volve about their center of mass in circular orbits. Show that the period 7 is propor-
tional to d/2 (Kepler's Third Law) and find the proportionality constant.
249. Two gravitationally bound stars with unequal masses m, and m2, separated by a dis-
tance d, revolve about their center of mass in circular orbits. Show that the period 7
is proportional to d³/² (Kepler's Third Law) and find the proportionality constant.
2-50. According to special relativity, a particle of rest mass mo accelerated in one dimen-
sion by a force F obeys the equation of motion dp/dt = F. Here p = mov/(1 –
v/c?) !/2 is the relativistic momentum, which reduces to mov for v²/c² «1. (a) For
the case of constant Fand initial conditions x(0) = 0 = v(0), find x(t) and v(1).
(b) Sketch your result for v(t). (c) Suppose that F/m, = 10 m/s² ( = g on Earth).
How much time is required for the particle to reach half the speed of light and of
99% the speed of light?
2-51. 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 bu², where b
is a constant. (a) Find and sketch v(t). How long does it take the boat to reach a
speed of /1000? (b) Find x(t). How far does the boat travel in this time? Let m =
200 kg, = 2 m/s, and b = 0.2 Nm-2s².
2-52. A particle of mass m moving in one dimension has potential energy U(x) =
U[2(x/a) 2 - (x/a) *], where U, and a are positive constants. (a) Find the force
F(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable and
unstable equilibrium. (c) What is the angular frequency w of oscillations about the
point of stable equilibrium? (d) What is the minimum speed the particle must have
at the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its ve-
locity is positive and equal in magnitude to the escape speed of part (d). Find x(4)
and sketch the result.
2-53. Which of the following forces are conservative? If conservative, find the potential
energy Ur). (а) F, 3D ауг + bx + с, F, %3D ахх + be, F, — аху + by. (b) F,
- ze*, F, = Inz, F̟ = e¯* + y/z. (c) F = e,a/r(a, b, c are constants).
2-54. 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 height of the potato if it has the same value of k,
98
2/ NEWTONIAN MECHANICS–SINGLE PARTICLE
but it is initially shot directly upward with a student-made potato gun with an initial
velocity of 120 m/s.
2-55. A pumpkin of mass 5 kg shot out of a student-made cannon under air pressure at
an elevation angle of 45° fell at a distance of 142 m from the cannon. The students
used light beams and photocells to measure the initial velocity of 54 m/s. If the air
resistive force was F= - kmv, what was the value of k?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5695cfc0-3856-40bc-8508-515c1aa613d3%2Fe846ebab-8240-48a9-b428-0541ea0b8c62%2F6jdx8qa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ll Jazz LTE
2:50 PM
@ 76%
( Classical-Dynamics-of-Particles-and-...
248. Two gravitationally bound stars with equal masses m, separated by a distance d, re-
volve about their center of mass in circular orbits. Show that the period 7 is propor-
tional to d/2 (Kepler's Third Law) and find the proportionality constant.
249. Two gravitationally bound stars with unequal masses m, and m2, separated by a dis-
tance d, revolve about their center of mass in circular orbits. Show that the period 7
is proportional to d³/² (Kepler's Third Law) and find the proportionality constant.
2-50. According to special relativity, a particle of rest mass mo accelerated in one dimen-
sion by a force F obeys the equation of motion dp/dt = F. Here p = mov/(1 –
v/c?) !/2 is the relativistic momentum, which reduces to mov for v²/c² «1. (a) For
the case of constant Fand initial conditions x(0) = 0 = v(0), find x(t) and v(1).
(b) Sketch your result for v(t). (c) Suppose that F/m, = 10 m/s² ( = g on Earth).
How much time is required for the particle to reach half the speed of light and of
99% the speed of light?
2-51. 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 bu², where b
is a constant. (a) Find and sketch v(t). How long does it take the boat to reach a
speed of /1000? (b) Find x(t). How far does the boat travel in this time? Let m =
200 kg, = 2 m/s, and b = 0.2 Nm-2s².
2-52. A particle of mass m moving in one dimension has potential energy U(x) =
U[2(x/a) 2 - (x/a) *], where U, and a are positive constants. (a) Find the force
F(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable and
unstable equilibrium. (c) What is the angular frequency w of oscillations about the
point of stable equilibrium? (d) What is the minimum speed the particle must have
at the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its ve-
locity is positive and equal in magnitude to the escape speed of part (d). Find x(4)
and sketch the result.
2-53. Which of the following forces are conservative? If conservative, find the potential
energy Ur). (а) F, 3D ауг + bx + с, F, %3D ахх + be, F, — аху + by. (b) F,
- ze*, F, = Inz, F̟ = e¯* + y/z. (c) F = e,a/r(a, b, c are constants).
2-54. 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 height of the potato if it has the same value of k,
98
2/ NEWTONIAN MECHANICS–SINGLE PARTICLE
but it is initially shot directly upward with a student-made potato gun with an initial
velocity of 120 m/s.
2-55. A pumpkin of mass 5 kg shot out of a student-made cannon under air pressure at
an elevation angle of 45° fell at a distance of 142 m from the cannon. The students
used light beams and photocells to measure the initial velocity of 54 m/s. If the air
resistive force was F= - kmv, what was the value of k?
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