Download Problems for Mathematics of Motion: week 6

Problems for Mathematics of Motion: week 6
Module F12MR2
2006-07
Important: Questions 1 and 2 on this sheet (with changed numerical values) will constitute
the second test for this module. This test will take place on Monday 19 February. You do
not need to hand in group solutions for these questions. Question 3 should be discussed in the
groups, and handed in as usual before the class on Monday 19 February.
1 At time t seconds, a particle of mass m = 1 kg has position vector ~r(t) metres relative to some
~r
fixed origin O. The particle moves under the influence of a central force F~ = −4 3 Newtons.
r
(a) Write down the equation of motion for the particle’s position.
1
4
(b) Show that the total energy E = ~r˙ ·~r˙ − is conserved during the motion.
2
r
~ = ~r × ~r˙ is conserved during the motion.
(c) Show that the total angular momentum L
(d) If the particle’s initial position is ~r(0) = ~i metres and its initial velocity ~r˙ (0) = 2~j m/s,
~ Show that the motion takes place in the ~i~j-plane.
compute E and L.
(e) In terms of polar coordinates (r, θ) in the ~i, ~j-plane the position vector can be written
~r = r cos θ ~i + r sin θ ~j where both r and θ are functions of t. Show that the angular
~ = r2 θ̇~k and that 1 ~r˙ ·~r˙ = 1 (ṙ2 + r2 θ̇2 ).
momentum can be expressed as L
2
2
4
1 2 L2
(f) Use the results of (e) to derive the formula E = ṙ + 2 − for the total energy (L =
2
2r
r
~
|L|). Insert the values for E and L found in (d) to obtain a differential equation for r.
2 At time t seconds, a particle of mass m kg has position vector ~r(t) metres relative to some fixed
origin O. The particle moves under the influence of the force
F~ = 4~r˙ × ~k Newtons,
where ~k is a constant unit vector.
(a) Write down the equation of motion for the particle’s position.
(b) Show that kinetic energy E = 1 m~r˙ ·~r˙ is conserved during the motion.
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(c) Suppose that the particle’s mass is m = 1 kg, its initial position is given by ~r(0) = 4~i and
its initial velocity is ~r˙ (0) = −16~j + ~k. Show that the equation of motion and the initial
conditions are satisfied by
~r(t) = R cos(ωt)~i + R sin(ωt)~j + vt~k,
where R, ω, v are constants which you should determine.
continued overleaf
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(1)
3 A particle of mass m = 1 kg moves under a central force of attraction
fixed point O where r is the distance from O and α is a constant.
α
r2
directed towards a
(a) In the lecture we showed that the orbit of the particle is an ellipse if the total energy E is
negative. If L is the magnitude of the angular momentum about O express the semi-latus
rectum ` and eccentricity of the ellipse in terms of α, E and L .
(b) Use energy
to show that the particle’s speed v is related to the distance r by
conservation
2
α
2
v2 = m
+ `−1 .
r
(c) If the
of the particle is a circle with centre O and radius a, show that its speed is
ppath
α
u = ma
.
(d) Assuming that the orbit is an ellipse with semi-major axis a and semi-minor axis b show
.
that the period of the motion is given by T = 2πabm
L
2
(e) In the case of an earth satellite for which O is the
q centre of the earth and α = mgR where
R is the radius of the earth show that T =
2π
R
a3
.
g
(f) A satellite has maximum and minimum heights of 750km and 250km respectively, above
the surface of the earth. Find the period of revolution. (Radius of the earth = 6400km,
g = 9.81ms−2 ).
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