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Université des Sciences et Technologies de Lille 1 2011/2012 Master degree in Mathematical Engineering Refresher Course in Physics Semester 3 Classical Mechanics Exercise 1 A body (assimilated to a point M) of mass m is submitted to the action of a central force of the form ~F = f(r)~ur, We use (r, ?) for the polar coordinates of a point M in (xOy) and (~ur, ~u?) for the associated orthonormal basis (see the course for precise definition). (1) Show that ~? = ~OM ? (m~v) and r2?˙ do not depend on t. (2) Denote C = r2?˙. Give an expression of f(r) in terms of m, C, u = 1/r and d 2u d?2 . (3) Suppose M desribes a spiral of the form r? = K. Give an expression for f(r). (4) Suppose M desribes a spiral of the form r = e?a?, where a > 0. Give an expression for f(r). Exercise 2 A body (assimilated to a point M) of mass m has initially velocity v0 and is falling on the vertical direction (on the surface of the earth). We make the assumption that the friction due to the air acts on M as ~F = ?bm~v, where b is a positive constant.
- friction
- d? √
- ?02 sin
- limit velocity
- sin2n ?02
- sin ?2
- sin ?
- dm dt
- positive constant
M m
~F =f(r)~u ;r
(r;) M (xOy) (~u ;~u )r
2~ _~ =OM^(m~v) r t
2d u2_C =r f(r) m C u = 1=r 2d
M r =K f(r)
a M r =e a> 0
f(r)
M m v0
M
~F = bm~v;
b
V M tl
1 1b = 25s v = 0;1m:s M0
99
r0
v t = 00
r(t) =r (1+bt)0
dm
dt
v
dv r0
+3b v =g:
dt r
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v(r) 1 2
M m
r
Kf(r) = 2r
K P
~kOP^m~v(P)k n~ n
~
31 28 2 34m = 9;110 kg K = 2;310 N:m ~ = 1;110 J:s
0
E
E
r v E n = 11 1 1
E1
m M
x() =R(+sin) y() =R(1 cos); 2 [ ;]:
dy
s O M
dx
R
M s
S S k k1 2 1 2
m l0
S y =a a> 2l S1 0 2
y = 0
y
S y =a a> 2l S1 0 2
y = 0
x xa
S y =a a> 2l S1 0 2
S S1 2
y
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2m x_q
at 2 kF(t) =F e ! = t = 0 x = 0 x_ = 00 0 m
x(t)
x(t) =!0
2 2! +a0x(t) =
2a
l O0
(xOy) m
mg~u y
~uy q
l0t = 0 = v = 0 T = 20 0 g
m
p Z
02 d
pT = T :0
cos cos0 0
0sin = sin sin
2 2
Z
2T 2 d
= q :
T 2 20 00 1 sin sin
2
Z +1 2n 0X 2T (2n)!sin 2 2n2= sin d :
n 2T 4 (n!) 0 0n=0
20T’T (1+ )0 16
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