5.68) Maximum P and E of a Rocket
The rocket begins at rest with mass M , ejecting exhaust at speed u. We can use
equation 5.54 to nd the rockets current speed as a function of its remaining
mass m(t)
M
v (t) = u log
m(t)
the rockets forward momentum is m
Phys 3201
Quiz #3
Spring 2006
Each problem carries equal weight. Please begin each problem on a fresh sheet
of paper. To ensure maximum partial credit, explain your reasoning whenever
possible. Good luck.
1. A rocket starts from rest in free space and the
Phys 3201
Quiz #2
Spring 2006
Answer all three problems. Each problem carries equal weight. Please begin
each problem on a fresh sheet of paper. To ensure maximum partial credit,
explain your reasoning whenever possible. Good luck.
1. Consider the periodi
6.25) Spring on a T
We can express the masses position in cartesian coordinates as
x
y
= l cos
= l sin
+
r sin
r cos
where r is the displacement of the mass along the cross-bar, and is the angle the
arm from the origin to the cross-bar (with lenght l)
Triangular Pendulum
dene as the pendulums angle of rotation. We can nd the kinetic energy of the pendulum from its
velocity. Since both masses are moving in polar coordinates at constant radius l, their velocity is simply l
and
1
T = m(l2 2 + l2 2 ) = ml2
7.13) Intersecting Orbits
The two masses will orbit about their common center of mass, with the mass m always twice as far from the
center of mass as the mass 2m.The minimal eccentricity for the two orbits to intersect will have the closest
approach for t
8.58) Pendulum Collison
The energy of the stick can be expressed as a sum of the energy of the eective point mass at
the stick center of mass, and the sticks rotational kinteic energy about that point. The rotational
energy is
1
1
Trot = I 2 = ml2 2
2
24
PHYS 3201 TEST 2
April 14, 2010
1.) Disk in a Bowl
0
0.1
0.2
0.3
R
0.4
0.5
0.6
0.7
0.8
r
0.9
1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
A disk of mass M , moment of inertia I , and radius r is placed in a bowl of radius R, where R r.
Assume the disk rolls wi
1.11) Waves On a String
String has mass m, length l and tension . The speed of wave has dimensions
[v ] =
L
T
while the quantitites listed above have dimensions
[m] = M
[l] = L
[ ] = M L/T 2
So to match dimensions
[m l ] = [v ]
we must have
+
+
2
L
M
T
=
3.29) Atwoods Three
@
@
'$'$
&%&%
'$
u
m
u
3m
&%
u
2m
6
x+
Dene the positions of the three masses as x1 , x2 , and x3 , with the positive xdirection dened as up in the gure. Summing the forces on each mass reveals
mx1
2mx2
3mx3
=
=
=
2
mg
2mg
3mg
with the
4.17) Eective Spring Constant
First consider two springs with constants k1 , k2 in parallel. When they are
stretched from their equilibirum length by a distance x, the force is
Fx = k1 x k2 x = ke x
so we see
ke = k1 + k2
Now consider two springs in serie
5.32) Cart in a Valley
The carts initial potential energy is mgh1 There is no initial kinetic energy
(cart is said to be at rest), so this is the total energy of the cart. As a quantity
m of sand leaks out of the cart, the carts energy will be reduced by
Phys 3201
Quiz #1
Spring 2006
Answer all three problems. Each of the three problems carries equal weight. Please begin each
problem on a fresh sheet of paper. To ensure maximum partial credit, explain your reasoning
whenever possible and as clearly as you