Environmental Engineering Reference
In-Depth Information
PROBLEMS 2
2.1
sfasfd
(a) Two masses
m
1
and
m
2
have position vectors
r
1
and
r
2
respectively.
Write down the position vector of the centre of mass of this system.
Find the vector equation for the straight line through the two masses and
hence show that the centre of mass lies on this line.
(b) Write down the position vector of the centre of mass for three equal
masses with position vectors
a
,
b
and
c
.
Show that the centre of mass lies at the intersection of the medians of the
triangle defined by the positions of the masses. [A median of a triangle
is a line running from one of its corners to the midpoint of the opposite
side.]
2.2 A system is composed of three isolated particles with masses
m
1
=
10 g,
m
2
=
30 g and
m
3
=
40 g at positions
r
1
=
2
i
−
2
j
,
r
2
=−
3
i
+
j
and
r
3
=
5
i
6
j
respectively (all distances are measured in metres). Calculate the posi-
tion vector of the centre of mass. If a force
F
+
3
i
(newtons) is applied to
one of the particles, work out the acceleration of the centre of mass. Does it
matter to which particle the force is applied?
2.3 A hose sends a stream of water at a rate of 2 kgs
−
1
and a velocity of 10 ms
−
1
against a wall. If all the water runs down the wall, what is the force on the
wall?
2.4 A prisoner of mass 80 kg plans an escape using a rope made of strips of
bed sheets. If his window is 20 m from the ground, and the makeshift rope
will not support a tension of greater than 600 N without breaking, find the
minimum speed at which the prisoner hits the ground assuming he descends
by sliding vertically down the rope without making contact with the wall.
2.5 Ancient Egyptians push a 5.00 tonne block of stone on rollers up a slope with
an acceleration of 0.30 ms
−
2
. The slope is inclined at 20
.
0
◦
to the horizontal.
Assuming that the rollers produce a frictionless surface, and that the mass of
the rollers can be ignored, calculate the force applied by the Egyptians to the
block. Calculate the normal force acting on the block.
2.6 A skier of mass 75.0 kg skis over a hemispherical mound of snow of radius
10.0 m. At the top of the mound the skier's velocity vector is horizontal with
a magnitude of 30.0 km hr
−
1
. Assuming the snow to be frictionless, calculate
the magnitude and direction of the force exerted by the skier on the snow at
the top of the mound.
2.7 Two blocks of mass
m
1
=
=
1
.
0 kg rest in contact on a fric-
tionless horizontal surface. If a force of 3.0 N is applied to
m
1
such that both
blocks accelerate, deduce the contact force between the blocks.
2.8 A 2 kg block rests on a 4 kg block that rests on a frictionless surface. The
coefficients of friction between the blocks are
µ
s
=
3
.
0kg and
m
2
=
0
.
2.
(a) What is the maximum horizontal force
F
that can be applied to the 4 kg
block if the 2 kg block is not to slip?
(b) If
F
has half the value found in (a), find the acceleration of each block
and the force of friction acting on each block.
(c) If
F
has twice the value found in (a), find the acceleration of each block.
0
.
3and
µ
k
=
