S A 0 27
As part of a packaging proccss, a box slides down a delivery chute as
shown in Figure 86. The cudkient of friction baweai the box and the
chute is 0.5. Estimate the aoceleration of the box.
SA028
In another part of the plant the box is raised on a wnveyor belt as shown
in Figure 87. The W e n t of friction in this carc is 0.6. What is the
maximum acdemtion which the belt may have without the box slipping?
S A 0 PO
At thb end of the proocss the bax is delivered to a level table ss shown in
Figure 88. The co&cieat of friction between the box and the table is 03
and it d v e a on the level d o n with a speed of 2 m a-'.
(a) What is the aocekration of the box on the level table?
(b) Doca the acceleration change as the box moves along the table?
(C) Sketch the motion curves from the instant the box arrives on the table
and estimate how far the box trawls befofe wmbg to rest.
Figwe 86
L
m,
Figure 87
\
Figwe 88
5.3 Vehicle dynamker problems
The method we an going to use to estimate the accelerations of vehicles in
thin acotion depends upon an assumption which you should a p p h t e
from thc outset. The motion of, say, a car is produced by a torque at the
axle producing a friction force at the contact between wheel and road It is
thin force that propeh the vehicle. However, aome of the torque will be
nquimd to produce angulat aderation of the wheels. The signiflcancc of
thin~willdependuponthe~tudeofthcdmomentofmass,
I.dthew~AtthinstPgearearegoingtoa~lmcthatitisrrmallglough
to be m g h t d Later, when dealing with rotatiom you will haw an
opporhmity to iw how thb simple model comparca with thc more
complex one which tnkw into account the e W of the rotating masses of
thew~.Wewillalroi~oreforthetimeboingtheddUaionintbe
available toque needed to wrroome feaom such as friction in the
bearings and tolling mistanca.
m-6
The dragster shown in Figure 89 can attain a starting acmlcration of
11.0 m B-' without the rear wheels slipping. The total maas of vehicle and
driver is 760 kg.
(a) What is the total friction force exerted by the road on the driviqg
wheels to produce this acceleration?
(b) If the radius of the rear whccls is 0.4 m, what is the torque delivned to
each rear wheel by the axle?
Sduual
Step I
Sketch (Figure 90)
Step 2
Fm-body diagram with rures (Figure 91)
W - weight force
N,, N, - mctkmn at front and rear wheels
F - friction force
Step3
R,=F=ma,
We need to go no further since this equation providss all the information
we rquire.
(a) F=760kgxlIms-'=8360N
-
(b) The torque T (F/2)
X
r
(force at cach wheel is F/2)
It is helpful to indicate clearly whether the torque T is clockwise. or
anticlockwise hcre. That is why I have txpnsscd the final solution for the
torque in vactor form as T=1.67 kN m>, indicating a clockwise torque.
In this analysis we have &o ignored the backward directed friction fonrs
generated on the front wheel(s) by ground contact, which were illustrated
in Figure 65(b). For the various reasons listed he= the actual torque
rcquirod would be a little greater than our estimate.
A small saloon car with driver has a mass of 700 kg. The starting torque
delivered to cach of the driving whals is 500 N m and the wheel diameter
is 560 mm. The driving whccIs start rolling with no slipping. Determine
the s t a r t i n g d m t i o n (a) on a level road, (b) on an upward incline at 20"
to the horizontal. The modelling assumptions outlined above should be
wedhsrc.
The drag cocfficicnt of road vehicles may be measured experimentally by
driving the vcbide up to spcod, dimmeoting the drive and measwing the
resultant *tiolL
A large van of mass MOO kg is mvelliag at 55 km h-' whm the drive is
dimmeated. Thc initial daleration is found to be 0.25 m S-'. In a
separate crpcrimcnt the rcsistancc from sources other than aerodynamic
drag (rolling resistance of the tgTw, bearing friction, etc.) is found to be
5M) N. The frontal a m of the van is 4 m'. Estimate the drag coe6icimt C
of the van. Use.p, = 1.225 kg m-3.
A saloon car of mass 1000 kg and frontal a m 2 ma has a rolling mistancc
of 120 N which m a i n s approximately constant with speed. The d r q
d c i w t C = 0.5. Detmint the fora which must be applied by the
driving wheels to produce an acceleration of l m sda when the car is
travelling ac (a) l0 km h-' (2.8 m S-'). (b) 60 km h-' (16.7 m S-').
p,, = 1.25 kg m-'.
(b)
K
#J
f
Figure 92
You may 5 d it inatmotive to compare the percentage m r that would
have arim in the two ceses of SAQ 32 had the drag fora h neglected.
.
The dynamic anatysis of certain problems involving rotation may be
.
performed quite accurately by approximating the motion as u m h e a r
tradation. As an example, consider the pmdulum ia Flguc 92 Figure
92(@ shows the aotoal motion wbere the pendulum bob is in 6xcd axis
rotation about 0.Thc system may, however, be a n a l y d by neglecting the
mass of the arm and modelling the bob as undergoing translation as
shown in Figure 92(b). In a latcr section you will be given the oppomtnity
to compare the results of analysis using this assumption with those
provided by a more complex model including the rotation.
Figure 92(b) also shows the co-ordinate system which is generally most
useful in this type of analysis. The X-y co-ordinate system has been
replaced by a system instantaneously n o d (n) and tangential (t) to the
motion. (Remember that the coordinates arc not moving with the
pendulum, they are fixed relative to the Earth. They happen to be aligned
with the pendulum at this instant.) This meam that, as before, we have
axes aligned with the known components of the acceleration of G, the
centre of mass of the bob. Thee acceleration components (which form step
4 of the procedure) are given from Unit 5 by:
ceaatioll CO-@
for drcalu motion
@=W
a. = w2r = v2/r
I
Ex.mp1. 7
A 5 kg ball swings in a vertical circle as shown in Figure 93. The length of
the string is 1m and at the instant shown the tension in the string is 40 N.
Determine the velocity and acceleration of the ball at this instant.
Step 1
Sketch (Figure 94)
Step 2
Free-body diagram (Figure 93, including the normal (n) and
tangential (t) co-ordinate axes system.
T - string tension (40 N)
W - weight
Step3
R,=Wsin60°=ma,
R.=T- W W S ~ O " = ~ ~ .
Step 4
q = v:/r
Step 5
W = mg
This provides sufiicient equations to solve the problem.
a. = (Tlm) - g cos 60'
= (4015) - 9.81 cos
60"
= 3.095 m S-'
a, = 3.10 m s-2
a,=gsin 60'
= 8.496 m s-l
a, = 8.50 m S-'
C o m b i i g the acceleration components:
a = C(3.095)' + (8.496)2]111
= 9.042 m S-'
iT=9.04 m ~ - ~ - 9 4 0 "
Figure 95
SA0 33
The pendulum of an impact testing machine is shown in Figure 96. It is
released from rest when 0 = 75". The mass of the bob is 50 kg and the mass
of the rod may be neglected. At the moment of r e k (a) what M the
tension in the rod? (b) what is the accelemtion of the centre of mass of the
bob? (c) what is the angular acceleration of the rod?
A locomotive of mass 100 X 103kg moves at a constant s p d of
70 km h-' around a bend of radius 300 m. Estimate the fora exerted by
the rails upon the locomotive.
5.5 Gravitation and orbital motion
To conolude this section, let us look at a few problems away from the
surface of the Earth. Remember that now you cannot use the speoial case
of the weight force W = mg, with g = 9.81 m S-', but must usc the general
relationship for gravitational forces P = Gm,m,/?.
In considering orbital motion, it may seem strange that satellites, both
natural and man-made,maintain orbits without expmding power. I t h i i
attempting the problems will sort this out. They src really just like the
mass on the string and the pendulum that you saw earlier. In fact, they are
easier to analyse since there is only one significant force acting.
(a) Derive an equation for the acceleration, a, due to gravitational forces
acting on a spacecraft which is at a distance x from the centre of the
Earth on a straight line joining the centres of the Moon and the Earth.
Your equation should contain the following trrms:
m,
- the mass of the Earth
& - the
mass of the Moon
l - distance between the centms of Moon and Earrh
x - distana from antre of Earth to spactcraR
G - gravitational constant
Use appropriate data from the 'Table of uscful physical data' on page
74.
Hint: Let the mass of the craft be m,. You should find however, that
the final equation for a in independent of m,.
(b) Evaluate the awleration for the instant when the spacecraft is
midway between the centres of Earth and Moon.
A lunar module is in circular orbit 30 km above the surface of the Moon.
Determine the time taken to describe one orbit. The radius ofthe Moon is
1740 km; use any other data you require from the table on page 74. (The
only significant gravitational force is that exerted by the Moon.)
A communications satellite is to be placed in a c i ~ ~ u lorbit
a r around the
Earth. In order that it remains over the same position, it orbits once eveq
24 hours (a gostationary orbit). At what height must the satellite be
stationed? The mdius of the Earth is 6.37 X 106 m: usc any other data you
r c q u i ~from
~ the table on page 74.