Lesson 14 notes – Analysing circular motion
Objectives
Be able to use equations for centripetal acceleration and force in different
situations.
Outcomes
Be able to apply the equations for circular motion to a number of different real life
situations in 2D.
Be able to apply the equations for circular motion to a number of different real life
situations in 3D.
When we release an object swung in a circular path, it takes a net force (the
resultant of all forces) acting inward that keeps the object spinning in a circle; if
you let go, the net force is no longer inward, so the object flies outward.
For example, If a car travelling around a level curve. Where does the net force
acting toward the center of the curve come from? Static friction. So Newton's
2nd law for this situation is determined as follows:
F = ma
and, static friction, assuming car is not skidding
Ff =
FN =
mg
Centripetal acceleration:
ac = v2/r
In this situation,
Ff = F
mg = ma = mac = mv2/r
Therefore:
g = v2/r
As a result, we see that the greater the
of the road is, the faster a car can
travel without skidding. And the car can travel in a small radius of a curve
without skidding.
Example
If a road's static coefficient is 0.5. And the radius of the curve is 50 m. What is
the greatest speed a car can travel without skidding out of the road?
As we have derived from the curve section:
g = v2/r
As a result, we see that the greater the
of the road is, the faster a car can
travel without skidding. And the car can travel in a small radius of a curve
without skidding.
= 0.5
r = 50 m
9 = 9.8 m/s2
We have:
g = v2/r
Rearrange it:
v2 =
gr
vmax =
vmax =
vmax =15.65 m/s
Extension
Wall of Death
A. THE DANGERS:
The frictional force needs to be high enough on the wall ( so don't try it with wet
walls )
Vehicle speed needs to stay above a calculable minimum ( so don't go too
slowly! )
Motorcycles must lean at a speed dependent angle to prevent tipping over: the
higher the speed, the smaller the needed lean angle.
Cars cannot lean but they do not need to (see why below).
Riders in cars on the wall should lean to reduce nausea and muscle strain.
THE PHYSICS CONCEPTS:

the maximum static frictional force (FF) is proportional to the normal
reaction (N), where the normal reaction is the always observed
perpendicular push back by a surface when it receives a force

the force causing motion in a circle acts towards the centre of the circle
and can be written as Fc= mv2/r, where m and v are the mass and velocity
of the rotating body and r is the radius of the rotation;

the force (down) due to gravitational attraction can be written F = mg;

the centre of mass of a body is the point that, if all the mass were
concentrated there, would behave translationally just like the real body
does;

if force is applied to a body at a distance from a pivot point, then the body
will tend to rotate about that pivot point, with the size of the rotation
determined by multiplying the force by the perpendicular distance (the
answer being a number we call "torque");

any force which does not act along a line through the centre of mass will
produce a torque (and hence a tendency to rotate);

precession (like that seen in a spinning top) needs a torque applied to stop
it.
THE PHYSICS DETAILS:
For a body of mass m moving in a horizontal circle on a wall of death, the normal
reaction from the wall is what supplies the force necessary to obtain motion in a
circle.
So we can write: N = mv2/r
If the body does not slide down the wall, then the pull of gravity downwards must
be balanced by the frictional force (due to the ‘roughness' of the wall) upwards.
So the force diagram looks like:
This frictional force can vary as needed up to a maximum of
.
But how much frictional force is available depends on the speed, which means
there will be a MINIMUM speed for revolution at a constant height above the
ground. This minimum speed will be the one for which the maximum frictional
force exactly equals the gravitational pull on the mass (since any lower frictional
force will not be strong enough to provide the balance).
Higher speeds will also produce no slipping, but as the speed rises steering and
control will become more difficult.
So when
, then v is the lowest it can be (ie vMINIMUM ), for
no slipping to occur. Which means the minimum safe speed is
.
BUT:
The above analysis assumes a point mass (i.e. with all the mass concentrated in
a single point located at the centre of mass and all the forces acting on that
point).
However for a motorcycle on the wall of death, although the weight acts through
the centre of mass, the friction acts at the wheels (on the wall).
These three forces (friction forces on each of the tyres, a weight force through
the centre of mass) balance but are not in the same line. Which means the
motorcycle will tend to rotate and tip over.
The normal reactions from the wall (acting where the tyres touch the wall) cannot
help to balance this torque because they will not produce any turning effect if the
motorcycle is perpendicular to the wall:
But if the motorcycle leans at an angle to the vertical to the wall then the normal
reactions from the wall, will produce a tendency to rotate (a torque) in the
opposite direction.
So if the rider leans the motorcycle at the correct angle then the torques will be
equal and no rotation will occur.
For other angles of lean there will be unbalanced torques causing the motorcycle
to rotate and fall.
At these other angles of lean the rider's muscles will need to push more strongly
in order to supply the extra torque to maintain balance. The rider may also
experience nausea (because the endolymph fluid in the ear will experience an
unbalanced torque and rotate).
With cars however the precession can't occur. Cars do not need to lean as long
as they have a relatively wide wheelbase and a low centre of mass. To see why
read the analysis below.
WHY DO CARS NOT NEED TO LEAN?
Summary : When a car is running around a vertical wall, the tyres closer to the
ground experience stronger normal reaction force than the tyres opposite them
(closer to the top of the wall). So, provided the speed is high enough, the
counterclockwise torque from the normal reaction of the lower tyres can provide
the extra balancing torque without the need to lean. Details: Consider the
diagram below showing the relevant forces and distances.
Define:
Nupper (lower) = sum of the normal reactions of the upper ( lower ) front and rear
wheels;
Fupper (lower) = sum of the friction forces of the upper ( lower ) front and rear
wheels;
L = the wheelbase (distance between the front or back wheels, assumed the
same)
H = perpendicular distance from the car's centre of mass to the Wall;
v = speed of the centre of mass of the car;
R = radius of the circular path travelled by the centre of mass of the car;
= static friction coefficient between the Wall and the tyres.
Assume the centre of mass of the car to be on the plane that perpendicularly
bisects the wheelbase.
Force analysis reveals:
(i) total normal reaction from the Wall provides the circular motion force (Fc):
...............................Nupper + Nlower = Fc= mv2/r .........................................…… (A)
(ii) total frictional force balances the weight:
...............................Fupper + F lower = mg ......................................................…… (B)
Torques (about the car's centre of mass) yield:
...............................(N upper)(L/2) + (Fupper + Flower )(H) = (Nlower )(L/2)........... ......(C)
Hence:
...............................Nupper = (1/2) (mv2/r) – (H/L)(mg) Flower = (1/2) (mv2/r) +
(H/L)(mg)
Since Fupper can never be negative (because walls can only push, never pull),
then:
(i)
For safety, at maximum friction ,
(ii)
..............{"no flip"}
(Nupper + Nlower ) > mg, so:
................{"no slip"}
Both conditions must apply simultaneously - meaning the minimum no-lean
speed required for travel on a Wall of death of given radius is determined by
whichever is the greater of (1/
) and (2H/L).
For bodies with a wide wheelbase and a low positioned centre of mass (say,
standard cars), the ratio 2H/L will be small, so the reciprocal of the friction
coefficient becomes the most important factor (i.e. do it on dry days only!) and
the minimum no-lean speed is easily attainable.
For bodies with a short wheelbase and a high positioned centre of mass (say,
rollerblade riders), the ratio 2H/L will be large so the minimum no-lean speed
becomes impracticably high. A motorcycle would be the limiting case of this,
since L is effectively zero and hence no no-lean speed possible.
Which is what we found when riding our real Wall of Death: cars do not lean,
motorcycles do!