FORCE AND MASS
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Fundamentally, a force (N – Newton) is a push or a pull
When you push or pull something, you exert a force
Holding something out in your hand exerts a force to
oppose the downward pull of gravity
There are two quantities that characterise a force
Magnitude and direction
Therefore force is a vector, which is a mathematical
quantity with both magnitude and direction
Generally there are several forces acting on an object
A book resting on a table experiences a downward
force due to gravity and an upward force due to the
table
The total, net, force being exerted is the vector sum of
the individual forces acting on an object
The mass (kg) of an object is a measure of how difficult
it is to change its velocity
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S FIRST LAW OF MOTION
(1)
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When you push a book across a table, it stops moving
when you stop pushing it
But it is wrong to say that a force is needed for an
object to move
A force is required to change an object’s motion
It is the force of friction between the book and the table
that causes the book to come to rest
You would travel much further sliding on ice where the
frictional force is a lot smaller than you would sliding
with the same initial speed on sand
If you were to slide on a frictionless surface, you would
keep moving with constant velocity forever
You would only stop moving if another force was
applied to stop you
Newton’s 1st Law: An object at rest remains at rest as
long as no net force acts on it. An object moving with
constant velocity continues to move with the same
speed and in the same direction as long as no net force
acts on it
No net force: no force acts on an object, or forces act
on an object but they sum to zero
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S FIRST LAW OF MOTION
(2)
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Newton’s 1st Law also known as law of inertia
Inertia literally means “laziness”
An object (matter) is “lazy” because it will not change its
motion unless forced to do so
An object at rest will not start moving on its own
If an object is already moving with constant velocity, it
will not alter its speed or direction unless a force
causes the change
Being at rest and moving at constant speed are
equivalent
A more compact statement of the first law is that if the
net force on an object is zero, then its velocity is
constant
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S SECOND LAW OF
MOTION (1)
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To hold an object in your hand, you must exert an
upward force to oppose gravity
Suddenly removing your hand causes the object to
accelerate downwards as the only force acting on it is
gravity
Unbalanced forces cause accelerations
Newton’s 2nd law can be demonstrated using a force
meter, which contains a spring inside
The scale gives a reading F exerted by the spring
Two equal weights exert twice the force of one
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S SECOND LAW OF
MOTION (2)
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A calibrated force meter can be used to perform
experiments that demonstrate the 2nd law
It can be used to accelerate a mass on a “frictionless”
air track. If the force is doubled, the acceleration is also
doubled – acceleration is proportional to the force
If the mass of an object is doubled but the force
remains the same, the acceleration is halved –
acceleration is inversely proportional to mass
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S SECOND LAW OF
MOTION (3)
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There may be several forces acting on a given mass,
and these forces maybe in different directions
r
r
Sum of force vectors = Fnet = ∑ F
Acceleration is also a vector: it has magnitude, direction
r
Thus Newton’s 2nd Law is:
r
F
r
a=
∑
m
or
r
F
=
m
a
∑
In words:
r If an object of mass m is acted on rby a net
force ∑ F , it will experience an acceleration a that is
equal to the net force divided by the mass. Because the
net force is a vector, the acceleration is also a vector.
The direction of an object’s acceleration is the same as
the direction of the net force acting on it
Vector components: ∑ Fx = max , ∑ Fy = ma y , ∑ Fz = maz
r
r
F
When ∑ = 0 , then a = 0 . When acceleration is zero,
then the velocity is constant, thus Newton’s first and
second laws are consistent with one another.
1. Newton’s Laws of
Motion and their
Applications
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FREE BODY DIAGRAMS
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When solving problems involving forces and Newton’s
laws, it is necessary to make a sketch that indicates
each and every external force acting on an object
This is called a free body diagram
In such diagrams, each object of interest is treated as a
point particle and each of the forces acting on the
object are applied to that point
1. Newton’s Laws of
Motion and their
Applications
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FREE BODY DIAGRAMS: EXAMPLE
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Mark, Larry and Colin push on a 752kg boat that floats
next to a dock. They each exert an 80.5N force parallel
to the dock. What is the acceleration of the boat if they
all push in the same direction? Give both direction and
magnitude. What is the magnitude and direction of the
boat’s acceleration if Larry and Colin push in the
opposite direction to Mark’s push?
1. Newton’s Laws of
Motion and their
Applications
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FREE BODY DIAGRAMS: REAL
WORLD EXAMPLE
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Foamcrete is a substance designed to stop an
aeroplane that has run off the end of a runway, without
causing injury to passengers. It is solid enough to
support a car, but crumbles under the weight of a large
aeroplane. By crumbling, it slows the plane to a safe
stop. Suppose a 747 with a mass of 1.75×105kg and an
initial speed of 26.8m/s is slowed to a stop inr122m.
What is the magnitude of the retarding force F exerted
by the Foamcrete on the airliner?
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S THIRD LAW OF MOTION
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Forces always come in pairs
Forces in a pair are equal in magnitude, but opposite in
direction
Newton’s 3rd law in words: For every force that acts on
an object, there is a reaction force acting on a different
object that is equal in magnitude and opposite in
direction
r
If object 1 exerts ra force F on object 2, then object 2
exerts a force − F on object 1
1. Newton’s Laws of
Motion and their
Applications
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NEWTON’S THIRD LAW OF MOTION:
EXAMPLE
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Two groups of canoeists meet in the middle of a lake.
After a brief visit, a person in canoe 1 pushes on canoe
2 with a force of 46N to separate the canoes. If the
mass of canoe 1 and its occupants is m1 = 150kg, and
the mass of canoe 2 and its occupants is m2 = 250kg,
find the acceleration the push gives to each canoe.
What is the separation of the canoes after 1.2s of
pushing?
1. Newton’s Laws of
Motion and their
Applications
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CONTACT FORCES
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When objects are touching one another, the actionreaction forces are often referred to as contact forces
Example: A box of mass m1 = 10.0kg rests on a
smooth, horizontal floor next to a box of mass m2 =
5.00kg. If you push on box 1 with a horizontal force of
magnitude F = 20.0N, what is the acceleration of the
two boxes? What is the force of contact between the
boxes? (The force exerted by one box on the other is
different, depending on which one you push – prove
this by calculating the contact force when pushing on
box 2 with the same force of 20N)
1. Newton’s Laws of
Motion and their
Applications
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THE VECTOR NATURE OF FORCES:
FORCES IN 2D
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If more than one force acts on an object, then its
acceleration is in the direction of the vector sum of the
forces
Force and acceleration have both magnitude and
direction
Mass is simply a positive number with no direction
Example: Suppose two astronauts are using jet packs
to push a 940kg satellite toward the space shuttle. With
the coordinate system indicated in the figure, astronaut
1 pushes in the positive x direction and astronaut 2
pushes in a direction of 52°above the x axis. If
astronaut 1 pushes with a force of magnitude F1 = 26N,
and astronaut 2 pushes with a force of magnitude F2 =
41N, what are the magnitude and direction of the
satellite’s acceleration?
1. Newton’s Laws of
Motion and their
Applications
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THE VECTOR NATURE OF FORCES:
FORCES IN 2D: EXAMPLE
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Jack and Jill lift upwardr on a 1.3kg bucket of water, with
F1 of magnitude 7.0N and Jill
Jack exerting a force
r
exerting a force F2 of magnitude 11N. Jill’s force is
exerted at an angle of 28°with the vertical. At wha t
angle θ with respect to the vertical should Jack exert
his force if the bucket is to accelerate straight upward?
1. Newton’s Laws of
Motion and their
Applications
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WEIGHT
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When you step onto a scale to weigh yourself, the scale
gives a measurement of the pull of Earth’s gravity – this
is your weight
The weight, W, of an object on the Earth’s surface is
the gravitational force exerted on it by the Earth
The greater the mass of an object, the greater its
weight
W = mg; Unit it the newton (N), g = 9.81m/s and is
constant for objects falling due to the Earth’s gravity
Weight, W, and g are both vector quantities
r
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Thus W = mg
Example: A 97kg fireman slides 3.0m down a pole to
the ground floor. Suppose the fireman starts from rest,
slides with constant acceleration, and reaches
the
r
ground in 1.2s. What is the upward force F exerted by
the pole on the fireman?
1. Newton’s Laws of
Motion and their
Applications
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APPARENT WEIGHT
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If you are in a lift moving
downward which comes to
a rest by accelerating
upward, you feel heavier
You feel lighter when a lift
moving upward comes to a
rest by accelerating
downward
Motion of lift gives rise to
an apparent weight that
differs from our own weight
Consider an elevator that is
moving with an upward
acceleration a
Sum of forces acting is
ΣFy = Wa – W
By Newton’s 2nd law, sum
must equal may (ay = a)
Wa – W = ma
The apparent weight Wa is
Wa = W + ma = mg + ma
If the lift accelerates
downward, a is –a, thus
Wa = W – ma = mg - ma
1. Newton’s Laws of
Motion and their
Applications
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APPARENT WEIGHT: EXAMPLE
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As part of an attempt to combine physics and biology in
the same class, a lecturer asks students to weigh a 5kg
salmon by hanging it from a fish scale to the ceiling
r of a
lift. What is the apparent weight of the salmon, Wa , if
the elevator is at rest, moves with an upward
acceleration of 2.5m/s2, and moves with a downward
acceleration of 3.2m/s2?
1. Newton’s Laws of
Motion and their
Applications
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NORMAL FORCES
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When an object is resting on a surface its acceleration
is zero, so the net force acting on it is zero
Thus the downward force of gravity is being opposed
by an upward force exerted by the surface
r
This force is called the normal force N
The reason why this force is called normal is that it is
perpendicular to the surface. In mathematical terms,
normal simply means perpendicular
In the figure below, the magnitude of the normal force is
equal to the weight of the tin
However, in general the normal force maybe be greater
or less than the weight of an object
1. Newton’s Laws of
Motion and their
Applications
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EXAMPLE OF NORMAL FORCE
DIFFERING FROM WEIGHT
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Consider pulling a 12kg suitcase across a smooth floor
by exerting a force of 45N at angle 20°above the
horizontal
Weight of suitcase is mg = 12×9.8 = 118N
The suitcase does not move in the y direction, so its y
component of acceleration is 0: ay = 0 → ΣFy = may = 0
r
So we can solve for the one force that is unknown N
ΣFy = Wy + Fy + Ny = - mg + Fsin20 + N = 103N
1. Newton’s Laws of
Motion and their
Applications
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NORMAL FORCES: EXAMPLE
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A 6kg block of ice is acted on by two forces as shown in
the diagram. If the magnitude of the forces are F1 =
13N and F2 = 11N, find the acceleration of the ice and
the normal force exerted by it on the table.
1. Newton’s Laws of
Motion and their
Applications
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FRICTIONAL FORCES
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Previously we had assumed that that surfaces were
smooth – objects could slide without resistance to their
motion
No surface is perfectly smooth
Atomically, smooth surfaces are jagged and rough
To slide one surface across another requires a force
large enough to overcome the resistance of
microscopic hills and valleys colliding into one another
This is the origin of the force called friction
Some friction is desirable, such as when walking the
force that propels you is the friction between your
shoes and the ground (harder to walk on ice!)
Two types of friction: kinetic and static
1. Newton’s Laws of
Motion and their
Applications
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KINETIC FRICTION
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Kinetic friction is the friction encountered when surfaces
slide against one another
The force generated by this friction is designated with fk
fk acts to oppose sliding motion at the point of contact
The force of kinetic friction is proportional to the
magnitude of the normal force, N
Thus: fk= µkN → not a vector equation
µk is the coefficient of kinetic friction (dimensionless)
The greater µk, the greater the friction; the smaller µk,
the smaller the friction
The force of kinetic friction is independent of the
relative speed of the surfaces, and the area of contact
between the surfaces
1. Newton’s Laws of
Motion and their
Applications
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KINETIC FRICTION: EXAMPLE
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A trained sea lion slides from rest with constant
acceleration down a 3.0m long ramp into a pool of
water. If the ramp is inclined at an angle of 23° above
the horizontal, and the coefficient of kinetic friction
between the sea lion and the ramp is 0.26, how long
does it take for the sea lion to make a splash in the
pool?
1. Newton’s Laws of
Motion and their
Applications
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STATIC FRICTION
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Static friction keeps two surfaces from moving relative
to one another
The force of static friction is fs
Consider moving a brick, as shown below
At rest, fs = 0, then fs = F1 but the brick does not move
When fs = F2, the brick stays at rest, but any bigger
than F2, the brick will move
Thus there is an upper limit that can be exerted by
static friction, fs,max → 0 ≤ fs ≤ fs,max
This maximum is also proportional to magnitude of
normal force → , fs,max = µsN (non vector)
µs is dimensionless and is generally greater than µk
fs independent of area of contact between surfaces
1. Newton’s Laws of
Motion and their
Applications
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STAIC FRICTION: EXAMPLE
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A flatbed truck slowly tilts its bed upward to dispose of
a 95kg load. For small angles of tilt, the load stays put,
but when the lift angle exceeds 23.2°, the crate begins
to slide. What is the coefficient of static friction between
the bed of the truck and the load?
1. Newton’s Laws of
Motion and their
Applications
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STRINGS AND SPRINGS
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A string pulled from
either end has a tension
T. If the spring were to
be cut at any point, the
force required to hold
the ends together is T
In a heavy rope, the
tension is noticeably
different at points 1, 2
and 3 (higher tension).
As the rope becomes
lighter, the difference in
tension decreases.
We assume that ropes
are massless, hence the
tension is uniform
Pulleys are used to
redirect a force exerted
by a rope
An ideal pulley
(massless and
frictionless) simply
changes the direction of
the tension in a rope
without changing its
magnitude
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Motion and their
Applications
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TENSION: EXAMPLE
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A traction device employing three pulleys is applied to a
broken leg as shown. The central pulley is attached to
the sole of the foot, and a mass m supplies the tension
in the ropes. Find the value of the mass m if the force
exerted on the sole of the foot by the central pulley is to
be 165N.
1. Newton’s Laws of
Motion and their
Applications
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SPRINGS AND HOOKE’S LAW
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A spring exerts a force
that is proportional to
the amount, x, by which
it is stretched or
compressed
F = kx
k is the force constant
or spring constant
Its units are N/m
The larger the k, the
stiffer the spring
More precisely, the
force exerted by the
spring is opposite to the
pull (or push)
Hence Fx = -kx (gives
magnitude and
direction)
Ideal springs are
massless and obey
Hooke’s Law exactly
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Motion and their
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TRANSLATIONAL EQUILIBRIUM
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An object is in translational
equilibrium when the net
r
force acting on it is 0: ∑ F = 0
From Newton’s 2nd law, it is equivalent to saying that
the object’s acceleration is zero
Equilibrium allows us to calculate unknown forces
For the bucket: ΣFy = 0 if lifted with constant speed
Thus T1 – mg = 0 (W = mg)
On the pulley: T2 – T1 – T1 = 0 or T2 = 2mg
1. Newton’s Laws of
Motion and their
Applications
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TRANSLATIONAL EQUILIBRIUM:
EXAMPLE
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To hang a 6.2kg pot of flowers, a gardener uses two
wires – one attached horizontally to a wall, the other
sloping upward at an angle of θ = 40°and attached to
the ceiling. Find the tension in each wire.
1. Newton’s Laws of
Motion and their
Applications
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CONNECTED OBJECTS
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If there are two objects connected by a string, and the
force and the masses are known, the acceleration and
tension can be found
Each object is considered as a separate system
Box 1: F – T = m1a
Box 2: T = m2a
Both boxes have the same acceleration
Adding both equations: F = (m1 + m2)a
Thus a = F/(m1 + m2)
Finally T = m2a = (m2/m1 + m2)F
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Motion and their
Applications
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CONNECTED OBJECTS: EXAMPLE
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If there is a pulley, it is easier to have the coordinate
system follow the string
A block of mass m1 slides on a frictionless table. It is
connected to a string that passes over a pulley and
suspends a mass m2. Find the acceleration of the
masses and the tension in the string.
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Motion and their
Applications
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CIRCULAR MOTION (1)
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When an object moves, a force is required to change its
speed, direction or both
If you drive a car with constant speed on a circular
track, the direction of the car’s motion changes
continuously
A force must act on the car to cause this change in
direction
Swinging a ball on the end of a piece of string causes a
tension in the string pulling outward
At the other end of the string where the ball is attached,
the tension pulls inward, towards the centre
To make an object move in a circle with constant
speed, a force must act on it that is directed toward the
centre of the circle
1. Newton’s Laws of
Motion and their
Applications
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CIRCULAR MOTION (2)
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From the previous example of the swinging ball, the ball
is acted on by a force toward the centre of the circle
It must therefore be accelerating toward the centre of
the circle
However, how can a ball moving at constant speed
have an acceleration?
Remember that acceleration is a vector, and is
produced whenever speed or direction changes
In circular motion, the direction changes continuously
This centre-directed acceleration is called centripetal
acceleration
When an object moves in a circle with radius r, constant
speed v, its centripetal acceleration is acp = v2/r
A force must be applied to an object to give it circular
motion
For an object of mass m, the net force acting on it must
have a magnitude given by fcp = macp = mv2/r
This force is called the centripetal force
1. Newton’s Laws of
Motion and their
Applications
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CIRCULAR MOTION: EXAMPLES
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A 1200kg car rounds a corner of radius r = 45m. If the
coefficient of static friction between the tires and the
road is µs = 0.82, what is the greatest speed the car can
have without skidding?
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If a road is banked at the proper angle, a car can round
a corner without an assistance from friction between
the tires and the road. Find the appropriate banking
angle for a 900kg car travelling at 20.5m/s in a turn of
radius 85.0m
1. Newton’s Laws of
Motion and their
Applications
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