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Forces and Motion
Forces in Two Dimension
Vectors

Vectors in Multiple Dimensions

We can add vectors even when they don’t point along same straight
line

Draw the vectors at correct angle and measure direction and
magnitude of resultant vector. Add vectors by placing them tip-to-tail
and then drawing resultant of vector by connecting tail of first vector
to tip of second vector
Vectors

Vectors in Multiple Dimensions

If adding two vectors at right angles, use Pythagorean theorem to find
magnitude of resultant, R

Sum of squares of magnitudes equal to square of magnitude of
resultant vector
Vectors

Vectors in Multiple Dimensions

If two vectors at angle other than 90°, use law of cosines or law of
sines
Law of cosines

Square of magnitude of resultant vector equal to sum of magnitude of
squares of two vectors, minus two times product of magnitudes of
vectors, multiplied by cosine of angle between them
Law of sines

Magnitude of resultant, divided by sine of angle between two vectors,
equal to magnitude of one vectors divided by angle between that
component vector and resultant vector
Vectors

Vectors in Multiple Dimensions

Find magnitude of sum of 15-km displacement and 25-km
displacement when angle between them 90° and when angle between
them 135°
Vectors

Components of Vectors

We start by choosing coordinate system similar to laying a grid on
transparent sheet atop vector problem

Choose where to put center of grid (origin) and establish directions
axes point
Vectors

Components of Vectors

When motion on surface of Earth, x-axis east and y-axis north

When motion of object moving through air, positive x-axis horizontal
and positive y-axis vertical (upward)

If motion on hill, place positive x-axis
in direction of motion and y-axis
perpendicular to x-axis
A = Ax + Ay
A
Ay
Ax
Friction

Components of Vectors
Vectors

Algebraic Addition of Vectors

If you have two or more vectors, they are added by first resolving each
vector into x- and y-components

X-components added to form x-component of resultant:
Rx = Ax + Bx + Cx

Y-components added to form y-component
of resultant:
Ry = Ay + By + Cy
Vectors

Algebraic Addition of Vectors

The components of a vector are projections of the component vectors
Vectors

Algebraic Addition of Vectors

Rx and Ry at right angle (90°) so magnitude of resultant calculated
using Pythagorean theorem, R2 = Rx2 + Ry2

To find angle or direction of resultant, tangent of angle vector makes
with x-axis is:
Angle of Resultant Vector
Vectors

Vectors in Multiple Dimensions

Jeff moved 3 m due north, and then 4 m due west to his friend’s house.
What is the displacement of Jeff?
Vectors

Vectors in Multiple Dimensions

Calculate the resultant of the three vectors A, B, and C as shown in the
figure.
(Ax = Bx = Cx = Ay = Cy = 1 units and By = 2 units)
Vectors

Vectors in Multiple Dimensions

If a vector B is resolved into two components Bx and By, and if  is the
angle that vector B makes with the positive direction of x-axis, which
of the following formulae can you use to calculate the components of
vector B?
Vectors

Vectors in Multiple Dimensions

A small plane takes off and flies 12.0 km in a direction southeast of the
airport. At this point the plane turns 20.0° to the east of its original
flight path and flies 21.0 km. What is the magnitude of the plane’s
resultant displacement from the airport?

A hammer slides down a roof that makes a 32.0° angle with the
horizontal. What are the magnitudes of the components of the
hammer’s velocity at the edge of the roof if it is moving at a speed of
6.25 m/s
Friction

Friction

When you push your hand across a surface you feel the force called
friction opposing the motion of your hand
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Friction is a force that resists motion whenever the surfaces of two
objects rub against each other
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There are two types of friction, and both always oppose the initial
motion of an object sliding across a surface
Friction
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Kinetic Friction

When you push a book across the desk, it experiences a type of
friction that acts on moving objects

This force is known as kinetic friction, and it is exerted on one surface
by another when the two surfaces rub against each other because one
or both of them are moving
Friction

Static Friction

To understand the other kind of friction, imagine trying to push a
heavy couch across the floor. You give it a push, but it does not move

Because it does not move, Newton’s laws tell you that there must be a
second horizontal force acting on the couch, one that opposes your
force and is equal in size

This force is static friction, which is the force exerted on one surface
by another when there is no motion between the two surfaces
Friction
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Static Friction

You might push harder and harder, as shown in the figure below, but if
the couch still does not move, the force of friction must be getting
larger

This is because the static friction force acts in response to other forces.
Finally, when you push hard enough, as shown in the figure below, the
couch will begin to move
Friction

Static and Kinetic Friction
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There is a limit to how large the static friction force can be. Once your
force is greater than this maximum static friction, the couch begins
moving and kinetic friction begins to act on it instead of static friction

Frictional force depends on the materials that the surfaces are made of

For example, there is more friction between skis and concrete than
there is between skis and snow

The normal force (FN) between the two objects also matters. The
harder one object is pushed against the other, the greater the force of
friction that results.
Friction

Static and Kinetic Friction
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If you pull a block along a surface at a constant velocity, according to
Newton’s laws, the frictional force must be equal and opposite to the
force with which you pull

You can pull a block of known mass along a table at a constant
velocity and use a spring scale, as shown in the figure, to measure the
force that you exert
Friction

Static and Kinetic Friction
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You can then stack additional blocks on the block to increase the
normal force and repeat the measurement

Plotting the data will yield a graph like the one shown here. There is a
direct proportion between the kinetic friction force and the normal
force
Friction

Static and Kinetic Friction
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The different lines correspond to dragging the block along different
surfaces
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Note that the line corresponding to the sandpaper surface has a steeper
slope than the line for the highly polished table

You would expect it to be much harder
to pull the block along sandpaper than
along a polished table, so the slope must
be related to the magnitude of the
resulting frictional force
Friction

Static and Kinetic Friction

The slope of this line, designated μk, is called the coefficient of kinetic
friction between the two surfaces and relates the frictional force to the
normal force, as shown below
Kinetic Friction Force
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The kinetic friction force is equal to the product of the coefficient of
the kinetic friction and the normal force
Friction

Static and Kinetic Friction

The maximum static friction force is related to the normal force in a
similar way as the kinetic friction force

The static friction force acts in response to a force trying to cause a
stationary object to start moving. If there is no such force acting on an
object, the static friction force is zero

If there is a force trying to cause motion, the static friction force will
increase up to a maximum value before it is overcome and motion
starts
Friction

Static and Kinetic Friction

The static friction force is less than or equal to the product of the
coefficient of the static friction and the normal force
Static Friction Force

In the equation for the maximum static friction force, μs is the
coefficient of static friction between the two surfaces, and μsFN is the
maximum static friction force that must be overcome before motion
can begin
Friction

Static and Kinetic Friction

Note that the equations for the kinetic and maximum static friction
forces involve only the magnitudes of the forces

The forces themselves, Ff and FN, are at right angles to each other. The
table here shows coefficients of friction between various surfaces

Although all the listed coefficients are less than 1.0, this does not mean that they must
always be less than 1.0
Friction

Static and Kinetic Friction
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You push a 25.0 kg wooden box across a wooden floor at a constant
speed of 1.0 m/s. How much force do you exert on the box (μk = 0.20)
?
Friction

Static and Kinetic Friction
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A force of 110 N is required to move a table of 120 kg. What is the
coefficient of static friction (μs) between the table and the floor?

A worker has to move a 17.0 kg crate along a flat floor in a warehouse.
The coefficient of kinetic friction between the crate and the floor is
0.214. The worker pulls horizontally on a rope attached to the crate,
with a 49.0 N force. What is the resultant acceleration of the crate?
Force in Two Dimensions

Equilibrium Revisited

When net force on an object is zero, the object is in equilibrium

According to Newton’s laws, the object will not accelerate with no net
force acting on it
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Object in equilibrium is motionless or moves with constant velocity
Force in Two Dimensions

Equilibrium Revisited

Equilibrium can occur no matter how many forces act on an object

If resultant zero, net force zero and object in equilibrium
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Figure shows three forces exerted on a point object. What is the net
force acting on the object?

Vectors may be moved if direction
(angle) or length is not changed
Force in Two Dimensions

Equilibrium Revisited

Figure shows addition of three forces, A, B, and C

Vectors form closed triangle

No net force; thus, sum of the forces is zero and object in equilibrium
Force in Two Dimensions

Equilibrium Revisited

Suppose two forces are exerted on object with sum not zero

How could you find third force that adds to other two totals zero, and
causes object to be in equilibrium?
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First find sum of two forces already exerted on object
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Single force that produces same effect as two summed individual
forces, is resultant force

Force to find has same magnitude as resultant force, but in opposite
direction

Equilibrant – Force that puts object in equilibrium
Force in Two Dimensions

Equilibrium Revisited

The figure below illustrates the procedure for finding the equilibrant
for two vectors
Force in Two Dimensions

Motion Along an Inclined Plane
Force in Two Dimensions

Motion Along an Inclined Plane

Because an object’s acceleration is usually parallel to the slope, one
axis, usually the x-axis, should be in that direction

The y-axis is perpendicular to the x-axis and perpendicular to the
surface of the slope

With this coordinate system, there are two forces—normal and
frictional forces. These forces are in the direction of the coordinate
axes. However, the weight is not
Force in Two Dimensions

Motion Along an Inclined Plane

This means that when an object is placed on an inclined plane, the
magnitude of the normal force between the object and the plane will
usually not be equal to the object’s weight
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You will need to apply Newton’s laws once in the x-direction and once
in the y-direction

Because the weight does not point in either of these directions, you
will need to break this vector into its x- and y-components before you
can sum your forces in these two directions
Force in Two Dimensions

Motion Along an Inclined Plane

A child shoves a small toboggan weighing 100.0 N up a snowy hill,
giving the toboggan an initial speed of 6.0 m/s. If the hill is inclined at
an angle of 32.0° above the horizontal, how far along the hill will the
toboggan slide? Assume that the coefficient of sliding friction between
the toboggan and the snow is 0.15.