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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
Frictional Force (𝑓):
- a resistance to motion. This force is directed along the surface, opposite the
direction of the intended motion.

Frictional Forces can change in magnitude so that the two forces still balance.
e.g. Push on a crate, it does not move. Push a little harder on the crate, it does not
move. The frictional force opposing this movement will adjust in magnitude to
counter balance the force you are exerting on it. However, there is a maximum
magnitude of the frictional force as can be witnessed when you push hard enough
on the crate to begin moving it.

There is a maximum magnitude of the frictional force that has to be exceeded before
an object will move. After the object starts moving, there is a frictional force that
continues to oppose the objects motion. Therefore, there are two types of frictional
force:
1. Static frictional force, 𝑓𝑠 – the force causing an object to not move.
2. Kinetic frictional force, 𝑓𝑘 – opposes the movement of an object already in
motion.
Frictional Force – the vector sum of many forces acting between the surface atoms of
one body and those of another body.
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
Properties of Friction:
Property 1: If a body does not move, then the static frictional force ⃗⃗⃗
𝑓𝑆 and the
component of the force 𝐹 that is parallel to the surface balance each other. They are
⃗⃗⃗𝑆 is directed opposite that component
equal in magnitude, and the static frictional force 𝑓
of force 𝐹 .
Property 2: The magnitude of the static frictional force ⃗⃗⃗
𝑓𝑆 has a maximum value 𝑓𝑆,𝑚𝑎𝑥
that is given by:
𝑓𝑠 ≤ 𝜇𝑠 𝐹𝑁
𝜇𝑆 = the coefficient of static friction.
𝐹𝑁 = the magnitude of the normal force on the body from the surface.

If the magnitude of the component of force 𝐹 that is parallel to the surface exceeds
𝑓𝑆,𝑚𝑎𝑥 then the body begins to slide along the surface.
Property 3: If the body begins to slide along the surface, the magnitude of the frictional
force rapidly decreases to a value 𝑓𝑘 given by:
𝜇𝑘 = the coefficient of kinetic friction.
𝐹𝑁 = the magnitude of the normal force on the body from the surface.

Thereafter, during the slide, a kinetic frictional force 𝑓𝑘 opposes the motion.
Note: The magnitude of the normal force 𝐹𝑁 appears in the frictional equations as a
measure of how firmly the body presses against the surface.
⃗⃗⃗𝑆 or the kinetic frictional force 𝑓
⃗⃗⃗𝑘 is
Note: The direction of the static frictional force 𝑓
always parallel to the surface and opposed to the attempted sliding, and the normal
force 𝐹𝑁 is perpendicular to the surface.
Note: The coefficients of the static frictional force 𝜇𝑆 and the kinetic frictional force 𝜇𝑘
are dimensionless and must be determined experimentally.
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
EMPHASIS ON THE COEFFICIENT OF SLIDING FRICTION:

µ (the Greek letter mu) is the coefficient of friction; static or kinetic.

The coefficient of friction (µ) is a dimensionless quantity.

The coefficient of friction (µ) values depend on the properties of the two surface in
contact and is used to calculate the force of friction

µ is valid only for the pair of surfaces in contact when the value is measured; any
significant change in either of the surfaces (such as the kind of material, surface
texture, moisture, or lubrication on a surface, etc.) may cause the value of µ to
change.

µ usually is expressed in decimal form, such as 0.85 for rubber on dry concrete (0.60
on wet concrete).
PROBLEM-SOLVING TECHNIQUES:
Most of the time when static friction is involved then the equation becomes:
𝜃 = tan−1 (𝜇𝑆 )
This is related to finding the maximum of a function but is a good formula to remember
if short of time or what you are coming up with doesn’t seem right.
PROBLEM SOLVING TECHNIQUES:
If a problem says that an object (say a block) is on the verge of moving than that means
that the static frictional force must be at its maximum possible value which is 𝑓𝑆 =
𝑓𝑆,𝑚𝑎𝑥 .
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
The Drag Force and Terminal Speed:
Fluid – anything that can flow (gas or liquid and in some cases, cats. They are weird
creatures).
⃗⃗ ) – a force that opposes the relative motion and points in the direction in
Drag Force (𝑫
which the fluid flows relative to the body.
⃗⃗ ) – the force exerted by a fluid (air for instance) on the object moving
Drag Force (𝑫
through the fluid.
Drag force is dependent on the motion of the object, the properties of the object, and the
properties of the fluid that the object is moving through.
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
The Equation for Drag Force is:
C = Drag coefficient
ρ = density
A = the effective cross-sectional area of the body (The area of a cross-section taken
perpendicular to the velocity v
⃗.

From the equation, we can see that as the speed of an object increases, so does the
magnitude of the drag force.

The size and shape of the object also affects the drag force as you can see from the
equation having Area in in it.

The drag force is also affected by the properties of the fluid, such as its viscosity and
temperature, as represented in the equation by density.
NOTE: Drag force is similar to a frictional force for liquids or gases.
⃗ will eventually equal the
NOTE: If a body falls long enough, the drag force ⃗𝑫
⃗⃗ is an upward force that opposes the downward
gravitational force ⃗⃗⃗
𝐹𝑔 . Drag force 𝑫
gravitational force ⃗⃗⃗
𝐹𝑔 on a falling body.
Terminal Speed (𝐯𝐭 ) – when a body falls at a constant speed that means it is no longer
accelerating (𝑎 = 0).
𝐷 − 𝐹𝑔 = 𝑚𝑎
Where D = 𝐹𝑔
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
Uniform Circular Motion:

Centripetal acceleration (𝑎
⃗⃗⃗⃗𝐶 ) – is the rate of change of a tangential velocity.
-
The direction of the centripetal acceleration is always inwards along the radius
vector of the circular motion.
𝐹𝐶 = 𝑚𝑎
⃗⃗⃗⃗𝐶
NOTE: Because the speed v in uniform circular motion is constant, the magnitudes of
the acceleration and the force are also constant. HOWEVER, the directions of the
centripetal acceleration and force are not constant; they vary continuously so as to
always point toward the center of the circle (that is radially inward).
NOTE: A centripetal force can be a frictional force, gravitational force, or any other
force.
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
NOTE: It is good to look at Uniform Circular Motion problems in two separate
components:
-
Radial Components
Vertical Components
NOTE: If a problem is asking for an angle with the vertical concerning uniform circular
motion of if it concerns an angle when talking about uniform circular motion then the
quick solution is often:
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tan θ = v ⁄rg
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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal
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