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, f s – the force causing an object to not move.

2. Kinetic frictional force, f k – 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.

Properties of Friction:1

Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal

Property 1: If a body does not move, then the static frictional force f⃗ S and the component of the force F⃗ that is parallel to the surface balance each other. They are equal in magnitude, and the static frictional force f⃗ S is directed opposite that component of force F⃗.Property 2: The magnitude of the static frictional force f⃗ S has a maximum value f S , max that is given by:

f s≤μsFN

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 F⃗ that is parallel to the

surface exceeds f S , maxthen 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 f k 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 f k 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.Note: The direction of the static frictional force f⃗ S or the kinetic frictional force f⃗ k is always parallel to the surface and opposed to the attempted sliding, and the normal force F⃗Nis 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. EMPHASIS ON THE COEFFICIENT OF SLIDING FRICTION:

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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal

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 f S=f S ,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).Drag Force (D⃗) – a force that opposes the relative motion and points in the direction in which the fluid flows relative to the body.

Drag Force (D⃗) – the force exerted by a fluid (air for instance) on the object moving 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.

The Equation for Drag Force is:

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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal

C = Drag coefficientρ = density A = the effective cross-sectional area of the body (The area of a cross-section taken perpendicular to the velocity v⃗.

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.

NOTE: If a body falls long enough, the drag force D⃗ will eventually equal the gravitational force F⃗g. Drag force D⃗ is an upward force that opposes the downward gravitational force F⃗g on a falling body.

Terminal Speed (v t) – when a body falls at a constant speed that means it is no longer accelerating (a⃗=0¿ .

D−Fg=ma

Where D = Fg

Uniform Circular Motion:

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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal

Centripetal acceleration (a⃗C¿ – 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.

F⃗C=m a⃗C

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.

NOTE: It is good to look at Uniform Circular Motion problems in two separate components:

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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal

- 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:

tanθ= v2

rg

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Notes 4.2: Forces and Motion II – Frictional, Drag, Centripetal

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