Simplifying Problems

Simplifying Problems

• used to isolate a system of interest and to identify and analyze the external forces that act directly upon it

Free-Body DiagramsFree-Body Diagrams

• common forces in free-body diagrams include:• tension forces• gravity (weight)• normal force• friction

Free-Body DiagramsFree-Body Diagrams

• ideal strings... • have no mass; therefore,

do not affect acceleration• do not stretch

Ideal StringsIdeal Strings

• ideal strings... • exert only pulling forces—

you can’t push on a string!

• exert forces only in line with the string

Ideal StringsIdeal Strings

• hold objects at fixed distances

• all objects connected by the string are pulled with the same speed and acceleration

Ideal StringsIdeal Strings

There is no single, uniform way to solve every problem

involving mechanics and connected objects.

Free-body diagrams can be very helpful in the analysis of

these problems.

Connected ObjectsConnected Objects

Drawing a “world diagram” is a good way to start. It should include all

connections and include an arrow showing the direction

of motion (if known).

Connected ObjectsConnected Objects

When drawing individual free-body diagrams for each object, include force vectors showing all forces acting on

the object.

Connected ObjectsConnected Objects

Select a coordinate system for each object.

It is not necessary for all objects to use the same

coordinate system.

Connected ObjectsConnected Objects

Example 8-1Example 8-1• Draw the world diagram.• Draw a free-body diagram

of block 2.• Calculate the acceleration

of the system.• Calculate the tension force

for block 2.

Transmitting Mechanical

Forces

Transmitting Mechanical

Forces

Ideal PulleysIdeal Pulleys• used to change the

direction of tension in a string

• has the following characteristics:

Ideal PulleysIdeal Pulleys• It consists of a grooved

wheel and an axle. It can be mounted to a structure outside the system or attached directly to the system.

Ideal PulleysIdeal Pulleys• Its axle is frictionless.• The motion of the string

around the pulley is frictionless.

Ideal PulleysIdeal Pulleys• It changes the direction of

the tension in the string without diminishing its magnitude.

Example 8-2Example 8-2The free-body diagrams are drawn first.Take special note of the coordinate system used for each block!Check all directions when you have finished.

Example 8-3Example 8-3The free-body diagrams are drawn first.Be especially careful with the components this time!Are the pulleys moving in the direction you calculated?

InclinesInclines• Since coordinate systems

are chosen, it is usually wisest to make the x-axis parallel to the incline.

• Of course, the x-axis and y-axis must be perpendicular.

Normal ForceNormal Force• This is the force exerted by

a surface on the object upon it.

• It is always exerted perpendicular to the surface (hence, “normal”).

• It is notated N.

Normal ForceNormal Force• On a flat surface, the

normal force has a magnitude equal to the object’s weight, but with the opposite direction.

• N = -Fw norm = -Fwy

Normal ForceNormal Force• On an inclined surface, the

normal force has a magnitude smaller than the magnitude of the object’s weight.

• Trigonometry is needed to find N’s components.

Normal ForceNormal Force• If an object is not moving,

the normal force can be used to measure the object’s weight.

• This is simplest with an unaccelerated reference frame.

Normal ForceNormal Force

• If they are accelerating upward, the apparent weight on the scale will be greater than the actual weight (see Ex. 8-6).

But what if the object and scale are accelerating??

Normal ForceNormal Force

• If they are accelerating downward, the apparent weight on the scale will be less than the actual weight (see Ex. 8-7).

But what if the object and scale are accelerating??

Normal ForceNormal Force

• If they are in free fall, the apparent weight on the scale will be zero (see Ex. 8-8).

But what if the object and scale are accelerating??

FrictionFriction

• Definition: the contact force between two surfaces sliding against each other that opposes their relative motion

What is Friction?What is Friction?

• explained by Newton’s 3rd Law

• necessary for forward motion

• necessary for rolling and spinning objects

What is Friction?What is Friction?

• friction that makes walking, rolling, and similar motions possible

• notation: ft

• also describes friction that prevents unwanted motion

TractionTraction

• opposes motion• rougher surfaces tend to

have more friction• very smooth surfaces have

increased friction

FrictionFriction

• What affects its magnitude?• mass• area of surface contact

does not affect it• greater on level surfaces

than slopes

FrictionFriction

• Friction is proportional to the mass and to the normal force on the object

• f = μN• μ is called the coefficient of

friction

FrictionFriction

• μ is unique for each particular pair of surfaces in contact

• μ is also dependent on the object’s state of motion

FrictionFriction

• More force is needed to start an object moving, than to keep it moving

• μk is the coefficient of kinetic friction—the object is already moving

Kinetic FrictionKinetic Friction

• Properties of the kinetic frictional force (fk = μkN):• is oriented parallel to the

contact surface• opposes the motion of

the system of interest

Kinetic FrictionKinetic Friction

• Properties of the kinetic frictional force (fk = μkN):• depends in some ways on

the kinds of materials in contact and the condition of the surfaces

Kinetic FrictionKinetic Friction

• Properties of the kinetic frictional force (fk = μkN):• is generally independent

of the relative speed of the sliding surfaces

Kinetic FrictionKinetic Friction

• Properties of the kinetic frictional force (fk = μkN):• is generally independent

of the surface area of contact between the surfaces

Kinetic FrictionKinetic Friction

• Properties of the kinetic frictional force (fk = μkN):• is directly proportional to

the normal force acting on the sliding object

Kinetic FrictionKinetic Friction

• friction between stationary objects

• friction will prevent objects from sliding until the force parallel to the surface exceeds the static friction

Static FrictionStatic Friction

• 0 ≤ fs ≤ fs max

• If the applied force parallel to the surface is less than fs max, static friction will cancel out applied force. No movement occurs.

Static FrictionStatic Friction

• If F > fs max, the surfaces will begin to slide.

• magnitude for maximum static friction between two materials in contact:

• fs max = μsN

Static FrictionStatic Friction

• Properties of static friction:• can be any value between

zero and a maximum value characteristic for the materials in contact

Static FrictionStatic Friction

• Properties of static friction:• is oriented parallel to the

contact surface• opposes the motion of

the system of interest

Static FrictionStatic Friction

• Properties of static friction:• depends on the kinds of

materials and condition of the contact surfaces

• is normally independent of contact surface area

Static FrictionStatic Friction

More Applications of Friction

More Applications of Friction

Rolling FrictionRolling Friction• Defined: the sum total of all

points of friction that retard the freedom of motion of the wheel, including the friction forces between the wheel and the surface over which it rolls

Rolling FrictionRolling Friction• notation: fr

• magnitudes: Fprop = Fapp – fr • forces: Fprop = Fapp + fr

• Assign coordinate systems to each system element so that the x-axis is aligned to the sliding surface and pointing up the slope. If there are multiple objects, axes should point in the same general direction relative to their motion.

Inclined-Plane Dynamics

Inclined-Plane Dynamics

• Resolve all forces acting on each element of the system into their components relative to the coordinate system for that system element.

Inclined-Plane Dynamics

Inclined-Plane Dynamics

• Determine the maximum static friction possible for the two materials at the angle of incline.

Inclined-Plane Dynamics

Inclined-Plane Dynamics

• Sum the nonfriction forces parallel to the sliding surface for the entire system and compare to the maximum static friction for the system to determine the dynamic state of the system.

Inclined-Plane Dynamics

Inclined-Plane Dynamics

• If the system is accelerating, calculate the kinetic friction.

• Sum the x-component forces, including kinetic friction, to find acceleration according to Newton’s 2nd Law.

Inclined-Plane Dynamics

Inclined-Plane Dynamics