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Simplifying Problems

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Simplifying Problems

Free-Body Diagrams

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

Free-Body Diagrams

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

Ideal Strings

- ideal strings...
- have no mass; therefore, do not affect acceleration
- do not stretch

Ideal Strings

- ideal strings...
- exert only pulling forces—you can’t push on a string!
- exert forces only in line with the string

Ideal Strings

- hold objects at fixed distances
- all objects connected by the string are pulled with the same speed and acceleration

Connected Objects

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 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 Objects

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

Connected Objects

Select a coordinate system for each object.

It is not necessary for all objects to use the same coordinate system.

Example 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

Ideal Pulleys

- used to change the direction of tension in a string
- has the following characteristics:

Ideal 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 Pulleys

- Its axle is frictionless.
- The motion of the string around the pulley is frictionless.

Ideal Pulleys

- It changes the direction of the tension in the string without diminishing its magnitude.

Example 8-2

The 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 8-3

The free-body diagrams are drawn first.

Be especially careful with the components this time!

Are the pulleys moving in the direction you calculated?

Inclines

- 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 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 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 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 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 Force

But what if the object and scale are accelerating??

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

Normal Force

But what if the object and scale are accelerating??

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

Normal Force

But what if the object and scale are accelerating??

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

Friction

What is Friction?

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

What is Friction?

- explained by Newton’s 3rd Law
- necessary for forward motion
- necessary for rolling and spinning objects

Traction

- friction that makes walking, rolling, and similar motions possible
- notation: ft
- also describes friction that prevents unwanted motion

Friction

- opposes motion
- rougher surfaces tend to have more friction
- very smooth surfaces have increased friction

Friction

- What affects its magnitude?
- mass
- area of surface contact does not affect it
- greater on level surfaces than slopes

Friction

- Friction is proportional to the mass and to the normal force on the object
- f = μN
- μ is called the coefficient of friction

Friction

- μ is unique for each particular pair of surfaces in contact
- μ is also dependent on the object’s state of motion

Kinetic Friction

- 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 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 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 Friction

- Properties of the kinetic frictional force (fk = μkN):
- is generally independent of the relative speed of the sliding surfaces

Kinetic Friction

- Properties of the kinetic frictional force (fk = μkN):
- is generally independent of the surface area of contact between the surfaces

Kinetic Friction

- Properties of the kinetic frictional force (fk = μkN):
- is directly proportional to the normal force acting on the sliding object

Static Friction

- friction between stationary objects
- friction will prevent objects from sliding until the force parallel to the surface exceeds the static friction

Static 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 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 Friction

- Properties of static friction:
- can be any value between zero and a maximum value characteristic for the materials in contact

Static Friction

- Properties of static friction:
- is oriented parallel to the contact surface
- opposes the motion of the system of interest

Static Friction

- Properties of static friction:
- depends on the kinds of materials and condition of the contact surfaces
- is normally independent of contact surface area

More Applications of Friction

Rolling 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 Friction

- notation: fr
- magnitudes: Fprop = Fapp – fr
- forces: Fprop = Fapp + fr

Inclined-Plane Dynamics

- 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

- 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

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

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

- 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.