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  Dynamics: Inclines & Machines

A. Forces on INCLINED Inclined Planes:

Review Topics Covered:  
This is a general review of essential topics.
More in-depth tutorials can be found using the quick search tool 

• Free Body Diagrams
• Forces
• Vector Components,
• Gravity, and Weight
• Friction,

Free Body Diagrams    (F.B.D.)
A free body diagram is a simple sketch showing all the forces acting on an object when it is on its own (i.e. removed from its surrounding).  Here are some guidelines for drawing Free Body Diagrams.

• Forces are represented by vectors
• A relative scale should be used to show the relative size of the forces whenever possible.
• Only the forces acting on the object should be included not the forces that the object exerts on other objects.
• All forces (vectors) should be drawn with respect to the center of gravity (or center of mass of the object).  This point is usually located at the geometric center of the object.

Also see FORCES

A force is usually described as a vector quantity with a definite magnitude and a definite direction which can be either a push or a pull on an object.

VECTORS

Also see Components of Vectors

Any vector is space can be defined by its coordinates in terms of its x, y, and z position.

In a two dimensional frame we need only consider its x and y co-ordinates.  Therefore any vector can be thought of as a resultant when its horizontal component in the x direction is added to its vertical component in the y direction.

The  component of a vector V in standard position is Vx  and the y component is Vy.

Where:

Vx = V cosQ       and      Vy = VsinQ

Then we can define V as

v = v_x + v_y

Gravity & Weight

_{The force of gravity as we know id expressed as Fg = mg.  Where g is the acceleration due to gravity close to the surface of the earth is (9.8 m/s2 ).  A "more common" name for force of gravity is Weight.  Weight should not be substituted as a synonym for mass.}

The acceleration of an object down an inclined plane (a) is equal to the value of the acceleration due to gravity (g)diluted by a factor corresponding to the sine of the angle of the incline with respect to the horizontal (sinq).

Friction

Friction is generally defined as the "force that opposes" motion.

When the applied force on an object is equal or less than the force of friction, the object will not move.  If the applied force is slightly bigger than the force of friction, the object will move in the direction of the applied force.  There are two types friction forces  acting on the object.

• Static friction: before the object starts to move

• Kinetic friction: while the object is moving

Static Friction is larger than Kinetic (moving) friction.

If an object is moving at constant speed , it will slow down and eventually come to a stop because the force of friction is constantly acting against it.

In our laboratory experiments we  found that friction depends on several factors:

1. The weight of the object i.e. the force of gravity ( Fg). Recall that Fg = m x g

2. Hence indirectly friction depends on mass 

3. Friction also depends on the type of surface

4. The presence of a lubricant

The general equation for calculating friction is   F_f = mmg
Where m is the coefficient of friction, m is the mass and g is 9.8 m/s²

Sample Problem: The inclined Plane

A skier goes down a smooth 30^o hill (frictionless) for a distance of 10 m to the bottom of the hill where he then continues on a frozen, frictionless pond.  After that, he goes up a hill inclined at 25^o  to the horizontal.  How far up this second hill does he go before stopping if the coefficient of friction on this hill is 0.10?

Analysis and Solution:

Part I - Accelerated Motion

a = (g)(sin30^o) 
    = 9.8 X 0.5 m/s²

^V₁ = 0     ( ^V₂ ) ² =  ( ^V₂ ) ¹ + 2(a)(d)      d = 10 m

 ( ^V₂ )     = 9.9 m/s

Part II - No acceleration

Therefore  Fnet = ma = 0  and V₁  is the same as V₂  from Part 1.

Part III - Friction up an inclined plane

From the Free Body Diagram

F_net = -F_gx - F_f