Equilibrium, Incline, Pullies and Trains

Equilibrium

Assumptions: 1. set positive axises

             2. no acceleration in both x and y direction becuase of the

                object is not moving

             3. no friction because friction can result in inequality.

                                

a = 0

                                     F = ma

Fx = ma                                                    Fy = ma              T1x - T2x = 0                                             -Fg + T1y + T2y = 0

T1x = T2x                                                  T1y + T2y = Fg

                                                           T1sinA + T2sinB = mg

                                                        (T1sinA + T2sinB) / g = m

Inclines - 1

Static - object have not started moving. No acceleration at all.

Assumptions:

1. fs = MFn

2. no acceleration. a = 0

3. set positive axises along the decline
4. no air resistance

No movement, acceleration is zero

Ms = ?

                               F = ma                      
Fy = ma                                              Fx = ma
Fy = 0                                               Fgx - f = 0
Fn - Fgy = 0                                         Fgsinθ- MFn = 0
Fn = Fgy                                             Fgsinθ = MFgy
Fn = Fgcosθ                                         Fgsinθ = MFgcosθ

                                                     Fgsinθ/Fgcosθ = M

                                                     tanθ = M 

Inclines - 2

Kinetic -  object is moving on x-axis thus there is acceleration.

Assumptions:

1. fk = MkFn

2. ax does not equal to zero, ay equal to zero. No movement on y-axis
3.  set positive axises along the direction of movement.

4.  no air resistance

a = ?
                               F = ma
Fy = ma                                               Fx = ma
Fy = 0                                                Fyx - fk = ma
Fn - Fgy = 0                                          Fyx - MkFn = ma
Fn = Fgy                                              Fgsinθ - Mkmgcosθ = ma
Fn = Fgcosθ                                          Fgsinθ - Mkmgcosθ/m = a
Fn = mgcosθ                                                                    

Pulley

1. two free body diagram with different positive axises.

2. Tension 1 = Tension 2

3. no horizontal force in pulley questions, however, if one mass is on a 

   horizontal surface

Assumptions:

1. no friction on rope

2. no air resistance

3. two free body diagrams

4. positive axises set along the movement

5. T1 = T2

6. acceleration is the same for either mass.

mass 1

                               F = ma

Fy = ma                                                   Fx = ma

Fg - T = ma                                             Fx = 0

m1g - T = m1a

T = m1g - m1a (1)

mass 2

                               F = ma

Fy = m2a                                                  Fx = ma

T - m2g = m2a                                         Fx = 0

T = m2g + m2a (2)

Find a

put (1) and (2) together

m1g - m1a = m2g +m2a

m1g-m2g = m1a + m2a

collect like terms:

m1g-m2g = a(m1+m2)

m1g - m2g / m1+m2 = a

Find T

use either equation (1) or (2)

(1): T = m1g - m1a

TRAINS

1. Three free body diagrams, each for different cart.

2.  no vertical movement, thus Fn = Fg

3.  only the leading cart that has the applied force. other carts' 

     force that cause movement is the tension force.

Assumptions:

1.  3 free body diagramss for T1 and T2

2.  acceleration is the same

3.  no air resistance
4.  set posivetion axises along the movement.

                               

mass 1

                                                      F = ma

Fy = m1a                                                                               Fx= m1a
Fn - m1g = 0                                                                         Fa - T1 - f = m1a
Fn = m1g                                                                               Fa - T1 - Mm1g = m1a

                                                                                                Fa - Mm1g - m1a = T1

mass 2

                                                     F = ma

Fy = ma                                                                                      Fx = ma

Fn = Fg                                                                                      T1 - f - T2 = m2a

Fn = m2g                                                                                    Fa - Mm1g - m1a - f -T2 = m2a

mass 3

                                                      F = ma

Fy = m3a                                                                                     Fx = m3a

Fy = 0                                                                                          T2 - f = m3a

Fn - m3g = 0                                                                               T2 = m3a + f

Fn = m3g                                                                                     T2 = m3a + Mm3g