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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

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Units, Constants and Useful Formulas

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Last modified: January 26, 2018


Inclined Plane
--------------



Friction
--------

Vertical force = mg
Normal force, F_N = mgcosθ
Parallel to slope force = mgsinθ
frictional force = F_f = μF_N  where μ is the coefficient of friction.
mass will slide if mgsinθ > F_f with acceleration = gsinθ - F_f/m

Ex: A box slides downwards at a constant velocity on an inclined surface that has a coefficient 
of friction μ = .58 The angle of the incline, in degrees, is calculated as follows:

F_f = μmgcosθ

mgsinθ ~ μmgcosθ

μ = tanθ

θ = 30 degrees 


With frictionless pulley:



m₂g - T = m₂a₂

T - m₁gsinθ - μ_km₁gcosθ = m₁a₁

For simplicity consider μ_k = 0

Now a₁ = a₂ = a.  Therefore, if we substitute for T we get:

T = m₂g - m₂a

Therefore,

m₂g - m₂a - m₁gsinθ = m₁a

Thus,

a = (m₂g - m₁gsinθ)/(m₁ + m₂)

Note:  if θ = 0 the equation reduces to:

a = m₂g/(m₁ + m₂)

Which the same as the table case.

      T
m₁ --->-------
------------ O 
  //////////|   |
                ^ T
                |
                m₁


F₁ = T = m₁a₁ - μ_km₁g = m₁a₁

F₂ = m₂g - T = m₂a₂

so,

m₂a₂ = m₂g - T

     = m₂g - m₁a₁

For simplicity assume μ_k = 0

Now a₁ = a₂ = a.  Therefore,

a = m₂g/(m₁ + m₂)

and 

T = m₁a = m₁m₂g/(m₁ + m₂)