# Redshift Academy

Wolfram Alpha:

Last modified: January 26, 2018
```Inclined Plane
--------------

Friction
--------

Vertical force = mg
Normal force, FN = mgcosθ
Parallel to slope force = mgsinθ
frictional force = Ff = μFN  where μ is the coefficient of friction.
mass will slide if mgsinθ > Ff with acceleration = gsinθ - Ff/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:

Ff = μmgcosθ

mgsinθ ~ μmgcosθ

μ = tanθ

θ = 30 degrees

With frictionless pulley:

m2g - T = m2a2

T - m1gsinθ - μkm1gcosθ = m1a1

For simplicity consider μk = 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 - μkm1g = m1a1

F2 = m2g - T = m2a2

so,

m2a2 = m2g - T

= m2g - m1a1

For simplicity assume μk = 0

Now a1 = a2 = a.  Therefore,

a = m2g/(m1 + m2)

and

T = m1a = m1m2g/(m1 + m2)

```