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Derive a relation between the angle of friction and angle of repose ?

Derive a relation between the angle of friction and angle of repose? 

Grade:10

3 Answers

Aarti Gupta
askIITians Faculty 300 Points
9 years ago
213-1012_MASS.JPG
Suppose angle of friction = alpha and angle of repose = theta
Let us suppose a body is placed on an inclined plane as in the above figure.
Various forces involved are:-
1. weight,mg of the body,acting vertically downwards.
2.normal reaction,R,acting perpendicular to inclined plane.
3.Force of friction,F, acting up the plane.
Now, mg can be resolved in two components:-
mgcos theta opposite to R
and mgsin theta opposite to F
In equilibrium,
F = mgsin theta -------- eq.1
R = mgcos theta ---------eq.2
On dividing eq.1 by eq.2 we get,
F/R = mgsin theta/mgcos theta
mu = tan theta
Where mu = coefficient of limiting friction.This is equal to the tangent of the angle of repose between two surfaces in contact.
Now, since mu = tan alpha
i.e coefficient of limiting friction between any two surfaces in contact is equal to tangent of the angle of friction between them.
Thus,
Theta = alpha
i.e angle of friction is equal to angle of repose.

Kushagra Madhukar
askIITians Faculty 628 Points
3 years ago
Dear student,
Please find the solution to your question below.
213-1012_MASS.JPG
Suppose angle of friction = alpha and angle of repose = thetaLet us suppose a body is placed on an inclined plane as in the above figure.Various forces involved are:-1. weight,mg of the body,acting vertically downwards.2.normal reaction,R,acting perpendicular to inclined plane.3.Force of friction,F, acting up the plane.Now, mg can be resolved in two components:- mgcos theta opposite to Rand mgsin theta opposite to FIn equilibrium,F = mgsin theta -------- eq.1R = mgcos theta ---------eq.2On dividing eq.1 by eq.2 we get,F/R = mgsin theta/mgcos thetamu = tan thetaWhere mu = coefficient of limiting friction.This is equal to the tangent of the angle of repose between two surfaces in contact.Now, since mu = tan alphai.e coefficient of limiting friction between any two surfaces in contact is equal to tangent of the angle of friction between them.Thus, Theta = alphai.e angle of friction is equal to angle of repose.
 
Thanks for asking the question.
Regards,
Kushagra
RISHIKA
550 Points
3 years ago
DEAR FRIEND
Relation between Angle of Friction and Angle of Repose
Thus, angle of friction is equal to the angle of repose.
 
Where mu = coefficient of limiting friction. This is equal to the tangent of the angle of repose between two surfaces in contact. i.e coefficient of limiting friction between any two surfaces in contact is equal to tangent of the angle of friction between them. i.e angle of friction is equal to angle of repose.
The major difference is that angle of friction is defined for rigid bodies while angle of repose is defined for granular particles like sand (Angle of repose).
BE HAPPY
THANK U

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