Question:
The co-efficient of friction ($a$) in terms of angle of friction ($f$) is given by:
The co-efficient of friction ($a$) in terms of angle of friction ($f$) is given by:
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The tangent of the angle of friction is always equal to the coefficient of static friction:
\[ \mu = \tan\phi \implies \phi = \tan^{-1}(\mu) \]
This relationship is used to measure the coefficient of friction experimentally on an inclined plane.
\[ \mu = \tan\phi \implies \phi = \tan^{-1}(\mu) \]
This relationship is used to measure the coefficient of friction experimentally on an inclined plane.
Updated On: Jul 23, 2026
- $\phi = \tan\mu$
- $\mu = \sin\phi$
- $\mu = \tan\phi$
- $u = \frac{1}{\tan\phi}$
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
Friction is the resistive force that opposes the relative motion of two surfaces in contact.
- The coefficient of friction (\(\mu\)) is a dimensionless constant representing the ratio of the limiting frictional force (\(F\)) to the normal reaction force (\(R_N\)) acting between the surfaces:
\[ \mu = \frac{F}{R_N} \]
- The angle of friction (\(\phi\)) is the angle that the resultant of the normal reaction force and the limiting frictional force makes with the normal reaction.
Step 2: Detailed Explanation:
Let us analyze the forces acting on a body resting on a horizontal surface at the point of sliding:
- The normal reaction force (\(R_N\)) acts perpendicular to the surface.
- The limiting frictional force (\(F\)) acts parallel to the surface, opposing any applied force.
- The resultant force (\(R\)) of these two perpendicular forces makes an angle \(\phi\) (the angle of friction) with the normal reaction \(R_N\).
- Using vector geometry, we can construct a right-angled triangle where:
- The side adjacent to the angle \(\phi\) is the normal reaction force (\(R_N\)).
- The side opposite to the angle \(\phi\) is the limiting frictional force (\(F\)).
- Taking the tangent of the angle of friction (\(\phi\)):
\[ \tan\phi = \frac{\text{Opposite Side}}{\text{Adjacent Side}} = \frac{F}{R_N} \]
- Since the ratio \(\frac{F}{R_N}\) is defined as the coefficient of friction (\(\mu\)):
\[ \mu = \tan\phi \]
This matches Option (C) perfectly.
Step 3: Final Answer:
The coefficient of friction in terms of the angle of friction is given by \(\mu = \tan\phi\).
Friction is the resistive force that opposes the relative motion of two surfaces in contact.
- The coefficient of friction (\(\mu\)) is a dimensionless constant representing the ratio of the limiting frictional force (\(F\)) to the normal reaction force (\(R_N\)) acting between the surfaces:
\[ \mu = \frac{F}{R_N} \]
- The angle of friction (\(\phi\)) is the angle that the resultant of the normal reaction force and the limiting frictional force makes with the normal reaction.
Step 2: Detailed Explanation:
Let us analyze the forces acting on a body resting on a horizontal surface at the point of sliding:
- The normal reaction force (\(R_N\)) acts perpendicular to the surface.
- The limiting frictional force (\(F\)) acts parallel to the surface, opposing any applied force.
- The resultant force (\(R\)) of these two perpendicular forces makes an angle \(\phi\) (the angle of friction) with the normal reaction \(R_N\).
- Using vector geometry, we can construct a right-angled triangle where:
- The side adjacent to the angle \(\phi\) is the normal reaction force (\(R_N\)).
- The side opposite to the angle \(\phi\) is the limiting frictional force (\(F\)).
- Taking the tangent of the angle of friction (\(\phi\)):
\[ \tan\phi = \frac{\text{Opposite Side}}{\text{Adjacent Side}} = \frac{F}{R_N} \]
- Since the ratio \(\frac{F}{R_N}\) is defined as the coefficient of friction (\(\mu\)):
\[ \mu = \tan\phi \]
This matches Option (C) perfectly.
Step 3: Final Answer:
The coefficient of friction in terms of the angle of friction is given by \(\mu = \tan\phi\).
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