#### IIT (NTSE/Olympiad)

# Newton's Laws Of Motion & Friction

__14. Angle of Friction__

__14. Angle of Friction__

**14. Angle of Friction (φ)**

**14.1Mathematical significance :**The angle of friction (f) may be defined as the angle between the normal reaction N and the resultant of friction force f and the normal reaction.

Thus tan φ = f/N

since f = μN . therefore tan φ = μ

**14.2 Physical Significance :**The angle of repose (φ) is that minimum angle of inclination of the inclined plane at which a body placed at rest on the inclined plane is about to slide down in equilibrium condition.

N = mg cos θ and mg sin θ = f ≤ f

_{max}

if θ = φ then f = f

_{max}

∴ mg sin φ = f

_{max}= μN

⇒ mg sin φ = μ mg cos φ

⇒ tan φ = μ

(a) When θ < φ (or tan

^{–1}μ) the body is in equilibrium

(b) When the angle of inclination is more than the angle of friction (θ > φ) the block starts sliding down.

**14.3 Conditions for equilibrium of block**(Depending upon the direction of applied force)

(a)

**Force parallel to the incline**

(b) Force normal to the incline

(c)

**External Horizontal force**

**(a) Force parallel to the incline**

(i) When θ > φ the friction force acting at its maximum value (f

_{max}= μN) is incapable of keeping the block stationary. Therefore, parallel force is required to keep the block in equilibrium

(ii) The minimum value of this force is

(F

_{p})

_{min}= mg sin θ – f

_{max}

(F

_{p})

_{min}= mg (sin θ – μ cos θ)

(iii) If a force slightly greater than (F

_{p})

_{min}is applied, then the block does not start moving up, but the force of friction force gets reduced.

(iv) It becomes to zero when the external force attains a value equal to F

_{p}= mg sin θ. When the force is further increased (Fp> mg sin θ). The block has a tendency to move upward and direction of friction force gets reversed

(v) The block does not start moving up unless the external force attains the maximum value. The maximum value of Fpis given by

(F

_{p})

_{max}= mg sin θ + f

_{max}= mg (sin θ + μ cos θ)

**Conclusion :**

The block remains stationary if (F

_{p})

_{min}≤ F

_{p}≤ (F

_{p})

_{max}

mg(sin θ – μ cos θ) ≤ F

_{p}≥ mg (sin θ + μ cos θ)

**(b) Force Normal to the incline (F**

_{N})(i) Force F

_{N}applied normal to the inclined plane increases the magnitude of the frictional force by increasing the normal reaction.

(ii) Therefore N = mg cos θ + F

_{N}. When F

_{N}has its minimum value (F

_{N}) minthe friction force acting at its maximum value is just capable of preventing the block from sliding down.

(iii) That is

mg sin θ = f

_{max}= μN

= μ [mg cos θ + (F

_{N})

_{min}]

or (F

_{N})

_{min}= m/μ g(sin θ – μ cos θ)

(iv) It is important to note that whatever may be the magnitude of F

_{N}, the block never attains a tendency to slide upward.

(v) When the magnitude of F

_{N}is more than its minimum value (F

_{N})

_{min}then only the magnitude of friction forces decreases.

**Conclusion :**

The block remains stationary if

F

_{N}≥ (F

_{N})

_{min}⇒ F

_{N}≥ mg/μ (sin θ – μ cos θ)

**(c) External Horizontal Force**

It serves two purposes :

(i) It supports the frictional force

(ii) It increases the normal reaction and thus increases the magnitude of the limiting force of friction

**Minimum Horizontal force (F**

_{H})_{min}:-(i) When the horizontal force acts as its minimum value, the friction force acts at its maximum value.

(ii) Applying the conditions of equation parallel and normal to the plane we get.

(F

_{H})

_{min}cos θ = mg sin θ– f

_{max}and (FH)minsin θ + mg cos θ = N and F

_{max}= μN

Solving these

**Maximum Horizontal Force : (F**

_{H})_{max}(iii) As the magnitude of F

_{H}is slightly increased from its minimum value, the block does not start moving up, it remains stationary

(iv) But the magnitude of the friction force starts decreasing and it becomes equal to zero when F

_{H}= mg tan θ.

(v) If F

_{H}is further increased, the block has tendency to move upward and it just starts moving up when F

_{H}attains its maximum value.

(vi) From the free body diagram of the block

(F

_{H})

_{max}cos θ = mg sin θ + f

_{max}and (F

_{H})

_{max}sin θ + mg cos θ = N

since f

_{max}= μN therefore

**Conclusion**

The box remains stationary if

(F

_{H}) min ≤ F

_{H}≤ (F

_{H})

_{max}

≤ F

_{H}≤

#### IIT (Class X)

- Unit, Dimension & Error

- Vectors

- Motion in One Dimension

- PROJECTILE MOTION

- NEWTON'S LAWS OF MOTION & FRICTION
- 1. First Law Of Motion
- 2. Second Law Of Motion
- 3. Third Law Of Motion
- 4. Reference Frames
- 5. Motion in a lift
- 6. Motion of a Block on a Horizontal Smooth Surface
- 7. Motion of bodies in contact
- 8. Motion of connected Bodies
- 9. Motion of a body on a smooth inclined plane
- 10. Motion of two bodies connected by a string
- 11. Friction and frictional force
- 12. Graphical representation of friction
- 13. Types of frictional force and their definition
- 14. Angle of Friction
- 15. Minimum force Required to move a block
- Points to Remember - Newton's Laws Of Motion & Friction

- WORK, POWER, ENERGY & CONSERVATION LAWS

- CIRCULAR MOTION & ROTATIONAL DYNAMICS

- GRAVITATION