JEE Physics Laws Of Motion Notes Set 06

Contact Force, Normal Reaction and Friction

When two bodies touch, each exerts a contact force on the other; the two forces are equal and opposite. The component perpendicular to the surface is the normal reaction N, and the component along the surface is friction f. The magnitude of the whole contact force is \( R=\sqrt{f^2+N^2} \). Friction arises from interlocking of surface irregularities and from molecular bonding between the surfaces.

Kinetic Friction

Kinetic friction acts only while there is relative sliding between the surfaces. Its direction on a body is opposite to the velocity of that body relative to the surface it touches, and has no connection with the direction of the applied force. Its magnitude is \( f_k=\mu_kN \), where \( \mu_k \) depends on the pair of surfaces. The friction forces on the two bodies are an action-reaction pair, so they are equal in size. Example: a 10 kg block with \( \mu_k=0.5 \) and initial speed 10 m/s stops after 10 m (deceleration 5 m/s²).

Static Friction

Static friction acts when there is a tendency to slide but no sliding. It is a self-adjusting force with \( 0\le f_s\le\mu_sN \), and the limiting value \( \mu_sN \) is reached just before sliding begins. In general \( \mu_s>\mu_k \); if neither is specified, take them as equal. The coefficient is dimensionless and can exceed 1. To find the direction of static friction, draw the free-body diagram (adding a pseudo force if the surface accelerates), find the resultant parallel to the surface of all other forces, and take friction opposite to it.

Rules for Solving Friction Problems

1. First check whether the body can move: compare the force trying to move it with \( \mu_sN \).
2. If it cannot, the friction equals that force and the acceleration is zero.
3. If it moves, use \( \mu_kN \) and Newton's second law.
4. For stacked blocks, find the maximum acceleration that friction can give the upper block (\( \mu g \)) and compare with the acceleration of the system to decide whether slipping occurs.
5. Friction always opposes the relative motion (or tendency of it) between the two surfaces in contact.

Inclined Plane and Angle of Friction

A block on an incline of angle \( \theta \) stays at rest while \( \tan\theta\le\mu_s \); the angle \( \theta=\tan^{-1}\mu_s \) is the angle of friction. Moving up the plane the retardation is \( g(\sin\theta+\mu\cos\theta) \), and sliding down the acceleration is \( g(\sin\theta-\mu\cos\theta) \). Hence the time of ascent is less than the time of descent. The least force that pulls a block of mass m along a horizontal surface is \( \mu mg/\sqrt{1+\mu^2} \), applied at \( \tan\phi=\mu \) above the horizontal.

Worked Example

A 10 kg block has \( \mu_s=0.5 \) and \( \mu_k=0.3 \). A horizontal force of 51 N exceeds the limiting friction 50 N, so the block moves. The kinetic friction is 30 N and \( a=(51-30)/10=2.1 \) m/s². A force of 50 N or less would leave the block at rest.