Complete Laws of Motion Summary

Newton's Laws, Friction, Centripetal Force, and Applications

Newton's Laws
Friction
Angle of Repose
Centripetal Force

1. Newton's Laws of Motion

Newton's First Law (Law of Inertia)

When there is no net force on an object:

  • An object at rest remains at rest, and
  • An object in motion continues to move with a velocity that is constant in magnitude and direction.
Newton's Second Law

Newton's second law states the relation between the net force and the inertial mass.

\[\sum \vec{F} = m\vec{a}\]

Note: The direction of acceleration is in the direction of the net force.

In terms of components:

\[\sum F_x = ma_x\] \[\sum F_y = ma_y\] \[\sum F_z = ma_z\]
Newton's Third Law

If the object exerts a force \( F \) on a second, then the second object exerts an equal but oppositely force \(-F\) on the first.

Forces exist in pairs.

Example (a): The force exerted by earth on the sun is equal and opposite to the force exerted by the sun on the earth.

\[\vec{F}_{earth} = -\vec{F}_{sun}\]

Example (b): The force exerted by the man on the ground is equal and opposite to the force acting on the man by the ground.

\[\vec{F}_{man} = \vec{F}_{ground}\]

2. Friction

Whenever the surface of a body slides over that of another, each body exerts a force of friction on the other, parallel to the surfaces. The force of friction on each body is in a direction opposite to its motion relative to the other body.

It is a self-adjusting force, it can adjust its magnitude to any value between zero and the limiting (maximum) value i.e.

\[0 \leq f \leq f_{max}\]

2.1 Types of Friction

1. Static Friction (\(f_s\))

The static friction between two contact surfaces is given by:

\[f_s \leq \mu_s N\]

Where:

  • \(N\) is the normal force between the contact surfaces
  • \(\mu_s\) is a constant called the coefficient of Static friction
2. Kinetic Friction (\(f_k\))

It acts on the two contact surfaces only when there is relative slip or relative motion between two contact surfaces.

\[f_k = \mu_k N\]

Where:

  • \(N\) is the normal force between the contact surfaces
  • \(\mu_k\) is a constant called the coefficient of kinetic friction

2.2 Laws of Friction

The limiting (or maximum) force of friction is proportional to the normal force that keeps the two surfaces in contact with each other, and is independent of the area of contact between the two surfaces.

\[f_{max} = \mu N\]

2.3 Properties of Friction

  1. If the body is at rest, then the static friction force \(f_s\) is parallel to the surface and the external force \(F\), are equal in magnitude and \(f_s\) is directed opposite to \(F\). So, if external force \(F\) increases then \(f_s\) increases.
  2. The maximum value of static friction is given by:
    \[f_{s(max)} = \mu_s N\]
    Where, \(\mu_s\) = coefficient of static friction and \(N\) is the magnitude of the normal response. If the external force is greater than \(f_{s(max)}\), the body slides on the surface.
  3. If the body starts moving along the surface, the magnitude of the friction force decreases to a constant value \(f_k\):
    \[f_k = \mu_k N\]
    Where, \(\mu_k\) is the coefficient of kinetic friction.

3. Angle of Repose

Suppose a body is placed on an inclined surface whose angle of inclination \(\theta\) varies between 0 to \(\pi/2\). The coefficient of friction between the body and the surface is \(\mu_s\). Then at a particular value of \(\theta = \phi\) the block just starts to move. This value of \(\theta = \phi\) is called the angle of repose.

3.1 Mathematical Derivation

If the block is just about to move, then:

\[mg \sin \theta = f\]

When \(\theta = \phi\), \(mg \sin \phi = f_{max}\)

or \(mg \sin \phi = \mu_s N = \mu_s mg \cos \phi\)

or \(\tan \phi = \mu_s\)

\[\phi = \tan^{-1} \mu_s\]

Definition:

The angle of friction 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.

4. Centripetal Force and Circular Motion

4.1 Centripetal Acceleration

A particle moving in a circular path with speed \(v\) has a centripetal (or radial) acceleration:

\[a_r = \frac{v^2}{r} = \omega^2 r\]

4.2 Tangential Acceleration

If there is angular acceleration, the speed of the particle changes and thus we can find the tangential acceleration:

\[a_t = \frac{dv}{dt} = \left( \frac{d\omega}{dt} \right) r = \alpha r\]

4.3 Net Acceleration

The net acceleration is the vector sum of radial and tangential components:

\[\vec{a} = \vec{a}_r + \vec{a}_t\]

The magnitude of acceleration is given by:

\[a = \sqrt{a_r^2 + a_t^2}\]

4.4 Centripetal Force

According to Newton's second law, the centripetal force required to keep an object moving in a circular path is:

\[F_c = ma_r = m\frac{v^2}{r} = m\omega^2 r\]

Key Points:

  • Centripetal force is always directed toward the center of the circular path
  • It is not a separate force but the net force component in the radial direction
  • Common sources of centripetal force include tension, gravity, friction, or normal force