L10. Statics and Friction

Friction

Friction plays a dual role in our lives:

• Impedes motion of objects, causes abrasion

• On the other hand, without it, we could not walk, drive cars, climb ropes or use nails.

• Friction is a contact force that opposes the relative motion of two bodies

• In 1748, Euler made a distinction between static and kinetic friction.

• If an object does not move, then the applied force must be exactly equal (in magnitude) to the force of static friction. (if these are the only 2 forces acting in the direction of motion, of course)

Experimentally, it is found that the limiting frictional force, F_f is proportional to the normal reaction force, N. Therefore: |F_f|= μ|N| where μ is the coefficient of friction.

F_f is perpendicular to the Normal Force

Often taken from Some point – Centre of Mass

You must distinguish between

• the coefficient of sliding (kinetic) friction μ_k

• the coefficient of static friction μ_s.

• Generally μ_k < μ_s.

Friction does work: W = F_f d cosθ

This energy becomes heat and sound and is usually not useful. F_f is an example of a dissipative force.

L11. Circular Motion

We can now extend the ideas of Newton’s Laws and the conservation of energy to motion in a circle.

F irst, define ω(t) = dθ/dt as the angular velocity.

• To derive the expression for the acceleration vector in circular motion, we define polar coordinates

• =cos θ

• = - sinθ

• = r =

Prove

• =

• =

Circular motion

The net force, which causes centripetal acceleration, is called the centripetal force.

The centripetal force is always supplied by some other force or set of forces (e.g. gravitation or tension). It is always directed toward the centre of the circle.

Moment of Inertia

And the total kinetic energy is:

I is called the moment of inertia of the body about the chosen axis of rotation:

Inertia is a measure of the distribution of mass about the axis of rotation

Angular momentum

• Similar to torque, the Angular Momentum of a particle of mass m rotating at a distance r about some axis is defined as:

Linear momentum: p = mv

It can be shown that the angular momentum: L = Iω

Conservation of momentum

1. Linear momentum is conserved if the net external force is zero;

2. Angular momentum is conserved if the net external torque is zero.

It can be shown that the work done by a torque τ that rotates a rigid object by an angle θ: W = τθ

Work-Kinetic Energy Theorem

The net work done by external torques rotating a rigid object about a fixed axis equals the change in kinetic energy of the object:

Linear and Angular motion equations

Periodic motion: Simple Harmonic Motion

SHM occurs when the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point.

Not every periodic motion is SHM.

However most periodic motions approximate to SHM for small displacements. Also assume no friction etc.

Horizontal motion of a spring-mass system

A mass on a frictionless surface is attached to the end of a horizontal spring as shown.

The mass is pulled to one side and then released (or compressed to the other side and then released).

Proof 1: Show that the subsequent motion is SHM.

The equation of motion is:

F = -kx = ma_x (where k is the force constant of the spring). And so it is SHM.