# Topic 2: Mechanics

This paragraph contains no physics. I have used ‘kinematics’ and ‘dynamics’ instead of ‘motion’ and ‘forces’ respectively, since they are a little more professional. I have also placed ‘work, energy, power’ after kinematics but before dynamics, since I believe it synergises better with kinematics than with dynamics.

## 2.1 Kinematics (Motion)

Only applicable when there is constant acceleration.

 Scalar Vector Distance Displacement Speed Velocity

Graphical methods

• The acceleration is the gradient of the velocity-time graph
• The velocity is the gradient of the displacement-time graph and the area under the acceleration-time graph.
• The displacement is the area under the velocity-time graph.

Three equations:

• v = u + at
1. From the definition of acceleration
• s = ut + ½ at2
1. From the area under the velocity-time graph
• v2 = u2 + 2as
1. By eliminating time from equations (1) and (2)

u: initial velocity, v: final velocity, a: acceleration, s: displacement, t:time

They key to using these equations are as follows: Write down all the known quantities. Write down the quantity you wish to find. Find which equation relates all the quantities you have written down, without any additional quantities which are not known.

## 2.2 Work, Energy, Power

When a force displaces an object, it does work on the object. Specifically, the work done is F.d or Fdcosθ, where θ is the angle between the force and the displacement. Work done has units of Joules, and is the associated change in energy. For example, if I were to lift an object of weight mg to a height h, the work done W would be mg * h. This is how the formula for the change in potential energy ΔEp = mgh is derived. Most teach it as Ep = mgh, which is a little inaccurate, as we have seen. In questions involving potential energy, we can sometimes choose a point where the potential energy is set to 0 by definition, but that is really not necessary. For example, if you fall one story, you will have a certain kinetic energy – this kinetic energy is the same whether you fell from basement 1 to basement 2, or whether you fell from the 34th floor to the 33rd floor. Caveat: of course, this assumes that the gravitational field g is constant, that is, h is small.

Try using WEP to solve kinematics problems, compare the two methods.

## 2.3 Dynamics (Forces)

Newton’s Laws:

• First Law: An object at rest will remain at rest and an object in motion will remain in motion at the same speed in a straight line unless an external net force acts on it.
• This defines the inertial frame of reference. All of us Newton’s laws are only valid in such a frame.
• Second Law: The rate of change of an object’s momentum is directly proportional to the net external force acting on it.
• In SI units, the constant of proportionality is the object’s mass.
• For solid objects (non-fluids), using the chain rule and setting dm/dt = 0, we get the usual F = ma.
• For fluids, we get F = v dm/dt, where v here refers to the change of velocity. For example, to calculate the force acting on a wall due to a jet of water: F = v dm/dt = (vafter colliding with the wall – vbefore colliding with the wall)*(mass of water hitting the wall every second)
• Third Law: For every action there is equal and opposite reaction
• Action and reaction pairs must satisfy 3 properties. Equal (in magnitude), opposite (in direction), and acting on different bodies.

Equilibrium: No net force, no net moment.

• Finding the magnitude, direction, and position of forces acting on an object in equilibrium
• If there are numerous unknown forces, choose the pivot which is on the line of action of some of these forces, which would remove their contribution to the net moment

Circular motion: Constant speed, changing velocity. Use Newton’s Second Law. (To put in Topic 6 later)

• Special name for the acceleration: centripetal acceleration
• Special name for the net force: centripetal force
• NO centrifugal force please, it’s fictional

## 2.4 Momentum and Impulse

The linear momentum p of a body is the product of its mass and velocity. Like velocity, momentum is a vector. As a result of Newton’s Second and Third Laws, momentum must be conserved. That is, a system with no net external force acting on it must have a constant momentum. Take special note of the words ‘net’ and ‘external’. External forces are fine, as long as they all add up to zero. Internal forces are also alright.

From Newton’s Second Law, we have F = dp/dt. We can integrate both sides with respect to dt to get integral F dt = Δp. That is, the area under the force time graph is equal to the change of momentum. This is like acceleration and velocity: the area under the acceleration-time graph is the change of velocity.

All collisions are taken to be head-on in the syllabus. This means that the velocity of the each particle involved must pass through the centre of the other particle.

Elastic collision: momentum and kinetic energy are conserved. Using the conservation of kinetic energy and conservation of momentum, one can derive the formula: speed of approach = speed of separation.

Inelastic collision: momentum is conserved but kinetic energy is not conserved.

• Perfectly inelastic. After colliding, the two bodies stick together and move as one body
• Partially inelastic. Some kinetic energy is lost after colliding
• Super-elastic. Super-elastic collision. After colliding, kinetic energy increases.