Mechanics

Motion

Distance: How far something travels

Displacement: The change in position of an object is known as displacement

Relative velocity: 

Relative velocity

An object's velocity is relative to another object.

Relative Velocity can be found by adding the vectors:

Velocity vectors of the train with respect to Earth, person with respect to the train, and person with respect to Earth. V sub T E is the velocity vector of the train with respect to Earth. It has value 10 meters per second and is represented as a long green arrow pointing to the right. V sub P T is the velocity vector of the person with respect to the train. It has value -2 meters per second and is represented as a short green arrow pointing to the left. V sub P E is the velocity vector of the person with respect to Earth. It has value 8 meters per second and is represented as a medium length green arrow pointing to the right.

Practise Question

Samantha walks along a horizontal path in the direction shown. The curved part of the path is a semi-circle.



The magnitude of her displacement from point P to point Q is approximately
A. 2 m.
B. 4 m.
C. 6 m.
D. 8 m.

Answer: D

Graphing motion

Image result for motion graphs acceleration velocity

Quantities represented by gradient and area of a graph

The Gradient Velocity-time: Acceleration

The area under Velocity-time: Displacement

The area under displacement-time: Velocity

Equations of uniformly accelerated motion

The equations for uniformly accelerated motion are also known as the kinematic equations. They are listed here:

Displacement with acceleration: S= u·t + at22 

Velocity equation: V= u+a·t

Timeless: V2= U2 + 2as 

Displacement with velocity: s=(u+v)t 2

You do not need to know these as they are in your formula booklet.

Symbol Definition
s = ... m Displacement
u = ... ms-1 Initial Velocity
V = ... ms-1 Final Velocity
a = ... ms-2 Acceleration
t = ... s Time

During exams use suvat, and make sure to fill out with the known variables, and figure out which equation uses all the known variables.

Practise Question

A ball is thrown up with a speed of 20ms-1. How long does it take to fall back down to the same starting point?

A: 2.04s

B: 5.09s

C: 4.08s

D: 6.12s

Answer: C

 

Forces

Forces are vectors as they have a magnitude and a direction

Types of forces

Force

Description

Properties

Symbol

Applied Force

Force caused when an object comes in contact with another object

Either a pull or a push

Fa

Weight

Force due to gravity.

mass *acceleration due to gravity

W

Friction

Force opposing motion and and

parallel to the contact surface

Ff

Normal force

Reaction force to weight

Equal To weight on a flat surface and perpendicular to surface.

R

Tension

Pull force caused by a rope being tight

Always a pull

T

Spring

Force caused by an elastic object, such as spring

Fe

Drag

Friction in a medium

Increases with speed.

Fd

Buoyancy 

Force caused by differences in density.

Fb



Free body diagrams

To draw a free body diagram draw a dot, and then draw vector arrows coming out from the dot to represent the forces.

There are three things you must do:

Newton's First Law

An object will remain at rest if there is no net force acting on it

If ΣF = 0, then v = const

ΣF = 0 is the condition for translational equilibrium

Inertia - matters tendency to not change its state of motion (or a state of rest)

Newton's Second Law

If the net forces are more than zero, the velocity will change. 

The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

Net force = mass times acceleration

ΣF = M*A

Momentum version

ΣF = △p/△t

Newton's Third Law

For every action force, there is an equal and opposite reaction force.

FA = -FB

 

Friction

Friction at the atomic level

At an atomic level, an object's surface is never perfectly smooth and is always made of small “peaks”. When objects come in contact, in reality only these peaks are touching and where this occurs weak bonds form, connecting the objects. When an object is pushed the forces are broken between the atoms. Breaking these bonds requires energy and this is what friction is.

Dynamic vs Static phase

When an object is not moving it is in the static phase and the bonds must be broken, for the object to move. However, after the initial bonds are broken an object can move fast enough that the bonds don’t have time to reform. This is the dynamic phase and since not all bonds have to be broken the friction force is lower.

Coefficient of friction

Friction force depends on two variables the normal force and the coefficient of friction.

The coefficient of friction is simply the ratio between the normal force and the friction force.

μ = friction coefficient (it has no units)

μ =Ff/R

There are a friction coefficient μs when it is static and a different one μd When it is dynamic

Therefore, friction depends on how hard an object is pushing on a surface and how slippery the materials are.

Dynamic vs Static Friction

Friction type

When?

Relation with applied force.

Equation

Symbol

Static

The object is in motion

Always equal to the applied force

Ff ≤ μs*R

Fs

Dynamic

Object is still

A constant force which does not change with applied force.

Ff = μd*R

Fd

Why is the friction force lower on an incline?

If an object is on a slope the weight force acts partly parallel to the surface and partly perpendicular to the surface. Weight force can be split into these two components FII and F⊥

As the normal force is equal to the force being exerted perpendicular to the surface. R = F⊥ . Therefore the normal force is decreased on a slope. The acceleration due to gravity is the parallel component FII and this is why an object will slide due to gravity.

Therefore as friction depends on the normal force and this is lower on slopes, things will have a lower friction force when on an incline

 

Energy

Law of conservation of energy: Energy cannot be made or destroyed; it can only be turned from one form into another.

There are two types of energy: potential and kinetic.

Potential energy - stored energy

Elastic potential equation = ½ * spring constant(k) * x2

Gravitational potential = Mass*Gravity*Height =m*g*h or Weight*Height = w*h

Kinetic Energy - the Energy of Motion

Kinetic energy equation = ½ *m*v2

Efficiency

In theory, things should transfer 100% of their energy from one form to another but in the real world, some energy is lost to things like heat, sound or light. The efficiency of an energy transformation is the percentage of energy in the output compared to the input.

Work

In Physics, we define work(W) as:

The unit for work is newton meters (Nm) or Jules(j)

Note: Work is the same as energy.

 

Momentum

Momentum = mass * Velocity

P = m*V

Proving Newton's second law momentum equation

Firstly, mass is substituted for momentum over acceleration in Newton's second law.

Fnet=m·a m =∆p∆vFnet=∆p∆v·a

Then, velocity and acceleration are replaced with their distance & time equivalents. This shows why acceleration mostly cancels out velocity, leaving just time.

Fnet=∆p∆ds-1*∆ds-2

Fnet=∆p∆t

Conservation momentum

In a closed system, if the net force remains the same before and after so will the momentum.

This can be easily proved as if you set net force to zero in the momentum equation you know either delta momentum or delta-time must be zero for both sides to be equal. However, as the change in time cannot be zero, so the change in momentum must be zero.

0 = △p/△t

0 = 0/△t

Momentum before = Momentum after

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