Vectors and Scalars

Vector: A quantity with both magnitude and direction

Scalar: A quantity with only magnitude

Examples

Vectors	Scalars
velocity	Mass
displacement	Time
Weight	Distance
Acceleration	Length

 

Ways of representing a vector

• A plus or minus next to the magnitude
• F = +5N
• An arrow next to the magnitude
• F = 5N ⟶
• A direction next to the magnitude
• F = 5N right
• An angle next to the magnitude (angles always starting from positive x axis like unit circle)
• F = 5N 0°

 

Vector addition and subtraction graphically:

S= A+B

To add two vectors, draw the first vector with its tail at the origin and then draw the second vector with its tail at the previous vectors head. FInally draw a line form the origin to the head of the second vector. This line is the new vector, as shown below.

S= A-B

To subtract a vector, do the same as with addition but flip the vector which you are subtracting.

 

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
Buyant	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:

• Tail of vector must touch the dot
• Arrows must be proportional to the forces. E.g if one force is 5N and the other is 10N, then the 10N arrow must be twice the size of the 5N one.
• Arrows must be labeled with magnitude of force (in newtons) and the type of force

Motion

Quantities of motion

Displacement(s):  Distance from starting point

Velocity(v):  displacement / time

Acceleration(A): change in velocity/time

Relative velocity: an object's velocity relative to another object.

• E.g two objects move towards each other at 5m/s. The velocity of the object on the left relative to the object on the the right is 10m/s.  

 

Equations of motion

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

S= ut + (½)at2 Displacement with acceleration

V= u+at          Velocity equation

V2= U2 + 2as Timeless

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

 

S = displacement

U = initial velocity

V = final velocity

A = acceleration = V/t

T=  time

 

Use suvat, make  table fill it out with the known variables, and figure out which equation uses all the known variables.

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 intermolecular bonds form, connecting the objects. When an object is pushed the intermolecular 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 intermolecular 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 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 is 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	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 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

Newton's laws

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 it’s state of rest)

 

Newton's second Law

If the net forces is 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 accelleration

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

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 - energy of motion

Kinetic energy equation = ½ *m*v2

 

Efficiency

In theory thing should transfer 100% of their energy form one form to another but in the real world some energy is lost to thing slike 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:

• force (F) times the displacement (s) in the direction of the force
• W=F*s*cosθ

 

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

 

Note: Work is the same as energy.

Momentum

Momentum = mass * Velocity

P = m*V

 

Proving the newton's second law momentum equation

Firstly, mass is subsituted for momentum over acceleratiion in newtons second law.

$$\begin{gathered} F_{net}=m·a \\ m =\frac{∆p}{∆v} \\ {F}_{net}=\frac{∆p}{∆v}·a \end{gathered}$$

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

$F_{net}=\frac{∆p}{∆d{s}^{-1}}*∆d{s}^{-2}$

$F_{net}=\frac{∆p}{∆t}$

 

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 change in time cannot be zero, so change in momentum must be zero.

0 =  △p/△t

0 = 0/△t

Momentum before = Momentum after

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