Kragte 11 e

Forces and

Newtons Laws of Motion

Module 1: Mechanics

Introduction to forces

A force is a pull or a push that an object experiences due to its interaction with other objects.

Definition

Properties Symbol: SI-unit: Vector

𝑭ሬሬԦ Newton (N)


Types of ForcesContact Forces

Non-contact Forces

Forces that are exerted by

objects that make contact

with each other

Forces exerted by objects that do NOT make

contact with each other


Contact Forces Applied Force ():

A force is exerted on an object by someone


Tension ():- A force exerted by a rope or cable when it is pulled.

- The same everywhere in the rope/cable.

Contact ForcesIntroduction to forces

Friction ():- A force between and object and the

surface on which it rests parallel to the surface.

- Always opposes motion

𝒗ሬሬԦ �⃗�


Normal force ():- The force that a surface exerts on the

object that rests on it.- Always perpendicular from the surface

on the object.


Weight ():- The force with which the earth

attracts an object- Always downwards-

where


Magnetic Force ():- The force that magnets exert on

other ferromagnetic objects.- Repulsive or Attractive

Non-contact ForcesIntroduction to forces

Electrostatic Force ():- The force that charged objects

exert on other objects- Repulsive or Attractive

Non-contact ForcesIntroduction to forces

Example 1Identify all the forces present in the following pictures:

On the trolley

a) On the balloon

b) On the basket


Representation of ForcesForce Diagrams

The object itself is represented diagramatically

Forces are shown with arrows where it really works on the object.

Lengths of the arrows shows the relative magnitudes of the forces

Representation of ForcesExample 2

Draw a force diagram that shows all the forces that are exerted on the trolley

𝑭𝒈ሬሬሬሬሬԦ

𝑭𝑵ሬሬሬሬሬԦ 𝑭𝑻ሬሬሬሬԦ

𝒇ሬԦ

𝑭𝑷𝒍𝒂𝒏𝒕𝒆ሬሬሬሬሬሬሬሬሬሬሬሬሬሬԦ

𝑭𝑻𝒙ሬሬሬሬሬሬԦ 𝑭𝑻𝒚ሬሬሬሬሬሬሬԦ

Freebody diagrams FREE OF A BODY Object represented by only a dot All arrows must point from the dot

outwards Lengths of the arrows shows the

relative magnitudes of the forces

Representation of Forces

Example 3Draw a free body diagram that shows all the forces exerted on the trolley𝑭𝑵ሬሬሬሬሬԦ

𝑭𝑻ሬሬሬሬԦ

𝒇ሬԦ 𝑭𝑷𝒍𝒂𝒏𝒕𝒆ሬሬሬሬሬሬሬሬሬሬሬሬሬሬԦ

𝑭𝑻𝒙ሬሬሬሬሬሬԦ

𝑭𝑻𝒚ሬሬሬሬሬሬሬԦ 𝑭𝒈ሬሬሬሬሬԦ

Representation of Forces

Triangle of ForcesForces in equilibrium

All the forces exerted on a certain point cancel each other

The resultant of the forces are zero The object remains in rest or moves

with a constant velocity When all the forces are drawn head

to tail it forms a CLOSED vector diagram.

Example

NO RESULTANT

Triangle of Forces

When three forces that are exerted on the same point

are in equilibrium, their magnitudes and direction can be shown by the three

sides of a triangle.

Triangle of Forces

FrictionTypes of Friction

Static Friction Kinetic Friction

Exerted on

Symbol

Properties

Formula

Objects in rest Moving Objects

Changes as the applied force

changesAlways constant

𝒇𝒌ሬሬሬሬԦ 𝒇𝒔ሬሬሬሬԦ

𝒇𝒌ሬሬሬሬԦ= 𝝁𝒌𝑭𝑵ሬሬሬሬሬԦ 𝒇𝒔 𝒎𝒂𝒙ሬሬሬሬሬሬሬሬሬሬሬሬԦ= 𝝁𝒔𝑭𝑵ሬሬሬሬሬԦ

WrywingTipes wrywing𝒇𝒔ሬሬሬሬԦ 𝑭𝑻ሬሬሬሬԦ

𝒇ሬԦ


𝒇𝒌ሬሬሬሬԦ

𝒗ሬሬԦ= 𝟎 𝒗ሬሬԦ> 𝟎

𝒇𝒔ሬሬሬሬԦ 𝒇𝒌ሬሬሬሬԦ 𝒇𝒔 𝒎𝒂𝒙ሬሬሬሬሬሬሬሬሬሬሬሬԦ

FrictionMaximum Static Friction

The static friction that an object experiences just before it moves. Static friction increases as the force it opposes increases, until it reaches a maximum. If the applied force increases further, the object starts to move. The object now experiences kinetic friction.

Friction

𝒇ሬԦ= 𝝁𝑭𝑵ሬሬሬሬሬԦ

Friction

Static/kinetic Coeficient of Friction

Normal Force

Formula

FrictionProperties

Strongly dependent on surface roughness. Directly proportional to the normal force. Indipendent on the surface area of the surfaces in contact. Kinetic friction is independent of the speed at which the object moves. Only the maximum static friction can be calculated with a formula 𝒇𝒌ሬሬሬሬԦ< 𝒇𝒔 𝒎𝒂𝒙ሬሬሬሬሬሬሬሬሬሬሬሬԦ

FrictionCoefisient of Friction

μ No unit Property of the surfaces in contact Mostly smaller than one The bigger μ, the bigger the friction 𝜇𝑘 < 𝜇𝑠

Net force of two or more forces

The net force of all the forces that are exerted on an object, is the vector sum of all the

forces that are exerted on the object.

Also known as the resultant force The net forces in the x-axis an

y-axis are calculated separately

Inclined forces are separated into perpendicular components.

𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ

𝑭𝑵𝑬𝑻 𝒙ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ= 𝑭𝒙ሬሬሬሬԦ 𝑭𝑵𝑬𝑻 𝒚ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ= 𝑭𝒚ሬሬሬሬԦ

Net force of two or more forces

Net ForceExample 1

A book is pulled over a rough surface with a constant force. The kinetic friction coefficient between the surface and the book is 0,5. All the forces are in equilibrium.a) Draw a free body diagramb) Determine the normal forcec) Calculate the kinetic frictiond) Calculate the applied force

Net forceExample 2

The same book as in the previous example is now being pulled across the table by an 5 N force, that makes an angle of 30° with the horizontal. a) Calculate the kinetic friction.b) How does the value of the friction

compare to the value in the previous exmaple? Explain.

c) Calculate the net force in the x-axis

Forces on an incline plane

𝑭𝒈ሬሬሬሬሬԦ θ

𝒇ሬԦ

𝑭𝑵ሬሬሬሬሬԦ


∥⊥

Forces on an inclined plane

θθ𝑭𝒈ሬሬሬሬሬԦ 𝑭𝒈⊥ሬሬሬሬሬሬሬԦ

𝑭𝒈∥ሬሬሬሬሬሬԦ

𝒔𝒊𝒏𝜽= 𝑭𝒈∥ሬሬሬሬሬሬԦ𝑭𝒈ሬሬሬሬሬԦ

𝑭𝒈∥ሬሬሬሬሬሬԦ= 𝑭𝒈ሬሬሬሬሬԦ𝒔𝒊𝒏𝜽

𝑭𝒈⊥ሬሬሬሬሬሬሬԦ= 𝑭𝒈ሬሬሬሬሬԦ𝒄𝒐𝒔𝜽

𝒄𝒐𝒔𝜽= 𝑭𝒈⊥ሬሬሬሬሬሬሬԦ𝑭𝒈ሬሬሬሬሬԦ

90°

∥⊥

Net ForceExample 3

All the forces exerted on the crate in the diagram are in equilibrium. Answer the following questions:a) Draw a free body diagram.b) Calculate and . c) Calculate the normal force.d) Calculate the kinetic friction.e) Calculate

𝑭𝒈∥ሬሬሬሬሬሬԦ 𝑭𝒈⊥ሬሬሬሬሬሬሬԦ

𝝁𝒌

Net forceExample 3

5 kg

25°

Newton’s Laws of Motion

Eureka!!!

Father of Mechanics, modern Calculus,

Astronomy and Optics

Sir Isaac Newton

1642-1727

Newton IIIAction-Reaction

If object A exerts a force on object B, object B will exert a force on object A that has the same magnitude, but opposite

direction.

Newton IIIAction-Reaction

A B

A:



𝒇ሬԦ

𝑭𝑩𝑨ሬሬሬሬሬሬሬԦ 𝑭𝑻ሬሬሬሬԦ

𝑭𝑵ሬሬሬሬሬԦ 𝒇ሬԦ


𝑭𝑨𝑩ሬሬሬሬሬሬሬԦ

B:


Newton IIIForce pairs

but opposite in direction Is NOT exerted on the same object Do NOT cancel eachother Works SIMULTANEOUSLY

- THEREFORE: For each action there is a reaction

𝑭𝑨𝑩ሬሬሬሬሬሬሬԦ= 𝑭𝑩𝑨ሬሬሬሬሬሬሬԦ

Newton IIIExample

Identify the Newton III force pairs in the following diagram.

A:



𝒇ሬԦ

𝑭𝑩𝑨ሬሬሬሬሬሬሬԦ 𝑭𝑻ሬሬሬሬԦ

𝑭𝑵ሬሬሬሬሬԦ 𝒇ሬԦ


𝑭𝑨𝑩ሬሬሬሬሬሬሬԦ

B:

Newton IIIForce pairs

Newton III Force Pairs

Newton ILaw of Inertia

An object will remain in a state of rest, or move with a constant

velocity in a straight line, unless a net force is exerted on the object.

Newton ILaw of Innertia

In Symbols:𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬԦ= 𝟎 𝑵

if

𝒗ሬሬԦ= 𝟎 𝒎∙𝒔−𝟏 ∴ 𝒂ሬሬԦ= 𝟎 𝒎∙𝒔−𝟐

�⃗�=𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

Newton IInertia

The property of an object that causes it to resist a change in its

state of motion

Newton IInnertia

A property of the object All objects with mass have innertia Bigger mass = More innertia

Newton IIForce and AccelerationIf a net force is exerted on an

object, the object will accelerate in the direction of the force. The

acceleration is directly proportional to the force and inversely

proportional to the mass of the object.

Newton IIForce and Acceleration

In Symbols:𝑭𝑵𝒆𝒕ሬሬሬሬሬሬሬሬԦ= 𝒎𝒂ሬሬԦ

where �⃗� 𝑵𝑬𝑻=𝑵𝒆𝒕𝑭𝒐𝒓𝒄𝒆 (𝑵)𝒎=𝒎𝒂𝒔𝒔 (𝒌𝒈)𝒂=𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏 (𝒎∙ 𝒔−𝟐)

Newton IIForce and Acceleration𝒂ሬሬԦ∝ 𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ

Direct proportionality =

Straight line through origin

𝒂ሬሬԦ

𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ

Newton II

𝒂ሬሬԦ∝ 𝟏𝒎

Inverse proportionality =

Hyperbole

𝒂ሬሬԦ

𝒎

Force and Acceleration

𝒂ሬሬԦ

𝟏𝒎

Newton II

𝒂ሬሬԦ∝ 𝟏𝒎

Straight line

Force and Acceleration

Newton’s LawsSolving of Problems

1) Separate inclined forces into perpendicular components

2) Draw free body diagrams3) Identify the applicable axis.

Handle the x-axis and y-axis separately.

4) Obtain an equation for𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ

𝒗ሬሬԦ= 𝟎 𝒎∙𝒔−𝟏

Newton I Newton II

𝒂ሬሬԦ≠ 𝟎 𝒎∙𝒔−𝟐

5) Decide which law you will use:

𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ= 𝟎 𝐍 𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ= 𝒎𝒂ሬሬԦ


�⃗�=𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

6) Check for friction:μ is given

YES NO

Calculate with:𝑭𝑵𝑬𝑻ሬሬሬሬሬሬሬሬሬሬԦ= 𝑭ሬሬԦ

Calculate 𝑭𝑵ሬሬሬሬሬԦ

Calculate with:𝒇ሬԦ 𝒇ሬԦ= 𝝁𝑭𝑵ሬሬሬሬሬԦ

𝒇ሬԦ


Example 1A Block, mass 5 kg, is being pulled across a horizontal table with a constant force of 50 N. The magnitude of the friction is, 20 N. Calculate:a) The acceleration of the block.b) The friction coefficient between the

block and the table.

Newton’s Laws

A boy pushes a crate of 50 kg over the floor with the aid of a rod that makes an angle of 40° with the horizontal. He exerts a force of 300 N on the rod. The friction coefficient between the crate and the floor is 0,2.

Newton’s Laws

Example 2

a) Calculate the friction between the floor and the crate.

b) Calculate the acceleration of the crate.

300 N

40°

Example 2Newton’s Laws

A person ski’s down a slope that makes an angle of 40° with the horizontal. The total mass of the skier and his ski’s is 50 kg. The friction coefficient between the snow and the ski’s are 0,1.a) Calculate the net force the person

experience parallel to the surface.b) Calculate the acceleration of the skier.


The diagram shows a 3 kg block (B) and a 2 kg block (A) that is being pushed forward by a force of 30 N so that the system accelerates to the right. The applied force makes an angle of 15° with the horizontal. Each block experiences a friction force of 5 N.


15°

a) Calculate the acceleration of the system.

b) Calculate the force that A exerts on B.c) Calculate the force that B exerts on A.

A B


B

6 kg

A4 kg

C6 kg

T1 T2


The friction coefficient between Block B and the table top is 0,034. Assume that the ropes have negligible mass and that the pulleys are frictionless. Calculate:a) The acceleration of the system.b) The tension in the ropes..


A man with a mass of 70 kg stands on a scale in a lift. Calculate the reading on the scale if the lift:a) Is in rest.b) Moves upward with a constant velocity

of 3,2 m·s-1.c) Accelerates upward at 3,2 m·s-2.d) Accelerates downward at 3,2 m·s-2.e) Freefall


Universal Gravitation

Between any two objects with mass there exist a gravitation force that is directly proportional to the product of

their masses and inversely proportional to the square of the

distance between their centre points.

Newton’s Laws

Universal GravitationSymbols

�⃗�𝑮=𝑮𝒎𝟏𝒎𝟐

𝒓𝟐

𝒓𝒎𝟏 𝒎𝟐

Universal GravitationSymbols

= Gravitation force (N) = Mass of objects (kg) = Distance between objects (m) = Universal Gravitation

Constant= 6,67 x 10-11 N·m2·kg-2

�⃗�𝑮=𝑮𝒎𝟏𝒎𝟐

𝒓𝟐

Universal GravitationSymbols: On a Planet

�⃗�𝑮=𝑮𝒎𝑴𝑹𝟐

= Massa van planeet (kg) = Massa van vorrwerp

(kg) = Radius van planeet (m)

Universal GravitationRelationship between G and g

�⃗�𝑮=𝑮𝒎𝑴𝑹𝟐

Attraction force by earth on object:

�⃗�𝒈=𝒎𝒈and

�⃗�𝑮= �⃗�𝒈but

𝑮𝒎𝑴𝑹𝟐 =𝒎𝒈

Universal Gravitation

therefore ÷𝒎

𝒈=𝑮𝑴𝑹𝟐

Relationship between G and g

Amount of matter Symbol: m Unit: kg Scalar The same everywhere

Universal GravitationMass

Universele GravitasieGewig

Force with which planets attract an object

Symbol: Unit: N Vector Function of the mass and

radius of a planet

Universal GravitationExample 1

Two spherical objects m1 and m2, with their centre points a distance r metre apart, exerts a gravitation force of 6 N on each other. Determine the magnitude of the force if:a) The mass m1 doubles.b) The distance between them halves.


Two metal spheres with masses 8 x 104 kg and 2 x 103 kg respectively is placed 340 cm apart. Calculate the gravitation force between them.


An astronaut with a mass of 80 kg on the earth lands on planet X with his spaceship. Planet X has a radius that is half that of the earth ant a mass that is double that of the earth.

a) Calculate the value of g on planet X.b) Calculate the garvitation force that the

man experiences on planet X.

Kragte 11 e

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