Module 5: Advanced Mechanics

Module 5: Advanced Mechanics 1

Module 5: Advanced MechanicsProjectile Motion

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analyse the motion of projectiles by resolving the motion into horizontal and vertical components, making the following assumptions: – a constant vertical acceleration due to gravity ✅ – zero air resistance �

Projectile Motion

A projectile motion is a two dimensional motion in which an object called projectile is launched and traces a trajectory which is determined by the projectile's initial velocity and the forces acting on it.

A projectile's typical trajectory follows a parabolic path. The projectile's horizontal and vertical components of the motion can be broken up, since they are independent of each other, to better understand a projectiles behaviour.

Vertical Component Of Motion

The only force acting on a projectile during flight is the force caused by gravity that acts downwards. Therefore, the vertical component of the projectile's acceleration has a constant acceleration of 9.8 downwards. m/s2


Horizontal Component Of Motion

Since the gravitation force, the only force acting on a projectile, acts only vertically: there is no horizontal force. Therefore, the horizontal component of the projectile's acceleration has an acceleration of 0 and a constant velocity.

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apply the modelling of projectile motion to quantitatively derive the relationships between the following variables: – initial velocity ✅ – launch angle � – maximum height � – time of flight � – final velocity � – launch height � – horizontal range of the projectile �

Initial Velocity

Initital velocity is the velocity at which a projectile is initially lauched at. We can separate the initial velocity vector into its horizontal and vertical component using trigonometry. Launch angle is the angle at which the projectile is launched.

Horizontal component of the initial velocity is:

m/s2

u =x u cosθ


Vertical component of the initial velocity is:

Maximum Height

Launch height is the height at which the projectile is launched. Launch height can be found when t = 0. Launch height is calculated using:

Maximum height is the maximum distance a projectile is from the ground beneath. Maximum height can be found when = 0. Maximum height is calculated using:

where g = 9.8

Time Of Flight

The time of flight is the total time the projectile takes to complete its trajectory. The time of flight can be found when = 0. The time of flight is calculated using:

where g = 9.8

Final Velocity

Final velocity is the velocity at which a projectile finishes is trajectory. Final velocity can be calculated by combinings its horizontal and vertical component.

The horizontal component of final velocity is:

The vertical component of final velocity is:

Range

u =y u sin θ

Vy

S =y 2gu sin θ2 2

ms−2

Sy

t =g

2u sin θ

ms−2

v =x u =x u cosθ

v =y u sin θ + gt


Range is the total horizontal distance a projectile covers. Range can be found when . Range is calculated using:

Circular Motion

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conduct investigations to explain and evaluate, for objects executing uniform circular motion, the relationships that exist between: – centripetal force ✅ – mass � – speed � – radius �

Uniform Circular Motion

Uniform circular motion involves objects travelling along a circular path at a constant speed. Even though the speed is constant, velocity is constantly changing since direction is changing. Therefore, the object is undergoing acceleration.

Period & Frequency

The period, T, is the time required to make one revolution aroudn the circle whereas the number of rotations each second is referred to as the frequency, f.

where:

f = frequency (Hz)

T = period (s)

S =y 0

S =xg

u sin 2θ2

f = and T =T

1f

1


Tangential Velocity

The tangential velocity (v) of an object in circular motion is the velcoity of the object. The velocity of any object in circular motion is tangential (at a tangent) to its path.

where:

v = tangential velocity (m/s)

r = radius of circular path (m)

T = period (s)

Angular Velocity

The angular velocity (ω) of an object in circular motion is the rate of rotation of the object - how quickly an object is rotating around the circle.

where:

ω = angular velocity (radians / s)

= angle (radians)

t = time interval (s)

v =T

2πr

ω =t

Δθ

θ


Centripetal Acceleration

Centripetal acceleration is the acceleration an object experiences undergoing circular motion. Since objects continually deviate inwards from their straight line direction, centripetal acceleration is always directed towards the centre of the circle.

where:

= centripetal acceleration ( )



where:

= centripetal acceleration ( )


T = period (s)

a =cr

v2

ac m/s2

a =cT

4π r2

ac m/s2


Centripetal Force

Centripetal force is the force experienced by an object in circular motion and responsible for accelerating an object towards the center. This force acts in the same direction as the centripetal acceleration, towards center of the circle.

where:

= centripetal force (N)

m = mass (kg)



where:

= centripetal force (N)


m = mass (kg)

T = period (s)

F =cr

mv2

Fc

F =cT 2

4π rm2

Fc


📖

analyse the forces acting on an object executing uniform circular motion in a variety of situations, for example: – cars moving around horizontal circular bends ✅ – a mass on a string � – objects on banked tracks �

Moving Around Horizontal Bends

When a vehicle turns around on a horizotal bend, the centripetal force is provided by the friction between the tyres and the road. Other forces experiences by the car are the typical normal force and weight force.

Mass On A String (Horizontal Motion)

When a mass attached to a string and swung around horizontally, the centripetal force is provided by horizontal component of the tension in the string. Other forces include gravity acting downwards.

ΣF = F =frictionr

mv2


The string makes an angle with the horizontal so that tension has a horizontal and vertical component. The verticle component of the tension force is equal to the gravitational force, as there is zero vertical acceleration.

The horizontal component of the tension force is the only force acting in the horizontal direction, meaning it is the centripetal force.

Mass On A String (Vertical Motion)

When a mass attached to a string and swung around vertically, the centripetal force is provided by the sum of tension in the string and gravity.

Since the string doesn't make an angle with the vertical, the tension force only has a vertical component. The vertical tension force and the gravitational force add the give centripetal force.

At the bottom of the circle, the net force is upwards while at the top of the circle, the net force is downwards.

Moving On Banked Tracks

When travelling on a banked track, the centripetal force provided is the vector sum of the gravitational force and normal force when the car is travelling at its design speed - speed at which there is no sidewards frictional force

θ

ΣF =y F + T y mg = 0

ΣF =x F =T xr

mv2

ΣF = mg + F =Tr

mv2


In the vertical direction, at design speed the car maintains its vertical height so the net force in the vertical direction must equal to zero.

The normal force experienced by a vechicle moving on a banked track can be determined by:

In the horizontal direction, at design speed the only force acting is the horizontal component of the normal force which is the centripetal force.

📖

investigate the relationship between the total energy and work done on an object executing uniform circular motion ✅

Kinetics Energy

Kinetic energy is the energy associated with motion, and is given by:

When an object is undergoing uniform circular motion its speed is constant, so its kinetic energy is also constant.

Potential Energy

Potential energy is the energy stored inside objects relative to other objects. For gravitational force, potential energy depends on how high an object is located.

ΣF =y mg + F =N y 0

F =N cosθmg

ΣF =x F =N xr

mv2

KE = mv21 2


For an object undergoing uniform circular motion in a horizontal place the potential energy is also constant.

Work

Work done on an object is equal to the total energy transferred to the object. So for an object moving in uniform circular motion in a horizontal plane there is no work being done since total energy is constant and also because the net force is perpendicular to the velocity.

For an object undergoing uniform circular motion and changing potential energy, there will be a change in energy. Because the object returns to initial position or initial potential energy, total work done in one revolution is still zero.

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investigate the relationship between the rotation of mechanical systems and the applied torque: 𝜏 = 𝑟⊥𝐹 = 𝑟𝐹sin𝜃 ✅

Torque

Torque is the turning moment of a force that exists when a force acts to produce a rotational movement. It is a vector quantity measured in Newton meters (N m). Direction convention used involves a clockwise rotation (negative) and a anticlockwise rotation (positive).

The axis of rotation is a centre line at which can object rotates about. The pivot point is the point through which the axis of rotation passes. The line of action of the force is the line through which the force vector passes.

When more than torque is acting on a single object then the net torque needs to be found

U = mgh


where:

𝜏 = torque (N m)

r = force arm (m)

F = force (N)

𝜃 = angle between force and pivot point (°)

Force & The Force Arm

A force applied directly towards or away from a pivot point will not create a turning movement. Torque is maximum when the force vector is perpendicular to the force arm and zero when parallel.

Torque on an object is directly proportional to the magnitude of the force. A larger force will result in a larger torque.

The force arm is the perpendicular distance between the pivot point and the line of action of the force. The length of the force arm is directly proportional to the torque. A larger force arm will result in a larger torque.

τ = r × F sin θ


Motion in Gravitational Fields

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apply qualitatively and quantitatively Newton’s Law of Universal Gravitation to: – determine the force of gravity between two objects 𝐹= 𝐺𝑀𝑚 / 𝑟2 ✅ – investigate the factors that affect the gravitational field strength 𝑔=𝐺𝑀 / 𝑟2 � – predict the gravitational field strength at any point in a gravitational field, including at the surface of a planet �

Newton’s Law of Universal Gravitation

Newton's law of universal gravitation proposes that massive particles attract other massive particles by a force acting along the line which intersects both particles.

This force is propertional to the two masses of the two particles and inversely propertionate to the square of the distance between the.

F = Gr2Mm


where:

F = gravitational force (N)

G = universal gravitation constant ( )

M = mass of central particle (kg)

m = mass of orbiting particle (kg)

r = distance between particles (m)

Newton's third law tells us whatever force one object exerts on another object, it will experience an equal and opposite force due to the interaction. This is also true for the gravitational force.

If a planetary body 'A' has gravitational force of F on another planetary body 'B', then 'B' will exert the exact magnitude of force but opposite direction on 'A'.

Gravitational Field Strength

The gravitation field strength is the gravitational force per unit mass. It is a vector field defined everywhere in space.

where:

g = gravitational field strength (N/kg)

F = gravitation force (N)

m = mass of orbiting particle (kg)



r = radius (m)

6.674 × 10 Nm kg−11 −2 −2

F = A on B −F B on A

g = =m

F

r2GM

6.674 × 10 Nm kg−11 −2 −2


📖

investigate the orbital motion of planets and artificial satellites when applying the relationships between the following quantities: – orbital radius ✅ – orbital velocity � – orbital period �

Orbital Motion

Orbital motion is considered as uniform circular motion where the centripetal force is provided by the gravitational force.

This immplies that:

more gravitational force is required to keep a larger mass in orbit.

value of acceleration due to gravity at some point, is equal to the value of thecentripetal acceleration at that point.

Orbital Radius

The orbital radius is the distance from the centre of the orbit to the object.

Orbital Velocity

Orbital velocity is the speed at which the object is moving through its orbit. The orbital velocity is independent of the mass of the orbiting body.

F =c Fg


where:

v = orbital velocity (m/s)



r = orbital radius (m)

Orbital Period

Orbital period is how long it takes for an object to complete one orbit. The orbital period is independent of the mass of the orbiting body.

where:

T = orbital period (s)

r = orbital radius (m)



📖

predict quantitatively the orbital properties of planets and satellites in a variety of situations, including near the Earth and geostationary orbits, and relate these to their uses ✅

v =r

GM

6.674 × 10 Nm kg−11 −2 −2

T = 2πGM

r3

6.674 × 10 Nm kg−11 −2 −2


Orbit Satellite

Two of the most common types of orbits are:

Low earth orbit

Geostationary orbit

Low Earth Orbit

Low earth orbits have very fast speeds due to short orbital periods. Satellites is not fixed in position relative to the surface of the Earth. These two features allow it to cover entire surface of Earth in one day. This makes it useful for:

imaging

remote sensing

surveillance & spying

Since satellites orbits at relativelly low altitudes, it can produce high resolution images with small field of view. and not much energy is required to launch from Earth. This makes is useful for communication satellites as it reduces the need for signal amplification.

The main advantage is the rapid orbital decay due to non neglible atmospheric drag. Friction between particles and satellite causes satellite to lose speed


resulting in decrease in altitude needing regular reboost to move satellites back into orbit.

Geostationary Orbit

Geostationary orbit is designed to remain fixed in the sky relative to observers on the ground. This is because the satellite matches the rotational period of Earht, Orbits directly above equator and follows direction of Earth's rotation.

Geostationary orbits are useful for:

weather satellites

communication satellites

boradcast satellites

This is because the receiving dish on Earth can easily maintain contact with satellite without rotating.

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investigate the relationship of Kepler’s Laws of Planetary Motion to the forces acting on, and the total energy of, planets in circular and non-circular orbits using: – 𝑣=2𝜋𝑟 / 𝑇 ✅ – 𝑟3𝑇2=𝐺𝑀 / 4𝜋2 �

Kepler's First Law: The Law Of Orbits

Kepler's first law also known as the law of orbits states that the orbit of every planet is an ellipse with the central planetary body (Sun) at one of the two foci.

Kepler's Second Law: The Law Of Area


Kepler's second law also known as the law of areas states that a line joining a plant to the central planetary body (Sun) sweep out equal areas over equal intervals of time.

This is due to conservation of angular momentum. The law of areas concludes that planets travel faster near perihelion - closest point to sun, and slower near aphelion - furthest point from sun.

Kepler's Third Law: The Law Of Periods

Kepler's third law also known as the law of periods state the square of the orbital speed of the plant is directly propertionate to the cube of the semi major axis of its orbit.

v =T

2πr

T α r2 3

=r13

T12

r23

T22


📖

derive quantitatively and apply the concepts of gravitational force and gravitational potential energy in radial gravitational fields to a variety of situations, including but not limited to: – the concept of escape velocity 𝑣esc=√(2𝐺𝑀 / 𝑟) ✅ – total potential energy of a planet or satellite in its orbit U=−𝐺𝑀𝑚 / 𝑟 � – total energy of a planet or satellite in its orbit U+K=−𝐺𝑀𝑚 / 2𝑟 � – energy changes that occur when satellites move between orbits �

Potential Energy Of A Satellite In Orbit

Gravity is a conservative force and so an isolated system acted upon by gravity alone conserves mechanical energy

Thus, the work done by a conservative force such as gravity on an object is the negative of the change in the potential energy. Equivalently, the change in gravitational potential energy between two points is the work done by a force opposed to gravity.

We conventially define the reference point of zero potential to be infinite, when we are outside the influence of the Earth's gravitational field.

where:

U = gravitational potential energy (J)



m = mass of orbiting particle (kg

ΔE =mech ΔK + ΔU = 0

W = ΔK = −ΔU

U = −r

GMm

6.674 × 10 Nm kg−11 −2 −2


r = distance from core (m)

Notes that U is always negative and that U 0 as r .

Total Energy Of A Satellite In Orbit

The kinetic energy of a satellite in circular orbit is given by:

Then the total energy of a satellite in orbit is:

As the orbital radius, r, increases:

gravitational potential energy increases

kinetic energy decreases

total energy increases

Energy Changes Between Orbits

Different orbits have different energies. However, the total energy is conserved within an orbit. This can be seen in Kepler's second law, the law of areas.

Escape Velocity

Escape velocity is the minimum speed required for an object to escape the gravitational influence of a body. The escape velocity is not a property of the escaping object, but of the larger body.

The escape velocity can be given by:

→ →∞

KE =2r

GMm

E =total U + K = −2r

GMm


where:

= escape velocity (m/s)



r = radius of central particle (m)

v =escr

2GM

vesc

6.674 × 10 Nm kg−11 −2 −2

Module 5: Advanced Mechanics

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