Physics fundamentals full semester notes PDF

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PHYS1002 DENSITY Density [r] is a material property: Ratio of mass to volume (kg/m3) r = m/v An object will float if its density is less than the density of the fluid. WEIGHT Weight refers to a force due to gravity and mass refers to the amount of matter. W = (m x g) N m = mass g = gravity 9/s2 CONS...

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PHYS1002( ! DENSITY( Density [ r] is a material property: Ratio of mass to volume (kg/m3) r = m/v

An object will float if its density is less than the density of the fluid.

WEIGHT( Weight refers to a force due to gravity and mass refers to the amount of matter. W = m x g) N m = mass g = gravity 9.8m/s2 CONSIDER: An object suspended on a rope. It is stationary. Tension force in rope = Weight force This is an example of a static equilibrium, where forces add to produce a zero resultant. Forces are represented by vectors with magnitude and direction.

BUOYANCY( Immersion of an object in a fluid produces a buoyant force which partially counteracts the weight. When an object floats Fb = Fg Archimedes’ Principle When an object is immersed in a fluid, there is an upward buoyant force equal to the weight of the volume of the fluid displaced by the object. If bag of water was immersed, the buoyant force would be equal to 0 as no water is displaced.

FLOTATION( How to determine how far a floating object is above waterline. Iceberg example 1. Iceberg is in static equilibrium. Fw = Fb 2. Buoyant force = Vsea x rsea x g 3. Weight force = Vice x r ice x g 4. Vsea/ Vice = rsea/ rice

PRESSURE( Buoyant force occurs due to pressure from surrounding fluid, a force that acts inwards from all directions on the object. Pressure occurs at a point in a fluid measured in pascals (Pa). Pressure is not a force. Pressure acting on a surface produces a force. The force is • Perpendicular to the surface • Proportional to fluid pressure p (Force) • Proportional to source area A

F=pxA

Pressure on all the above samples are the same as area is the same. Pressure at every point, at a given horizontal level in single body of fluid at rest must be equal. Example: What is the pressure acting on the floor of a pool 50m long, 25m wide, 2m deep? Weight of water = 50 x 25 x 2 x 9.8 = F Pressure = F/A Therefore if depth is double, pressure is doubled. So p = p0 + kd where d is depth. Depth is proportional to additional pressure. Pressure at a point at a distance d below the surface of the fluid is due to the weight of fluid above that point. This is in addition to any pressure p0 that may be acting on the surface. p = p0 + rgd

Gauge Pressure – minus atmospheric pressure

Absolute Pressure – plus atmospheric pressure Pressure differences at different depths produce an upward force (buoyancy Archimedes) on whatever was occupying the space between the two levels. DF = Dp(mxd/v) x A! DF = mfluidg = weight of fluid Pascal’s Principle: In a fluid at rest in a closed container, a pressure change in one part is transmitted without loss to every portion of the fluid, and to the walls of the container. E.g. Pressure change at surface of water (increase pressure) will lead to pressure changes in water leading to a smaller buoyancy force.

COMMON(QUESTIONS( 1. Archimedes’ Golden Crown • • • •

Calculate weight in air vs weight in water o Weight in water = weight force – buoyant force Express answer in terms of density of objects by Archimedes’ Principle Express in ratio of weight in water/actual weight Sub in values

2.

KINEMATICS( The satellite is an extension of projectile motion. A satellite is continuously free falling as any object undergoing projectile motion free falls as force direction keeps changing. Keep increasing the range (by increasing initial velocity) and we reach the case of a satellite orbiting the earth. In the case of the ball we can assume a flat earth while in the case of the satellite we can’t. The satellite has a large enough sideways (or horizontal) velocity that the change in direction of motion (while falling) is such that it stays in orbit. If the satellite is too slow, gravity will pull object back towards Earth. If the satellite is too fast, it will escape Earth’s gravity and completely fly out of orbit. The greater the altitude of the satellite, the slower its orbiting speed. This is as altitude increases, the satellite feels less of Earth’s gravitational pull. A communication satellite such as used for satellite television is in geosynchronous orbit and has a rotation period of 24 hours, the same as that of the Earth. It is known as a geostationary satellite such that its distance to the earth will lead to orbital speed that corresponds with spinning speed of the earth. It therefore moves with the same angular speed as a location on Earth and appears to remain at the same position in the sky. This results in a circular motion.

Satellite is being pulled down and everything inside it. Nothing is being pulled or pushed against each other. Free Fall Cabbage still has a weight since it has mass and gravity is still acting upon it. However, the instrument has lost capacity to register its weight during free fall as it falls together with the cabbage both due to gravity at an acceleration of 9.8ms-2. In this sense, there is no push or pull on the weighing instrument to produce a reading.

Even when cabbage is rising, both cabbage and scale are still being pulled down due to gravity. Therefore, they are still experiencing ‘apparent weightlessness.’

NEWTON’S(FIRST(AND(SECOND(LAWS( ! 1. Newton’s First Law/Law of Inertia If no net external force is applied to an object, its velocity will remain constant.

A body cannot change its state of motion without outside influence. 2. Newton’s Second Law Fnet = ma If sum of all forces on a body does not add to zero, then acceleration occurs If a body is accelerating, there must be a force on it -

Acceleration is always same direction as net force F=0 à body is in equilibrium

FRICTION( Frictional force is parallel to the surface and proportional to normal surface. Friction is due to roughness of both surfaces and microscopic “cold welding.” Acts in the direction opposite to slippage direction. The three types of friction are: 1. Static friction – Preventing motion 2. Kinetic friction – Points opposite motion of direction 3. Rolling friction – Wheels on surface

MEASUREMENT( Strength of friction is measured by µ which is a scalar. (Depends on both surfaces)

µs>µk>µr Once an object is in motion, it is easier to keep in motion.

Fs(max) is equal the amount of applied force needed to set an object in motion. This is equal to friction.

NEWTON’S(THIRD(LAW( Newton’s third law For every action (force) there is an equal and opposite reaction (counter-force).

The action and reaction force are not exerted on the same object. They act on different objects. For example book on a table:

Are not action-reaction pair forces as both Normal and Weight force are acting on the book. Weight force is reaction of gravitational attraction by book on earth. Normal force is force by book on table. For example walking:

This is an action-reaction pair. Force on floor from foot vs force on foot from floor.

APPARENT(WEIGHT( Apparent weight is given by the magnitude of the normal force. A person standing on a spring scale normally:

Fsp = Fw

If he weighs himself in a lift which is accelerating upwards, there is a net positive force. Fnet = Fsp – Fw Fsp = Fnet + Fw The scale reads heavier. The normal force is greater. When decelerating upwards, there is a net negative force. Body interprets greater normal force is feeling heavier. If body is accelerating with the same acceleration as gravity, body can experience weightlessness. i.e. astronaut in satellite in orbit.

CIRCULAR(MOTION(

After travelling a distance of s, an object moving around a circle with radius r has angular displacement:

Tangential velocity if motion is uniform:

Period of motion T = time to complete on revolution Frequency f = number of revolutions per second

Angular velocity:

Uniform circulation motion occurs when angular velocity is constant. All points on a rigid rotating object will experience same angular velocity. Velocity is a vector changing direction. There is an acceleration and a net force. Change in velocity points towards centre of circle.

Angle between velocity vector is

so:

Since acceleration points towards centre: Centripetal acceleration ac

As object accelerates, there must be force to keep it moving in a circle.

Centripetal force may be provided by friction, tension in string, gravity etc. Centripetal force is the resultant force provided by some other force not a separate force in the free-body diagram.

In this case, normal force is large enough is sufficient to maintain circular motion. Provides centripetal force. Therefore, not in equilibrium.

CENTRE(MASS(( Centre of mass The CM of an object or system of particle is the point that moves in agreement with Newton’s 3 laws of motion as though all of the mass were concentrated there. At equilibrium, a body free to rotate will hang so its CM is vertically under the suspension point. Centre of mass can be considered as the “average position.” It can apply to more than one object. This is weighted on the mass of the objects. If the ith particle is (xi, yi, zi) and mass is mi, the total mass M is: M = m1 + m2 + ... + mn

Then the x-coordinate of CM is;

Same can be applied to y and z coordinates. If there is an external force applied to an object, the centre of mass is accelerated. Therefore:

Since we have defined centre of mass as a single point within a body, we only focus on the external forces applied to an object and disregard any forces acting in within the object. These don’t affect the motion of the centre of mass. If an object breaks up due to internal forces, the CM will continue to follow the same trajectory subject to same external forces. Imagine a firework:

The CM of all exploded fragments will complete the projectile motion since they are all still subject to external force of gravity.

Example Problem:

They meet at centre of mass. CM does not have to be within the object. E.g. high jumping Athlete curves back in order for centre of mass to pass underneath bar, using less force to jump up to the bar. Remember CM is average position.

TORQUE(( Line of action Line of infinity in both directions in the direction of the force from a particular point.

Translation vs Rotation Rotation occurs if lines of action don’t pass through the same single point. e.g. seesaw, the heavier person on a seesaw makes seesaw rotate. Rotation only occurs since the object is not a single point. Archimedes’ Lever Rule

If there is a force on one end of lever opposing force on other end, a small force can be used to balance a big force if there is a large lever arm balancing a small lever arm. i.e. Large force and small lever arm is balanced by a small force and large lever arm at the fulcrum. This is expressed as:

This shows that the position of where the force is exerted is very important and not just the magnitude of the force. If force is exerted further away from the fulcrum, there is a greater turning ability. For general angles:

Torque is measured in newton metres (Nm) and refers to the rotational ability of forces that can change angular velocity of an object. Torque has a sign depending on direction of rotation e.g. clockwise. This must be defined. Torque occurs from a force with a “line of action” that does not cross the axis of rotation. i.e. the normal force at fulcrum has no torque. Consider a door:

Least amount of force is required to be applied when pushed at: • Largest r • 90o perpendicular angle Pushing on axis rotation will do nothing. Adding up Torques

1. 2. 3. 4. 5.

Choose sign convention – anti-clockwise is positive Choose rotation axis around – pick where torque disappears Draw line of action for each force Calculate resulting torque for each force Add up torques

EQUILIBIRUM(( For an object to be in static equilibrium:

Types of equilibrium: 1. Neutral No tendency to move due to no net torque Centre of mass is in line with normal force 2. Stable Net torque is anticlockwise Object will restore position Centre of mass is to left of normal force 3. Unstable Net torque is clockwise Object will fall Centre of mass is to right of normal force

MOMENTUM((

• • •

Also a vector Same direction as velocity Kg.m.s-1

Total momentum is the vector sum of individual momenta:

Newton’s 2nd law can be rewritten as:

When there is no net external force, total momentum is constant. Momentum is conserved. Consider breaking up the balls in pool. The momentum of the one white ball is transferred exactly without loss to other balls. Pinitial = Pfinal

IMPULSE(( Impulse J of a force is the change in momentum Dp caused by that force. If F is constant

If F is not constant

Impulse = area under F vs t curve.

Fav can be minimised during impact by increasing impact time Dt. e.g. absorbing impact vs. landing with locked knees

ENERGY(( Energy is needed to do useful work and is measured in joules. It is a scalar. 1J = 1 joule = 1kg.(m.s-1)2 1. Kinetic energy

2. Gravitational Potential energy

Kinetic energy + Potential energy = Mechanical energy Law of Conservation of Energy Energy can not be created or destroyed. It can only be changed from one form to another. Describes isolated system where there is no energy transfer into or out of the system.

WORK(( Work Process of transferring energy from environment to a system, or vice versa by application of forces. Change in energy = Amount of work done Doing work means using a force to: 1. Transfer energy from one object to another 2. Convert energy from one form to another Also a scalar and S.I. unit is joule. When force is constant

F=applied force d=object displacement while force is applied q=angle between applied force and displacement K=kinetic energy Work is done on an object and can be negative. Energy is transferred from the body. No work is done when: • When object does not move • If force is perpendicular to displacement

WORK(ENERGY(THEOREM( Work done by individual forces can be added together. Work-Energy Theorem: Change in kinetic energy of a system equals sum of work done by all individual forces on system:

Only applies to rigid objects.

POWER( Power means rate at which force does work on an object. i.e. powerful car is related to acceleration not velocity. Instantaneous power:

If F and q are constant then:

P = F.V Units: Watt (W) = joule.Second-1 (J.s-1)

GRAVITATIONAL(ENERGY( a) This is stored energy due to height in a gravitational field. Results from any force which depends only on position. b) Two bodies are attracted due to gravity caused by mass. Work is required to push them apart – energy is transferred to system when this happens. Known as gravitational potential energy, U. U is property of whole system, both objects. Work done by weight(gravity) on ball as it falls a distance h towards earth is.

Change in height is significant. Gravitational energy only depends on height of object above reference level, not on the path taken.

!

! CONSERVATIVE(FORCES( If work done by a force in moving an object from A to B does NOT depend on path taken, we call it a conservative force. e.g. gravity, ideal springs An object moving under influence of conservative force always conserves mechanical energy.

! Every conservative force has potential energy associated with it. Forces like friction which dissipate energy instead of storing it are non-conservative force.

Object which moves in a closed loop under conservative force – total work done is 0. Work done in moving an object form A à B under influence of a conservative force is exact negative of work done in going from B à A. i.e. reversible. Examples: Pendulum • Moves from all potential to all kinetic energy. • FT is always perpendicular to motion so does no work. • Gravity does work • Gravity is conservative, friction and drag are negligible unconservative forces • M.E = (K + U)A = (K + U)B • Undergoing circular motion o Net force towards centre

Loop the Loop •

•

•

•

•

At the o o o

top of the loop Weight force Normal force from track Net force = W + N § Minimum force to go around when N = 0. § Net force = W = mrw Minimum speed: o mrw = mg o V2 = rg At initial position: o V=0 o H is unknown At final point: o H = 2r (r = radius) o V = (rg)1/2 H = 5r/2

NON-CONSERVATIVE(FORCES( Mechanical energy is not conserved when non-conservative forces are acting. Friction convert work or ME directly into thermal energy. WE include this in our expression for conservation energy:

Examples Sliding • Block on horizontal surface slides to rest due to kinetic friction o Work done by friction is: § DME = DK = -Fkd § Can be figured out by time block takes to stop § Friction removes energy from object

•

Block o o o

sliding on horizontal surface at constant velocity by applied force Word done AGAINST friction by applied force Fapp DK=DU=0 Amount of thermal energy produced is equal to amount of work done.

Energy generated is not useful and dispersed to the system. When there is friction: • MEA does not equal MEB • MEB will be less and difference equal thermal energy produced

GRAPHING(MECHANICAL(ENERGY(

COLLISIONS( 1. Inelastic collisions • Energy is lost in collision – Kinetic energy not conserved o Converted to heat sound or deformation o Momentum is conserved

o •

•

§ Only equation to solve: Objects stick and move together after collision o Occurs in perfectly inelastic collisions o Treat objects as single object after collision: o Plastic/inelastic à object deformed by external force permanently deformed even after force is removed o Work done deforming object is irreversible o All or most of work done is converted to thermal energy o Lump of plasticine

2. Elastic collisions • Energy not lost in collision o Momentum is conserved during collision:

§

• •

•

§ Bounce of each other – each have its own separate velocity Elastic à object deformed by external force rapidly returns to original shape when force is removed. o Work done deforming object is reversible o Little to no thermal energy generated o Involves sub-atomic particles or highly rigid objects o Rubber band, steel spring, tennis ball If both objects are same mass, velocities swap after perfectly elastic collision e.g. Newton’s cradle

Most substances stretch or bend elastically until they reach their elastic limit beyond that they form plastically or break. Collisions and Impulse During collision, momentum is conserved – none is lost Consider:

Momentum lost by 1 = momentum gained by 2 Impulses are equal and opposite :

e.g. Consider:

Both have same magnitude in change in velocity: However, since truck has greater mass, change in velocity of car is much greater. So safer to be in truck.

IMPULSE(APPROXIMATION(( Momentum is absolutely conserved in collision only if there are no net external forces on system. However, usually has weight force acting. If external force is much smaller than collisional forces, and collision time is short, Momentum is very nearly conserved during collisions or explosions even with external force. Known as impulse approximation. e.g. hitting nail with hammer, recoiling gun

WHAT(IS(AN(OSCILLATION?(( Any motion that repeats itself Described with reference to: • Equilibrium position where net force is zero • Res...