Elastic collisions

Conservation of energy should include the word kinetic energy , a mistake ! pic taken from: http://jcphysics.wikidot.com/dynamics
If the collisions is elastic , show this relationship: u1 - u2 = v2 - v1

Electric potential energy

What is the potential energy of the assembly of three charges?
First consider a charge already there. Find the work done to bring the second charge to its location and then finally the third charge in its location. Sum the work done in bringing the second and third charge , it is equal to the electric potential energy of the system. Let label the 3 charges 1 , 2 and 3 and the location p and q.
The work done = kq1q2/r + kq3(q1/r + q2/r) . Hence electric potential energy gained is kq1q2/r + kq2q3/r + kq1q3/r. If q2 is negative the add negative to the equation when you substitute the q2 values.

Work done is stretching spring

The spring is horizontal and therefore the force applied is equal to the Tension in the spring.The work done by external force is equal to the elastic potential energy stored.
When the spring is vertical , the force applied is not equal to the Tension in the spring as weight is in the same direction as the force applied.The work done by external force is less that the elastic potential energy stored in the spring. There is epe stored in the spring due to the mass hanging on the spring.

Damping system

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Change in velocity vector

Change in velocity is vector subtraction . Take the final velocity vector , at the end of the arrow join the negative vector of the initial velocity, then join the start to the end and then that is the change in velocity vector (same as add vector). If use component method the do two times . First change in velocity (x-comp) is vf-vi. Then change in velocity (y-comp) is also vf-vi. Then use pythagoras theorem to find resultant change in velocity vector and tangent to find angle of the final vector.Seldom used because its very long .

For circular motion, draw a small sector , draw tangential velocity , one will be initial and the other final velocity. Bring the initial to the final (but change the arrow) as is -ve vector .Join the start to the end and the change in velocity point towards the centre of the circle (nearly) . As the sector is smaller and smaller , then it will point to the centre.

Acceleration down a slope

Acceleration down a slope is determined by weight component down the slope which is mgsin(angle), therefore a = gsin(angle). This information is frequently used and therefore you should not hesitate committing into your memory.If the angle is increased to 90' , then it would be just like falling freely where acc. is equal to g.

Some thoughts: Falling freely is often the theme of extreme sports like sky diving ,base jump etc. Before going into such ultimate sports, you would want to try some skiing , snowboarding, sliding in water theme parks, roller coaster etc which gives you an acceleration close to g . You can say its an appetizer before the main course.

Acceleration time graph of a bouncing ball

pic from citejournal( Fig.6)
A bouncing ping pong ball spend most of its time in the air , whether on the way up or down its acceleration due to gravity act downwards and has the unchanging value of 9.81 ms-2.Taking downwards as negative the acceleration time graph (graph 3) is a horizontal straight line (at -ve value) except at a very short time interval where it suffers a smack from the ground in a direction opposite to its weight and this explains the peak found in the graph 3. Obviously this force is large compared to its weight and thus the acceleration is positive and has a large magnitude. As it bounces a number of times its height decreases and the upward force decreases and the +ve acceleration decreases in magnitude too. As the height decreases the time spend in the air is shorter and the horizontal line showing -ve 9.81 ms-2 should get shorter (see graph 3 above).

Understanding gravitational potential

The way gravitational potential is defined is really convenient for scientist who look to outer space and want to leave the gravitational pull of the Earth and when that happens the potential is zero. The earth no longer pull you. From the Earth , you see the highest potential as zero and therefore lower than zero must assume negative values. This is another view of why potential has negative values other than the definition.

Binary star or twin star

Using gravitational force , the distance between the star is R. If you apply this force as centripetal force , the radius of the circular orbit is only R/2. A twin star circulate about each other as both are just as massive. The trap is here where most students think that the distance between the star is the radius of the circle . Here is not the same as Earth and Moon where the Moon circle around the Earth and the distance between them R is the radius of the circle,R.

Types of collision(improved)

For all types of collisions(where no external forces act) you can apply conservation of momentum.
Elastic and inelastic collisions included.

Apply conservation of kinetic energy only for perfectly elastic collision. Other types like inelastic means kinetic energy not conserved which normally means heat is produced as total k.e is less (k.e lost)



Therefore you can only apply relative speed of approach is equal to relative speed of separation when the collision is perfectly elastic . After all it is derived from COM an Cok.e.

What is head on? If the centre of approaching mass hit the centre of the other mass than motion will be straight line (linear or 1D)

Caution:What is elastic? The word perfectly is not used but it may be assumed to mean perfectly elastic in some TYS questions. As the syllabus is very much reduced only two types are asked elastic(actually perfectly) and inelastic only.

Perfectly elasitc collision between two bodies

link to detail calculation

Very mathematical ! Add in some physics .
Write two equations : COM and COk.e
Fast track to : u1-u2 = v2-v1 (dont worry about its mass)
Sub into COM (wiser choice that COk.e)
Then solve it, to find uknowns.
Important to take note of directions given, unknown must assume positive value (unless specified already)

Area of graphs

The following graphs are popular!
Area under force time graph is change in momentum . If area is positive , there is a increase in momentum and negative means decrease in momentum

Area under force displacement graph is work done . If positive work done there could be increase in other forms of energy like kinetic energy or potential energy (gravitational or elastic) and negative means decrease in these other forms of energy and being converted to heat energy.

Area under pressure volume graph is also work done . A increase in volume means work done by gas and decrease in volume means a work done on gas.

Gradient of energy time graph is power. Gradient at a point on the graph gives instantaneous power. Taking the energy /time taken gives average power. Gradient zero at any point in time means instantaneous power zero.

To calculate instantaneous power P=Fv (force and velocity at that instant). Total energy divide by time taken gives average power.

Making the right moves in projectile motion

Very much like in a game of chess , got to have some strategy and know what each piece can do. The x and y component quantities can only move horizontally and vertically. If you have time then you have the best piece in your game because it can go diagonally. Therefore always try to find time as it is the most versatile information without direction. Use time to find the quantity that is required. The quantites obtained are in component form and therefore needs to be added using phythagoras theorem to find the resultant magnitude and tan angle to find the resultant direction of either velocity or displacement.
very detail account by eiu

limiting speed in circular motion

The acceleration of circular motion is limited by acceleration due to gravity. Two common scenarios are roller coaster on top of the loop and a car on top of a curved bridge or a hump. At such situations normal reaction force is zero .There will be a feeling of weightlessness for the passenger in the roller coaster or the car . The minimum acceleration of a roller coaster inside a loop can only be g ,the centripetal acceleration must be greater than g , the speed must be greater than (square root of rg ). A car going over a bridge also has a maximum acceleration of g and therefore the centripetal acceleration must be smaller than g, the speed must not be greater that (square root of rg) .The speed (square root of rg) is the min. speed and also the max. speed depending on the direction the normal reaction is acting.
link : unsw clips showing how a car becomes airborne, click airborne automobiles.

Circular motion

Pic from Nascar racing
The centripetal force is the net force acting towards the centre of the circle. Since it is the net force , then it must be contributed by some forces like tension, normal reaction, friction, weight etc. Most of the time it is a combination of these forces . A roller coaster has weight and normal reaction. A car has normal reaction and weight. A planetary body has gravitational force. The problems in circular motion are successfully solved only if the forces are identified correctly and aknowledging that the resultant force is towards the centre of the circle. The acceleration a is v2/r and is related to period using v=rw. In gravitation , there is a simplified scenario due to the existence of only gravitational force around a massive body , the gravitational field strength is the centripetal acceleration. Therefore equating g = GM/(r2) = (v2)/r or (w2) r enable you to calculate period and velocity of the satellite circulating the massive body M.

Taking moments about the point O

Taking moments about a point O , where there is unknown force.
Go to MOE link
Link to free body diagrams provided by EIU on static equilibrium.

Sand on Platform executing shm

The sand on a platform that is oscillating with a certain frequency and amplitude will soon find itself becoming airborne if the acceleration approaches 9.81 ms-2. Taking the position of the platform at the highest point (max displacement) , the acceleration a is the greatest,but the only force on the sand is weight (down) and normal reaction(up). Therefore , writing the summation of force = ma , mg-R = ma, the max possible a is g , i.e when R=0 , the sand will become airborne. Equate g to the acceleration due to shm , you can calculate the max. amplitude for a given f or max f for a given amplitude.

It may be easy to imagine yourself on a trampoline that bounces with shm, is it likely that you become airborne sometime .Isn't it obvious where you will become airborne. To analyse such cases, always draw free body diagram and recognising that the acceleration always point towards the equilibrium. At the trough of the motion, the reaction force is always greater that the weight.

Forces : Hints

1. Three forces : two tensions and weight. Answers: 4.48N and 3.36N

2. Need to resolve tension before taking moments. Tensions have the same magnitude = 8.5N

3. Forces at the two supports : 5640N and 4414 N

4. Need to resolve tension before taking moments. Answer : 12.0N and Reaction at hinge can be obtained using pythagaros theorem and tangent of angle. Answer : 8.14 N and 18.6 degrees.

5. The tension in the strut should be called force by the strut on the shelf. Please ammend . The direction of the force F has to act upwards along the line of strut which angle can be found using trigonometry. Answer : F= 54.1 N and R at the hinge 72.0 N and 53 degrees.

6. Forces at support = 9250 N and 13750N.

Conservation of momentum

with links to video
It is important to memorise the statement.
The total momentum of a system is a constant provided no external force is acting.
Concept wise :
If you are running with a bowling ball , the system is you and bowling ball, momentum is mass (you+ball ) x velocity. You stop abruptly to allow all the momentum to go to the ball. In that case , the velocity of the ball would be large. Your coach would be happy to see you transferring your momentum to the bowling ball. In other words, initial momentum(before throw) is equal to final momentum(after throw) .Momentum is constant (not lost) in a system because the force on ball and force on you which is equal and opposite and therefore justifying the application of conservation of momentum(no external force is acting) .
If you recall , many throwing events , the athlete has to increase his momentum before he throws and transfers all this momentum to the shotput , discus, javelin, tennis ball , cricket and many other examples.

Rate of change in momentum

with links to video
Momentum is mass x velocity. A lorry traveling slowly may have the same momentum as a tennis ball served at high speed.
Force required to change the momentum of a fast traveling tennis ball to zero depends on how quick it is stopped. If the racquet is pulled back in the direction of motion upon contact , prolong the time to stop it , therefore reducing the force on the racquet. If the racquet is swung towards a fast moving tennis ball thereby reducing the time to stop it , therefore increasing the force on the racquet.

Therefore prolonging time of impact saves life in an accident as the driver is as innocent as the tennis ball for traveling as such great speed. Seat belts and air bag allows more time to stop , thereby reducing the force on the driver. The crumple zone set in the bonnet allows more time to elapse before the car finally stop , thereby reducing the force.

In motorcycle racing, riders fall off their vehicle and slide on the gravel prolonging the time to stop , therefore saving themselves of bone breaking injuries. They wear suits that can protect their skin.