Concepts of PhysicsMonday, November 30th 2009

BeatsOne more special case of overlapping waves to add. In situations where waves with the same frequency overlap, we have seen how the principle of linear superposition leads to constructive and destructive interference and how it explains diffraction. We will see that two overlapping waves with slightly different frequencies give rise to the phenomenon of beats. However, the principle of linear superposition again provides an explanation of what happens when the waves overlap. A tuning fork has the property of producing a single-frequency sound wave when struck with a sharp blow. Assume we have two identical tuning forks placed beside each other.

One has a piece of putty stuck on it which lowers the frequency be 2 Hz because of the higher mass. So we have 440 Hz from the first fork and 438 Hz from the second one. When the forks are sounded simultaneously, the loudness of the resulting sound rises and falls periodically - faint, then loud, then faint, then loud, and so on. The periodic variations in loudness are called beats and result form the interference between two sound waves with slightly different frequencies.

This effect is used for tuning musical instruments (it has a bell like sound). The number of times per second that the loudness rises and falls is the beat frequency and is the difference between the two sound frequencies.

Double slit - towards modern PhysicsThe wave-like properties of light were demonstrated by the famous experiment first performed by Thomas Young in the early nineteenth century. In original experiment, a point source of light illuminates two narrow adjacent slits in a screen, and the image of the light that passes through the slits is observed on a second screen.

the two slit experiment is key to understand the microscopic world

The dark and light regions are called interference fringes, the constructive and destructive interference of light waves. So the question is will matter also produce interference patterns. The answer is yes, tested by firing a stream of electrons.

• waves can interfere, for light this will make a series of light and dark bands

• matter particles, such as electrons, also produce interference patterns due to their wave-like nature

• so with a high flux of either photons or electrons, the characteristic interference pattern is visible

However, notice that electrons do act as particles, as do photons. For example, they make a single strike on a cathode ray tube screen. So if we lower the number of electrons in the beam to, say, one per second. Does the interference pattern disappear?

• if we lower the intensity of light, or the flux of electrons (the electric current), we should be able to see each photon strike the screen

• each photon makes a dot on the screen, but where is the interference pattern?

The answer is no, we do see the individual electrons (and photons) strike the screen, and with time the interference pattern builds up. Notice that with such a slow rate, each photon (or electron) is not interacting with other photons to produce the interference pattern. In fact, the photons are interacting with themselves, within their own wave packets to produce interference.

• the interference pattern is still there, it simply takes some time for enough photons, or electrons, to strike the screen to build up a recognizable pattern

• interference, or a wave phenomenon, is still occurring even if we only let the photons, or electrons, through one at a time

• so what are the individual particles interfering with? apparently, themselves

But wait, what if we do this so slow that only one electron or one photon passes through the slits at a time, then what is interfering with what? i.e. there are not two waves to destructively and constructively interfere. It appears, in some strange way, that each photon or electron is interfering with itself. That its wave nature is interfering with its own wave

The formation of the interference pattern requires the existence of two slits, but how can a single photon passing through one slit `know' about the existence of the other slit? We are stuck going back to thinking of each photon as a wave that hits both slits. Or we have to think of the photon as splitting and going through each slit separately (but how does the photon know a pair of slits is coming?). The only solution is to give up the idea of a photon or an electron having location. The location of a subatomic particle is not defined until it is observed (such as striking a screen).

• in order for a particle to interfere with itself, it must pass through both slits

• this forces us to give up the common sense notion of location

1.) Centripetal acceleration is caused by the centripetal forceʼsa)change in magnitudeb)change in directionc)change of direction and magnituded)sometimes magnitude and sometimes direction

2.) One fixed pulley doesa)change direction onlyb)distribute the forces equally to each ropec)nothingd)leave only one third of the force on the rope you are pulling on

3.) Total mechanical energy is the sum ofa)kinetic energy and potential energyb)rotational kinetic energy and linear kinetic energyc)gravitational potential energy and elastic potential energyd)all forces

Midterm A - multiple choice

1.) Centripetal acceleration is caused by the centripetal forceʼsa) change in magnitudeb) change in directionc) change of direction and magnitude d) sometimes magnitude and sometimes direction

2.) One fixed pulley doesa) change direction only b) distribute the forces equally to each ropec) nothingd) leave only one third of the force on the rope you are pulling on

3.) Total mechanical energy is the sum ofa) kinetic energy and potential energyb) rotational kinetic energy and linear kinetic energyc) gravitational potential energy and elastic potential energyd) all forces

Midterm A - multiple choice

4.) What is the unit for energy? a) Wattb) Joulec) Newtond) Energy unit

5.) When an acceleration is present, it needs to be caused by a)a counter accelerationb)the work donec)a forced)friction

6.) For constant speed, the acceleration isa)zerob)velocity over timec)change in distanced)always negative

4.) What is the unit for energy? a) Wattb) Joulec) Newtond) Energy unit

5.) When an acceleration is present, it needs to be caused by a) a counter accelerationb) the work donec) a forced) friction

6.) For constant speed, the acceleration isa) zerob) velocity over timec) change in distance d) always negative

7.) For a stone thrown horizontally from a building with height h, the vertical component can be treateda) the same as the horizontalb) as accelerated motion, but the horizontal part has to be accounted forc) independently from the horizontal and has no accelerationd) independently from the horizontal and has acceleration

8.) An empty and full cylinder made from the same material roll down an incline. The solid cylinder isa) slowerb) same speedc) fasterd) twice as fast

9.) Torque depends on the lever arm mathematicallya) quadraticb) linearc) not at alld) linear but like 1/x

10.) When a leave falls off a tree, the speed will be influenced by a) the time of the yearb) gravity and movement of the airc) movement of the aird) gravity only

11.) Formulate Newtonʼs first law in your own words!

12.) Draw a free body diagram for a book resting on a table.

13.) Describe potential gravitational energy in words and write as a formula.

14.) Explain the work-energy theorem in your own words. (2 points)

Short questions

11.) Formulate Newtonʼs first law in your own words!

In absence of an external force, an object will stay in the stage of movement (rest or moving with constant velocity)

12.) Draw a free body diagram for a book resting on a table.

Gravitational force downward, Normal force (perpendicular to surface) upward - same length

13.) Describe potential gravitational energy in words and write as a formula.

PE = mgh Energy stored in an object due to its location. Product of mass height and acceleration due to gravity.

14.) Explain the work-energy theorem in your own words. (2 points)

Work is given as the difference between initial and final energy

Short questions

15.) In linear motion with constant acceleration the following equation is true: vf2 = vi2 + 2ad with vi initial velocity, vf final velocity, a acceleration and d displacement. How does this expression change for rotational motion - write down the formula and name the parts as done above.

same as in second one.

16.) Give an example for conservation of angular momentum. (2 points)

Spinning skater - arms in versus arms out

17.) What do we mean, when we talk about the center of mass - describe in your own words.

Depends on distribution of mass in and object or system. If the object is uniform (or symmetric), the center of mass will be in the center. It is also the point where an object would

be balanced.

18.) Two objects collide head on - what can you say about the momentum of the system before and after the collision?

It is conserved - same.

19.) If the gravitational acceleration on Jupiter is larger than on earth, what does that mean for the weight of a person? (2 points)

Weight W = mg, so if g larger, W larger

20.) How is the normal force defined? How is the friction force defined? (2 points)

Normal force is perpendicular to surface and opposite to gravitational force (weight of an object).

Friction is a property of a surface, friction force counteracts movement.

21.) What is the connection between gravity and tides? Describe in two sentences.

Tides are caused by the gravitational pull between earth and moon. It causes the level to go up and down about twice a day

22.) Three springs of the same relaxed length (original equilibrium position) have spring constants k1 > k2 > k3. The springs are suspended from the ceiling and identical masses are then hung on each of the springs. In response to these identical stretching forces, the springs stretch amounts x1, x2 and x3, respectively. What is the relation between the amounts stretched x1 ? x2 ?x3 for the given relation of the spring constants? (4 points)

F = kx, therefore x1 < x2 < x3 for k1>k2>k3

23.) A force of 10 N works on a ball over a distance of 5 m. The force is perpendicular to the displacement of the ball. What is the total work done by the force?

24.) How is kinetic energy proportional to velocity?

25.) Describe under which conditions the center of mass of a system is located in the center of the system.

23.) A force of 10 N works on a ball over a distance of 5 m. The force is perpendicular to the displacement of the ball. What is the total work done by the force?

Perpendicular means Θ = 90° , therefore cos90 = 0 which means no work done.

24.) How is kinetic energy proportional to velocity?

KE = 1/2mv2 so kinetic energy is proportional to the velocity squared

25.) Describe under which conditions the center of mass of a system is located in the center of the system.

For a uniform object made from the same material (distribution of mass) or a symmetric system. For example two planets with the same mass.

A.) Elevator at a mine (could be here in Sudbury) 30 Points

The elevator going down the shaft in a mine is called a cage and consists of two cabins on top of each other, where each one is 5 m long, 3 m wide and 8 m high. The cabins have a total mass of 5.0 tonnes. The cage is held by 1 cable (made of stainless steel) with 2.5 cm diameter and it is 2.2 km long. Each cabin can hold 40 people and we will assume an average mass of 80.0 kg per person.

a)Calculate the volume of one of the cabins as well as the volume of both of them. Volume V = L x W x H = (5 m) x (3 m) x (8 m) = 120 m3 Therefore volume of two cabins is 2 x 120 m3 = 240 m3

b)The stainless steel cable has a density of 4000 kg/m3. What is the mass of the cable? Density formula was not given, but you can find it from the units: Density = mass/Volume Of course for the cable you have to take the volume of the cable, which looks like a cylinder, so it is V = Ah (Area of a circle time the lenght of the cable h), Therefore V = π r2 h [1.08 m3]

A.) Elevator at a mine (could be here in Sudbury) 30 Points

The elevator going down the shaft in a mine is called a cage and consists of two cabins on top of each other, where each one is 5 m long, 3 m wide and 8 m high. The cabins have a total mass of 5.0 tonnes. The cage is held by 1 cable (made of stainless steel) with 2.5 cm diameter and it is 2.2 km long. Each cabin can hold 40 people and we will assume an average mass of 80.0 kg per person.

c) If the engine of the system has to lift the cable and cabins full of people with v = 3.6 m/s, how much power is needed?

Power P = F * v (v given) F = mg need to calculate mass of cabins + mass of people + mass of cable

Total mass: 15270 kgForce F: 154056 N and Power P: 555 kW

(d) If the cable slips and the cage falls, what would be the work done by gravity?

Work W (G) = F(G) cosΘ d (d is not given, so the result will depend on it)[154056N *d]

A.) Elevator at a mine (could be here in Sudbury) 30 Points

The elevator going down the shaft in a mine is called a cage and consists of two cabins on top of each other, where each one is 5 m long, 3 m wide and 8 m high. The cabins have a total mass of 5.0 tonnes. The cage is held by 1 cable (made of stainless steel) with 2.5 cm diameter and it is 2.2 km long. Each cabin can hold 40 people and we will assume an average mass of 80.0 kg per person.

(e) In free fall, how much kinetic energy would the cage system have at the moment when it passes the 6800 feet level - assuming that it started at surface?

You have to calculate the distance - convert ft to mKE = 1/2 m v2 [319 MJ]

You can use vf2 = vi2 + 2 a d (where a =g, initial velocity zero) to get v2

It is also possible to use energy conservation (only PE on top, only KE at bottom)

(f) The cage is moving up with v = 3.6 m/s. When it passes the 3600 feet level, the cable disconnects from the cage (just on top of the cage) and it starts free falling.

What is its velocity when it passes the 6000 feet level?

Same formula for final velocity in accelerated movement, but this time there is an initial velocity and the distance is different - again has to be converted to m

[v = 119.83 m/s = 120 m/s]

B.) A ship (Grace) approaches an island from the west. The island has one big mountain in the center. The island forms a perfect circle and the mountain is 1500 m high. The diameter of the island is 7.0 km. On the east side of the island is another ship. In order to communicate with the other ship (Wildthing) the captain of the Grace wants to use a canon to shoot over a message. But he doesnʼt know which angle above the horizon to point the canon at, so that he reaches his goal. He knows that the velocity of the message once it leaves the canon is 250 m/s. He is still one km from the island and so is the other ship. Can you help him? (Make a drawing first including all the important information and comment on your calculations along the way)(15 points)

The main part here is to produce a drawing and to realize, that the message has to get over the mountain - which means the height and half the distance. Then the angle can be obtained by using the inverse tan function of the ratio of height over half the distance.

The result is 18.9 degrees for these numbers.

C.) Two ice skaters have masses m1 and m2 and are initially stationary. Their skates are identical. They push against each other and move in opposite directions with different speeds. While they are pushing against each other, any kinetic frictional forces acting on their skates can be ignored. However, once the skaters separate, kinetic frictional forces eventually bring them to a halt. As they glide to a halt, the magnitudes of their accelerations are equal, and skater 1 glides twice as far as skater 2. What is the ratio m1/m2 of their masses? i) make a drawing (before and after) (3 points)ii) write down reasoning (5 points)iii) solve problem (7 points)

We had done this example in class - just with numbers. So first step was to realize: Momentum conservation. Since they start from rest, we get

m1 v1 = m2 v2 with v = d/t for any point, for example just before they stopwe can solve for the ratio m1/m2 [1/2]