lab 1 - manual.doc

8/20/2019 Lab 1 - manual.doc

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11a, lab 1: Motion, Measurement and Statistics

In the first lab we will investigate the motion of a ball rolling down an inclined plane. We will

use this simple system to investigate the concept of systematic and random measurement

uncertainties to understand how much we can trust a measurement. We will then compare theexperimental outcome with theory.

I. Ball on an inclined plane: Measuring motion

We want to measure the motion of a ball that rolls down on an inclined plane and see whetherthis matches what we would expect. We will roll a ball down a tilted aluminum plate where

gravity will accelerate the ball. It then rolls along a level surface (Plexiglas) where it should roll

with approximately constant velocity as long as friction can be neglected.

• Set up the experiment. Start by placing the large wooden board onto the table, and use theadustment screws and the level to ensure the plane is level. Place the Plexiglas strip onto

the wood and use a steel ball to chec! again that it is indeed level.

• Place the ramp onto the metal inclines and use the magnets to secure it. "a!e sure the

ramp touches the hori#ontal surface, but does not flex at the edge of the inclines.

• $ne challenge is to launch the ball without giving it a !ic!. %his is possible with a

magnetic launch mechanism you can put together using an electromagnet, a power source

and a switch. &lamp the electromagnet to the metal surface approximately ' cm up the

ramp. When the circuit is closed, the current through the electromagnet will generate a

magnetic field that will hold the ball bearing at the electromagnet on the ramp. %o releasethe ball bearing, simply open the switch to brea! the current. ind the minimum voltage

needed to hold the ball. Set it slightly higher. Caution: If the voltage is too high, the

electromagnet will heat up and might get destroyed. Check your setting with the TF!

• %est your launch mechanism, and ma!e sure the ball rolls straight down the plastic strip



• %he goal now is to measure the velocity of the ball after it has rolled down the ramp.

*elocity is the length the ball travels per time.

1. Can you think of two different ways to measure the velocity? Hint: You can use a

ruler to measure/define length, and a metronome/clock to measure/define a time

interval. Descrie the two alternative methods. !"se head#hones for the metronome

on the com#uter $ otherwise your fellow students might get annoyed% &.

'. How do you decide what length or time interval you use to get the est #ossile

#recision? (n other words, why might your #recision e negatively affected if your

length/time is either !a& too short or !& too long?

). *ry oth methods, and discuss them with your *+. Decide which one you will use for

the following e#eriment. -lso, look around what others are using and make sure

not everyone is using the same method. *his way we can later com#are the statistical

error of oth methods.

. Can you think of a way to measure the velocity without a time kee#ing instrument

!sto#watch/ruler&, ust with a ruler? *here is actually an easy way % can you figureit out? Discuss it with your *+0 !(f you have time left at the end, try it out for a

onus0&

5.

• We now have a measurement techni+ue in place that we will use to measure velocity.

efore you ta!e a dataset, prepare a wor!sheet with the logger pro software that allows

you to input your data and calculates the velocity for you. ollow the instructions on page- of the logger pro file if you use a method in which the time interval is constant and the

length is measured, and page ' if your length is constant and you measure the time

interval.

. Check everything with your *+, and take measurements until you are confident inyour techni2ue. -s you record data, create a scatter #lot and monitor it. Continue

until you have enough data to characteri3e the random error. *his will #roaly

re2uire at least )4 #oints. Discuss this among your grou# and with your *+. 5lot a

histogram of your measured velocities. Does it resemle anything like a 6aussian at

all? +it it to 6aussian, and #rint and attach it. -re there 7utliers !check whether

there are no ty#os in the data in#ut&?

8. +ollow the instructions in the 9ogger #ro worksheet to generate a scatter #lot.

-naly3e the statistics on the data. hat is the mean of the velocity? hat is the

s#read of the data, characteri3ed y the standard deviation ;? How do you e#ect ;

to change as you take more data #oints? rite your method and standard deviation

on the lack oard and com#are it to the other methods. How does ; com#are to the

fitted width of the 6aussian? hy might it slightly differ?

<. How large is your standard error, i.e. with what #recision can you tell the velocity?

6ive the standard error in asolute units !m/s&, as well as the relative units != of the

mean&.

>. How many measurements would you have to carry out to achieve a 4.1= #recision

in your velocity measurement?



!4.1= is the standard error in relative units&



10. II. What do we expect theoretically

sing the ideas of !inematics, we can perform calculations and ma!e predictions about the

velocity of an obect that that has accelerated down a ramp.

F parallel / mg sin(θ)

F perpendicular / mg cos(θ)

where m is the mass, g is the acceleration due to gravity, and θ (theta) is the angle that the

inclined plane ma!es with the hori#ontal surface. In the absence of friction and other forces

(tension, magnetism, etc.), the acceleration a of an obect on an incline is0

a = F parallel1m / g sin(θ)

$nce we !now this acceleration we can loo! at how far the ball rolled down the incline to predict

its final velocity0

vfinal' / 'ad 2 vinitial

'

3ere, d is the length parallel to the inclined surface over which the acceleration acts. 4ssuming

the initial velocity is #ero, we find that0

vfinal / s+rt('ad )

4s shown above, g is related to acceleration a by the angle theta. 3owever, d is also related to

the angle θ.

d / h 1 sin(θ)

where h is the height. So

a5d / g5h

vfinal / s+rt(' gh)

We do not need to know the angle of the incline to predict the total final velocity. 4 deeperreason for this is that 6potential energy7 (characteri#ed by the height h) is converted into 6!inetic

energy7 (characteri#ed by the final velocity). %his general concept will be discussed in one of the

following class lectures.



e can now test the following hypothesis:

We expect that velocity is related to height according to the following relation0

v / s+rt(' gh)

%o test this theory, we do not need to collect more data. 8ather, we can use the data collected in

the observation experiment. %he only additional measurement re+uired here is the height from

which the ball bearing was released. We assume that g is constant and !nown through other,

independent methods.

11. easure the vertical distance etween the flat rolling surface and the ottom of the

all at its #oint of release as accurately as #ossile. *his is the initial height h of the

all. How large do you estimate the measurement error to e?

1'. Calculate the #redicted velocity, ased on the hy#othesis #resent aove. Don@t forgetto account for the uncertainty in the measurement that you have included.

8ecipe for 6poor man9s7 error propagation0 If you e.g. measure a height of 'mm

plusminus .:mm, calculate the velocity for h/'mm and for h/'.:mm. %he difference

of the two velocities you get is your velocity error. (or the experts0 ;ote that the relativeerror in the velocity is smaller then the one in height < this is because of the s+uare root

in the e+uation)

1). Com#are the velocity calculated from distance measurements to the velocity

calculated from the height measurement. Do the theoreticallyA#redicted velocity and

the measured velocity agree with one another Bwithin the error ars?

1. hat assum#tions are eing made in the theory? (f these assum#tions are false,

how would you e#ect the #redicted value to change?

! let"s get the ball rolling !

In this course, you will learn about the rotational motion of obects. 4lthough rotation has not

yet been addressed in lecture, we can build an intuitive model for the effect of rotation. =ou can

imagine that, as the ball is moving down the plane, it starts rolling faster and faster because the

friction force is always pushing bac! on the bottom of the ball (but not the top). 3owever, by pushing bac! on the ball, it is also slowing it down a little bit. So we would expect that the

rolling ball will, overall, be moving a bit more slowly at the bottom of the ramp than a ball thathas been sliding without rolling.

=ou can also understand it in terms of 6energy70 %he potential energy is turned into two forms of

!inetic energy < linear motion and rotation. So there is less energy left for the linear motion.

4gain, this will be addressed later in class.



When rotation is incorporated into the theory, we find that the slope still does not impact the

final velocity. or a solid sphere rolling down an incline we find0

v / s+rt((-1>) gh)

1. Calculate the new final velocity that you would e#ect for the all when rolling is

considered. hat is the uncertainty?1. Does the theoreticallyA#redicted velocity and the measured velocity agree with one

another Bwithin the error ars? (f not, what else might we have to consider? Does

this ring is into the right direction?

III. #recision measurement

In +uestion ? you saw that at some point statistical averaging is not a very effective way to

improve your measurement precision. If you want to get -x better precision, you need -,x

the number of measurements. In addition, there might be systematic errors that might forexample depend on the person carrying out the measurement (reaction time) etc. @et9s explore a

much more precise way of measuring velocity < photogates. We use two photogates that are

connected to the computer. %hey consist of a little light emitting diode and a photo detector. Ifthe spaced between the light emitter and the detector is bloc!ed, an electronic signal is generated

that is read out by the computer. %he computer then measures the time difference between the

arrival of the ball at the two photogates.

• "ount two of the gates onto the provided brac!et, and measure their distance as precise

as possible. ;ow, put the gates over the straight path, fairly close to the incline, such that

the ball rolls through both of the gates

18. Carry out )4 measurements and use 9ogger #ro !#age ), see instructions on that#age& to record them. Don@t forget to tell the com#uter the distance etween the

#hotogates you measured. hat does the histogram look like !attach it&? (s the

standard deviation smaller now? How large is it?

1<. How large is the standard error? How many measurements would it now take to get

4.1= #recision?

1>. Com#are the measurement to theory. hat dominates the total errors now, the

random error or the systematic error of the measurement, or the systematic error of

the height determination?

In the e+uations above you saw that the mass never played a role < it divided out. 3ence, you

would expect that the velocity does not depend on the mass of the ball. %ry it out0

'4. "se a glass marle instead of the steel all. "nfortunately you can now not use the

magnet to launch the all $ try whether you can use your fingers instead without

giving the all a kick. ake a few measurement $ what do you get?



4nd finallyA

'1. E7F"G: (f you have a it time left, try out the method with no clock, ust using a

ruler !see 2uestion &.

Have fun0

lab 1 - manual.doc

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