Experiment Investigating Newton's 2nd Law

[Jimmy87]

Homework Statement

We were given two methods to look into Newton's 2nd Law and evaluate them by looking at where sources of error may have come from. They both involved accelerating a car across a table. A plastic track was used to guide the car along a straight path to make sure it went through the light gate. String was attached to the car which was put over a bench pulley and we hung slotted masses to provide the force. We only had one light gate so we used it to measure the time it took the car to go through. We stuck card on the top of the car which was used to break the light beam. We found the velocity by dividing the length of the card on top of the car by the time recorded at the light gate. We started the car 20cm away from the light every time for method 1 (this was 's'). We used the equation below to work out the acceleration. The initial velocity was zero so the acceleration was just v^2/2s. The length of the card on top of the car in method one was 2cm.

The second method was very similar except that the card on top of the light gate started off right at the light gate instead of 20cm away. Therefore the distance 's' was just the length of the card which was 8cm in method 2. We used the time at the light gate and the length of the card to work out the average velocity instead of the final velocity. We then doubled this value to give the final velocity as it was a constant acceleration that started at rest. I just wanted some pointers on why the results were all over the place for method 1 but not for method 2. For both experiments we varied the masses on the end so we had a set of 5 different masses and 5 different accelerations. We used this data to find the mass of the car we accelerated. We measured the true mass afterwards which was 40 grams. Using method 1, all five values for the mass were between 130-260 grams! Using method 2 they were between 25-51 grams. Why is method 1 so awful?

Homework Equations

v^2 - u^2 = 2as

The Attempt at a Solution

The only thing I can think of is that there is a bigger distance for drag to slow down the car in method 1? Or maybe the length of the card in method 1 (2cm) is too big to give an accurate value for the final velocity which we should try to get as close to an instantaneous value as possible? This is all I can think of. The cars did accelerate very nicely and smoothly in both methods so I cannot see why method 1 gives such bad results. Here is one set of results for method 1 and 2:

Method 1:
Distance from car to light gate ('s') = 20cm
Length of card = 2cm
Time at light gate = 0.021s
Final velocity = 0.95 m/s
Acceleration = 0.95^2 / 0.4 = 2.26
Mass on hanger = 50g = 0.49N
Mass of car = 0.49/2.26 = 216g (true mass in both was 40g)

Method 2:
Length of card = 8cm
Distance ('s') = 8cm (as started right at the beam)
Time at light gate: 0.137s
Average velocity = 0.08/0.137 = 0.58 m/s
Final velocity = 0.58 x 2 = 1.16 m/s
Acceleration = 1.16^2 / 0.08 = 16.82
Mass on hanger = 50g = 0.49N
Mass of car = 0.49/16.82 = 29g (true mass in both was 40g)

As you can see method one gives a much smaller acceleration even though the force etc is the same as method 2.

Thanks for any help

 

[CWatters]

Have a think about the accuracy with which the light beam switch can detect the position of the edge(s) of the card. If the accuracy was constant (say +/- 5mm) for both experiments would that effect the accuracy of the two experiments in the same way?

Is the accuracy of the light switch the same for both transitions (eg air to card and card to air)?

 

[Jimmy87]

Have a think about the accuracy with which the light beam switch can detect the position of the edge(s) of the card. If the accuracy was constant (say +/- 5mm) for both experiments would that effect the accuracy of the two experiments in the same way?

Is the accuracy of the light switch the same for both transitions (eg air to card and card to air)?

Ok, so are you saying that +/- 5mm is quite a big error when the length of the card is small (25%)? Would the length of the card need to be much smaller if the first method is going to work? Does it not have much to do with drag then?

 

[andrevdh]

The accelerating force on the car is not equal to the weight on the hanger.
It is reduced by the tension in the connecting string between the car and the hanger.

 

[CWatters]

Ok, so are you saying that +/- 5mm is quite a big error when the length of the card is small (25%)?

Yes.

Would the length of the card need to be much smaller if the first method is going to work?

in the OP #1 you said...

We found the velocity by dividing the length of the card on top of the car by the time recorded at the light gate.

Making the card smaller would increase the error not reduce it.

Think about what happens if the error is different for one edge of the card compared to the other.

 

[CWatters]

Method 1:
Distance from car to light gate ('s') = 20cm
Length of card = 2cm
Time at light gate = 0.021s
Final velocity = 0.95 m/s
Acceleration = 0.95^2 / 0.4 = 2.26
Mass on hanger = 50g = 0.49N
Mass of car = 0.49/2.26 = 216g (true mass in both was 40g)

As Andrevdh points out that's not correct. See...



Even if you correct this mistake I think you still get a value for the mass of the cart that is too high (I made it about 160g).

Note that any error in the light switch detecting the edges of the card affects V which is then squared when working out the acceleration.

Sorry, had to edit this post to correct a mistake of my own. Being interrupted here.

 

[Jimmy87]

As Andrevdh points out that's not correct. See...



Even if you correct this mistake I think you still get a value for the mass of the cart that is too high (I made it about 160g).

Note that any error in the light switch detecting the edges of the card affects V which is then squared when working out the acceleration.

Sorry, had to edit this post to correct a mistake of my own. Being interrupted here.

Thanks guys, very helpful. Took me a while to read/watch video to understand. I get why method 1 is not accurate now. Why was method 2 accurate though? Surely you still have the light gate and the edge of the card problem? I know the card is bigger (4 times), would this really account for the much better accuracy? If I repeated method one but made the card 8cm like method 2 to calculate the final velocity would that give just as accurate values? But would this then not be a very good indication of the final velocity because the distance 's' is 20cm but I would actually be measuring a final velocity which is more like an average velocity since 8cm quite a lot in relation to 20cm?

Thanks again to all!

 

[andrevdh]

Something seems to be seriously "off" with your results.
The acceleration for the two methods should not differ by much when the mass on the hanger is the same for both. For example in method 1 above your acceleration is 2.26 m/s² and in method 2 it is 16.8 m/s² while the mass of the hanger was 50 grams for both.

How did the program report the "time at light gate" or how did you determine this measurement?
Didn't it have a "time in light gate" measurement.

 

[CWatters]

As a check let's calculate what the acceleration should be without any friction. With reference to that video..

a = M_hg / (M_h + M_c)
where M_h is the mass of the hanger and M_c is the mass of the cart.

= 50*9.81/(50+40) = 5.4 m/s²

So I agree the experimental results appear to some way out.

 

[CWatters]

Method 2:
Length of card = 8cm
Distance ('s') = 8cm (as started right at the beam)
Time at light gate: 0.137s
Average velocity = 0.08/0.137 = 0.58 m/s
Final velocity = 0.58 x 2 = 1.16 m/s
Acceleration = 1.16^2 / 0.08 = 16.82

If you use the SUVAT equation to work out the acceleration..

s = ut + 0.5at²

then because u =0

a = 2S/t²
= 2*0.08/0.137²
= 8.52 m/s²

which is roughly half the value you calculated but still some way off the theoretical value of 5.4m/s.

Edit: Error cannot be due to friction as that would reduce the acceleration not increase it. Does the 50g on the hanger include the mass of the hanger?

 

[CWatters]

Regarding Method 1..

If the acceleration had been the theoretical value of 5.4m/s that I calculated in post #9 then using method 1 the velocity at the light beam should have been...

v² = 2as

v = SQRT(2as)

= SQRT(2*5.4*0.2) = 1.47m/s

Time at light gate = 0.021s

If we also assume the timer data is correct then the effective length of the card in the experiment was about...

d = v*t = 1.47 * 0.21 = 3cm.

So was the card 3cm wide rather than 2cm? If the card was 2cm and the light beam error was +5mm on one edge and -5mm on the other that could appear account for the discrepancy in method 1.

 

[andrevdh]

Also the 2nd acceleration can't be correct since the maximum acceleration attainable is 9.8 m/s²

The light gate switches between 5 volt and 0 volt when something enters (blocks) and exits (unblocks) the gate.
So either the signal is zero volt while the card moves through the gate and is high before and after or vice versa.
You need the time that the card is in the gate for your calculations.
So how did you determine this time?

 

[CWatters]

You need the time that the card is in the gate for your calculations.

I assumed that's what he measured.

 

[andrevdh]

just wanted to make sure that is not where the problem lies

 

[Jimmy87]

If you use the SUVAT equation to work out the acceleration..

s = ut + 0.5at²

then because u =0

a = 2S/t²
= 2*0.08/0.137²
= 8.52 m/s²

which is roughly half the value you calculated but still some way off the theoretical value of 5.4m/s.

Edit: Error cannot be due to friction as that would reduce the acceleration not increase it. Does the 50g on the hanger include the mass of the hanger?

Hi guys, sorry for delay in response - I have had internet issues changing broadband. The 50g definitely includes the mass of the hanger. The time I measured was definitely the time at the light gate for both i.e. the timer starts when the front of the card goes through and stops when it leaves. Used the same light gate for both so really don;t get the discrepancy! So from what I have read is method 1 bad because there is such a large uncertainty when the width is only 2cm? If i changed the length of the card in method 1 to 8cm should I get better results?

 

[haruspex]

I considered the fact that the string inevitably will have some elasticity.
If my algebra is right, this will lead to an initial acceleration of $g\frac{M}{M+m}\frac{M+2m}{m}$, instead of $g\frac{M}{M+m}$, where M is the suspended mass and m is the mass of the car. With the 40g and 50g masses, that's (65/36)g instead of (5/9)g.
It may seem surprising that this is independent of the spring constant, but the duration of the excess acceleration will be very short, and the stiffer the spring the shorter its duration. E.g. if the initial stretch (before release) is 1mm then the deviation in the car's later position should never be more than 1 or 2mm.

Clearly the true acceleration should be about 5.4m/s² in each method. Method 1 gets half that while method 2 (corrected to 8.52m/s²) gets 1.6 times it. Elasticity might go some way to explaining the 8.52, but it seems unlikely to go far.

 

[andrevdh]

Referring firstly to method 1

A better way to analyze your data is to draw a graph of the acceleration of the cart (y-axis) against the tension in the string (x-axis).
The graph should be directly proportional if Newton's second law holds and the inverse of the gradient of the graph should then be the cart's mass.

You should also make sure that the card was 20 cm away from the light gate. That is if the card was in the middle of the cart and the front of the cart was 20cm away from the light gate it would mean that the card was actually further away.

 

[CWatters]

So from what I have read is method 1 bad because there is such a large uncertainty when the width is only 2cm? If i changed the length of the card in method 1 to 8cm should I get better results?

Problem then is that 8cm is large relative to the 20cm that the cart moves. So the velocity of the cart will be changing significantly while the 8cm card is passing the light gate.

Have you considered calibrating the light gate using the 2cm card? I would push the cart very slowly past the light beam and mark on the table the position of the cart where the timer starts and stops to measure how far the cart moves while the timer is running. It should be the same as the width of the card (eg 2cm). If it isn't then redo the calculations using the actual distance measured.