# Worksheet

## Inclined Plane

#### Introduction

In this practical you are going to investigate the motion of a trolley rolling down and inclined plane. Since the forces on the trolley are unbalanced it will accelerate down the slope at a constant rate, this means that its velocity, v will increase with increasing displacement, s down the slope. Since the trolley has uniform acceleration the usual equations apply (suvat) . In this case we are going to use the relationship between v^{2}=u^{2} + 2as to find the acceleration from the displacement and velocity. This can be simplified to v^{2} = 2as since the initial velocity is zero.

#### Research Question

*How does the velocity of the trolley depend upon the distance travelled down the slope?*

**Independent variable**: The distance down the slope

**Dependent variable**: The final velocity

**Controlled variables**: The angle of the slope, the surface of the slope, initial velocity.

#### Method

To measure the velocity of the trolley a photogate is going to be used. A photogate sends a signal to the computer every time an object breaks the beam and the computer records the times when it happens. So if a card of known length,d passes through the photogate in time t the velocity of the card is d/t, in this way the velocity of the trolley can be measured at different distances down the track. Set up the apparatus as in the diagram using a book to make the track inclined. Connect the photogate to the computer as you have been instructed previously, the photogate has several different timing options, choose "time in gate".

Show that the velocity and displacement are related by the equation v^{2} = 2as and use a graphical method to find the acceleration of the trolley (**a)**. Compare your result to that found by measuring the angle of the track.

This is all the information I would give if the practical was being used for assessment however since this topic is very early in the year student's are not quite ready to do the whole thing on their own, in this case I use this practical as practice and give them the following instructions:

- Set the photogate somewhere near the end of the track so that the marker goes through the photogate before the trolley leaves the track, make sure the photogate is adjusted to the correct height so the marker cuts the beam, there is a light on the photogate that indicates when the beam is broken.
- Start the interface software and perform a trial run to make sure everything is working.
- Measure the distance between the starting point of the trolley (the middle of the card) and the photogate.
- Send the trolley down the slope and record the time taken for the card to pass.
- Enter your results in an excel table like the one below, then repeat the procedure 4 more times.

- Repeat the procedure for 5 different distances filling in the table as you go.
- Estimate the uncertainty in your measurements, write your estimates in the table headers. You should explain in your report how you got these values.

#### Processing the Data

The first step in processing data is to get a more realistic value for the measurement of your dependent variable (time in gate). Your estimated value is based on the precision of the photogate and timer software, however your method will probably have introduced a bigger uncertainty than this. This can be found from the spread of data in other words 1/2 the difference between the maximum and the minimum values in a row of data.

- Add two columns to you table as shown below.
- Add equations to these colums to calculate the average value of the time =AVERAGE(B2:F2) and the the uncertainty in the values =(MAX(B2:F2)-MIN(B2:F2))/2 . The formulas can be copied into all the cells by dragging the bottom right hand corner of the cell downwards.

Now it is time to calculate the velocity by dividing the length of the card by the time taken for the card to pass through the gate, the PASCO card is 2.5cm long so this length will be used in this example.

- Make three more columns in your excel table as shown below.
- In the average velocity column insert the formula =0.025/G2
- To calculate the maximum velocity you take the distance/min time =0.025/MIN(B2:F2)
- To calculate the minimum velocity you take distance/max time =0.025/MAX(B2:F2)

As mentioned before the velocity and displacement are related by the equation v^{2}=2as so a graph of v^{2} against s will give a straight line graph with gradient 2a. Before you can plot this graph you need to calculate v^{2} and its uncertainty.

- Add two more columns to the table as shown below.
- Insert a formula to calculate (average velocity)
^{2}=I2*I2 - Insert a formula to calculate the uncertainty in v
^{2}, this is found from ((Max v)^{2}- (Min v)^{2})/2 the formula in this case is =(J2*J2-K2*K2)/2

#### Presenting Processed Data

Now you are ready to draw a graph of v^{2} against s.

- Copy the distance and (average velocity)
^{2}columns into the graphing programme (LoggerPro). - Make a new column in and copy the uncertainty in v
^{2}column into it. - Add error bars to the graph as you have done before.
- Plot the best fit line automatically and the steepest and least steep lines manually.
- Find the acceleration from the best fit line and the uncertainty in this value from the steepest and least steep lines.

#### Conclusion and Evaluation

- Was the acceleration uniform?
- Compare your answer with the value obtained by measuring the angle of the slope (a=gsinø), is your value bigger or smaller? Explain why this might be.
- Comment on the spread of data on the graph compared to the error bars.
- Comment on the intercept of the line.
- According to your results what were the weaknesses in your method
- Suggest ways of addressing the weaknesses

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