cart on inclined plane

FIRST YEAR PHYSICS LABORATORY MANUAL 35

CART ON AN INCLINED PLANE


Introduction

The purpose of this experiment is to find the mass of a cart and the coefficient of static friction for the cart on an inclined plane.

The exercise is designed to give you an understanding of:

What to do before you get to lab

1) Read and make notes on the following preliminary information2) Review the relevant section of your textbook (Chapter 5) for a more detailed account of the theory3) Complete the preliminary problems for this exercise on Moodle4) Read through the rest of this exercise so that you will know what to do in the laboratory5) Watch the video about this experiment, the link can be found on Moodle


36 FIRST YEAR PHYSICS LABORATORY MANUAL

Preliminary Problems

Background

Imagine a trolley of mass M and ramp set up as shown in the diagram below.

Making the assumption that the cart moves up the slope with acceleration a+, as shown in the diagram, we can use Newton’s second law to write two equations of motion for this system: ma+�� Ma+��

These equations can be solved simultaneously to give (make sure you can do this!):

If the cart is moving down the slope then the frictional force will act in the opposite direction. Newton’s second law for the masses become (in this case the cart is moving down the slope in the negative direction):

ma� = mg � T Ma� = T ��M g sin � + M g �s cos �

This can be solved to give:

Mg

Ff

T

T

mg

a+

N

a+ = �� g

M + m(1)

a� =�� g (2)

M + m



Your task is to calculate �s and M, the mass of the cart and block placed on the cart. You will measure a+ and a�. During the experiment you will vary the hanging mass, m. You may assume g = 9.797ms�2.



Equipment

For your work in the laboratory you will need the following

motion sensor

You will also need to use one of the computers.

In the Laboratory

Connect up your LabPro interface by plugging the USB cable into the USB port of one of the iMac computers. The USB port is on the right hand side of the computer, right near where the keyboard plugs in. Connect up the power of the LabPro interface as well. Connect the lead from the motion sensor into “DIG SONIC 1”.

The motion sensor works by sending out short bursts of sound. The sound reflects of the cart and this echo is analysed by the sensor. The distance between the bursts and the time it takes the compete burst to return are used with the speed of sound to calculate the velocity of and distance to the object.

Any objects (such as people) moving in front of the detector will also be detected and will return spurious readings. Please make sure that there is no one moving in front of your sensor when you take measurements. It can detect movement up to 6 meters away.

There is a template file on the computer for this experiment called “Cart on an inclined Plane”. You will need to open this to record data during this experiment. By selecting “Data Collection” from the “Experiment” menu ensure that you are recording 30 measurements every second.

DIG/SONIC

1



The string should be tied to the cart and the masses. Thread it over the pulley. Place the motion sensor at the bottom of the inclined plane. Choose a small angle for your track, around 2o or 3o, (this is important as it allows you to make the assumption later that sin�!0). Place the black unknown mass on the cart. Do not change the angle during any part of the experiment. The cart should roll down the track, if it does not you will need to increase the angle. The equipment should be set up as shown in the photos below.

You will now need to collect data to calculate a+ and a�. To start with find the smallest carrier mass, m, that will cause the trolley to move up the ramp. Use a mass 5g greater than this mass. Release the mass from rest and start the motion sensor. The acceleration should be constant. Acceleration can be calculated from the gradient on the velocity time graph. Use the “Linear Fit” tool to help you. Make sure that you select an appropriate part of your graph to calculate the acceleration from. Ask your demonstrator if you are unsure about

black unknown mass



how to do this. Your screen should look similar to the screenshot below.

You will need to repeat this at least twice. Remember that the mass carrier has a mass of 5g, this needs to be included in m. Record the information in the table below, include uncertainties (for the mean):

Angle, � : Hanging mass, m :

Calculate the uncertainty in this measurement. You may assume that the percentage uncertainty is the same for all accelerations up the slope.

Now keep the angle, �, constant and repeat this experiment for at least another four values of a+, make sure you include uncertainties. Note that uncertainties can be fitted into the cells in the table next to the values. To change a+ adjust the mass, increasing the mass in increments of 5g works well. You will be given a mark for accuracy in this experiment so the more values you collect the better!

Acceleration, a+ (ms-2)Trial 1

Trail 2

Trial 3

Average



m (g) a+ (ms-2)

You will now perform the experiment with smaller hanging masses in order to find a� . Do not change �. Start with 2g less than the largest mass for which the cart will move down the slope.

Hanging mass, m :

Acceleration, a!, (ms-2)

Trial 1

Trial 2

Trial 3

Average

Slowly decrease the mass (in increments of 2g), recording the values of a� for each mass. Complete the table below, include uncertainties. Remember to record the negative sign in front on you values for a�. You can assume all a! values have the same percentage uncertainty.



m (g) a� (ms-2)

You have now finished collecting data and it is time to analyse the results. Open the excel file called “Cart on an inclined plane graph”. This file will perform a least squares fit on the data to produce two straight lines, one for a+ and one for a�.

In the space below record the equations for the line a+ and a�.

a+: y = y = y =

a�: y = y = y =

Let m+ be the mass when a+ = 0 (the x-intercept of the a+ line) and m� be the mass when a� = 0 (the x-intercept of the a� line). Use the data you have recorded above to calculate m+ and m� with uncertainties.

Hint: What is the value of y for the x intercept? Insert this value into the equations above to find x. You can use the three values to calculate the uncertainty.



In the space below use equations (1) and (2), in the background information to show that:

m+�� m+ + m��

Use your values for m+ and m� to calculate M (mass of cart + block) and then �s. You should comment on the size of the uncertainties however you do not need to calculate them as this is very involved due to the sin� term. For details, look at the notes on calculating uncertainties for this experiment, which can be downloaded from Moodle.

It is also possible to calculate M in another way. If you take the derivative of equation (1) you get:

This is the gradient of the a+ line on your graph. This equation looks complicated but we can make some simplifications. The mass m is much smaller than M and � is small. Show that we can simplify this to:

da+ =g M

�� )dm ( m + M )2

da+ !g�� )dm M



Likewise from equation (2) we can obtain:

We can use these equations to get:

Use your gradients to calculate M and �s. Include uncertainties. Comment on how the size of the uncertainties compared to your value for �s.

da+ �da� =

2g��dm dm M

da+ +da� =

2gdm dm M

da� !g�� )dm M



Now weigh your cart. Record the value of M.M =

Which of these methods do you think is most accurate? Why?

Did you achieve the aim of this experiment? Explain. If not what could be changed?



Marking Guidelines

Get your demonstrator to tick these boxes as you go:

Recorded first piece of data for a+, and calculated uncertainty, units included

Completed table for a+ with uncertainties

Completed table for a� with uncertainties

Derived results using m+ and m�

Calculated m+ and m� and used to calculate M and �s

Derived result for da+/dm

Calculated M and �s with uncertainties and units

Answered questions intelligently

M lies within error bars of M calculated in experiment and these error bars are calculated correctly i.e. Accurate measurements

Finished and packed up 20 minutes before end of lab session

Total: /10

Demonstrator’s signature:______________________________

Student Name: _________________________________________

Student ID Number: z_______________________

cart on inclined plane

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