An investigation into the relationship between the force applied on a cantilever and its deflection

I aim to investigate the relationship between the force applied on a cantilever and its deflection.

Research – A cantilever is a projecting structure such as a beam, which is supported at one end and carries a load at the other end or along its length.

To measure the deflection, I will place a ruler, vertically at the end of the cantilever with the force applied. I will record the amount of deflection with no force (only gravity) and calculate the difference.

Before I did the experiment, I carried out some preliminary work so that I could find out the best length of the cantilever, material of the cantilever, and maximum and minimum force applied to the cantilever.

I learnt from my preliminary experiment what my ranges should be and how many results I would need to prove my prediction. Also I learned how to set up my apparatus and how big or small my overhang should be.

I chose to use a meter ruler for my cantilever, as it is a widely available object to use and it is already marked in measurements.

I will use a wooden meter ruler, as it can take more force that a plastic one, I know this from my preliminary work.

I will use a maximum load of 7 Newtons (700 grams) because the ruler will only take a maximum of 7 Newtons (700 grams) I also know this from my preliminary work.

The variables in this investigation are the mass of the ruler, the length of the ruler (overhang), thickness of the ruler, material of the ruler, position of force on the ruler and the amount of force exerted on the ruler.

I have chosen to use the amount of force exerted on the ruler as a variable, therefore I must keep the position of the weights on the ruler the same so that the weight is evenly distributed the same all the time, the material of the ruler the same because a ruler of different material may deflect more or less and affect my results, I will keep the thickness of the ruler the same because a thicker ruler will deflect less and a thinner ruler will deflect more, the length of the ruler the same because a longer ruler would have the weight distributed over a bigger length and a shorter ruler would have the same weight over a smaller distance. Also I shall keep the mass of the ruler the same..

The outcome variable that I am going to measure in this investigation will be the deflection of the ruler in mm, I will do this by looking at the reading horizontal to it so that the reading are not affected by the angle of which I read them from.

The equipment I will need for my experiment will be;

– Two meter rulers

– One G-clamp

– One set of weights measured in intervals of 1 N (100 grams)

– String

These are the methods I will use to obtain my results.

1. I will clamp the ruler to a secure hangover E.g.: a table edge, using the G-clamp.

2. Next, I will place a ruler vertically next to the ruler acting as a cantilever so that I can read the measurements clearly against it.

3. Then, I will record the deflection of the ruler acting as a cantilever against the vertical ruler.

4. I will add a force of 1 N (100 grams) to the end of the cantilever, by hanging on the string tied to the end of the ruler, and record the deflection of the ruler acting as a cantilever

5. I will then continue to repeat step 4 adding a force of 1 N each time until I reach a total of 7 N.

This is a diagram of how I will set up my apparatus.

To make my investigation safe, I will take care whilst handling the weights not too drop them and cause injury, I will make sure that I only use a maximum amount of weights which will not snap the ruler. I will do the investigation in a safe place where people are not trying to move around.

To make my investigation accurate, I will measure my results in mm, check that all the 1 N weights are accurate by weighing them, and use an accurate way of measuring the deflection.

To make my results reliable, I will repeat the investigation three times and take an average of them. When I have drawn my graphs, I will justify and explain them.

Prediction – I predict that when the force on the ruler is doubled, the deflection of the ruler will double. Hopefully you will be able to see this from my graph; by this I state that the cantilever will obey Hook’s law. I predict that there will be a definite increase with larger turning effect (moment) when more force is added. The amount of deflection will be directly proportionate to the amount of force applied. Also, I believe that doubling the force, will double the deflection, my reasoning for this are that the amount of force applied to the ruler will push it down due to gravitational force which is being applied. This will cause the ruler to bend and stretch due to its elasticity.

In the experiment, when force is applied to the cantilever the top side of the ruler, will become stretched, its molecules wider apart whereas the bottom side of the ruler will be compressed with its molecules closer together.

Earlier work that I have done with springs, shows that when something is under a force which makes that compress or extend they will obey ‘Hook’s law’ which states that deflection is directly related to the force, and that when a force is doubled, the compression or extension or in our case deflection will also be doubled.

Sources of information – I used information from our physics lessons, from our physics textbook “physics for you” and from the Hutchinson encyclopaedia pg. 762.

Results –

Set 1

Force applied (N)

Deflection (mm)

Difference (mm)

0.00

168

0

1.00

203

035.00

2.00

233

065.00

3.00

267

099.00

4.00

282

114.00

5.00

316

148.00

6.00

331

163.00

7.00

365

197.00

Set 2

Force applied (N)

Deflection (mm)

Difference (mm)

0.00

172

0

1.00

198

026.00

2.00

224

052.00

3.00

251

079.00

4.00

276

104.00

5.00

302

130.00

6.00

326

154.00

7.00

350

178.00

Set 3

Force applied (N)

Deflection (mm)

Difference (mm)

0.00

174

0

1.00

200

026.00

2.00

226

052.00

3.00

251

077.00

4.00

275

101.00

5.00

300

126.00

6.00

325

153.00

7.00

350

176.00

Averages

Force applied (N)

Deflection (mm)

Difference (mm)

0.00

171.33

0

1.00

200.33

029.00

2.00

227.33

056.00

3.00

256.33

085.00

4.00

277.66

106.33

5.00

306.00

134.67

6.00

327.33

156.00

7.00

355.00

183.67

Analysis – From my graph, I can see that the force is directly related to the deflection. I can tell this because the graph is in a straight line.

Also from my graph, I can see that when the force is doubled the deflection is doubled. I can see this because when I drew a straight line from 2 N up to the line of best fit, it read 50 mm deflection then when I read up from 4 N the deflection was 100 mm. This means that my results agree with hook’s law.

Stiffness constant – K = F (N)/D (mm)

So F = K D

K = 1/22.33 = 0.04 N/mm K = 0.04 at 1N

K = 2/49.67 = 0.04 N/mm K = 0.04 at 2N

K = 3/78.33 = 0.04 N/mm K = 0.04 at 3N

K = 4/99.67 = 0.04 N/mm K = 0.04 at 4N

K = 5/128.0 = 0.04 N/mm K = 0.04 at 5N

K = 6/149.33 = 0.04 N/mm K = 0.04 at 6N

K = 7/177.00 = 0.04 N/mm K = 0.04 at 7N

This proves that my prediction is right, because the constant stays the same all the way through.

Evaluation –

My procedure was good enough to make my measurements reliable as I got another person to read them as well as me.

I can tell that my results are reliable as my line of best fit is a straight one which passes through the point 0, 0 and all of the points lie close to the line of best fit. I can tell that they are also reliable as they graph agrees with Hook’s law.

I did not have any anomalous results in my investigation as all of my results fit in with my pattern.

From the above, I believe that my results were accurate enough, although if I was to do the investigation again I would take more results at more regular intervals. E.g. Every 0.5 N so that my results were even more accurate because I would be taking more results.

My graph shows me what I needed to see, but if I had more results, I would be able to see a better line. This would allow me to get a better conclusion of my results. From my graph I can see that when the force is doubled so is the deflection.

Overall, I believe that my investigation was conclusive enough for what I wanted to know but could have been more accurate if I wanted it to be. My results show that the deflection is directly related to the Force, and that when the force is doubled the deflection is doubled this agrees with Hook’s law.