ENGG1300 Rolling Disc Lab (z5225431) PDF

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Following experiment has been designed and conducted to determine the motion of a rolling disk experimentally and theoretically. An inclined plane and a steel disk having two concentric axles have been used for the experimentation. The disk was placed at specific distances from the end point of the ...

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ENGG1300 Lab report: Motion of a Rolling Disk Mehul Kumar Agarwal, z5225431 Allocated Lab Session: Tuesday / 3:00pm

Abstract Following experiment has been designed and conducted to determine the motion of a rolling disk experimentally and theoretically. An inclined plane and a steel disk having two concentric axles have been used for the experimentation. The disk was placed at specific distances from the end point of the inclined plane and released allowing it to roll to the end point. Meanwhile, the time taken by the disk to roll from the specified point to the end point of the inclined plane was recorded. Simplified kinematics equations and summation of moments using the mass moment of inertia formulae has been used to calculate the experimental values and theoretical values. These formulas will further be discussed in details.

1. Introduction For this experiment to have been conducted, a steel disk with two concentric axles has been used along with an inclined plane with a constant slope. Only the distance travelled by the disk was changed. The disk was placed at specific distances from the end point of the inclined plane and released, allowing it to roll to the end point. Meanwhile, the time taken by the disk to roll from the specified point to the end point of the inclined plane was recorded. This step was repeated thrice for each distance considered and the average of 3 recorded timings was taken as the final experimental value, which will further be used for analysis. 2. Motion of a rolling disk experiment The general motion of a body which is rolling down an inclined plane is basically made of two parts: First is the translational motion- in which the body moves as if its entire mass is concentrated at the center of gravity of the body. Second, is the rotational motion, of the body about an axis through the center of gravity of the disk. Formulas will be derived and used, using this basic concept in the theoretical calculation section. 2.1.

Apparatus: Motion of a Rolling Disk experimental setup

Apparatus only involves a steel disk with two concentric axles and an inclined ramp with an adjustable slope as shown in Figure 1. Apparatus. However, the slope remained constant and only the distance along the slope that the disk had to be rolled down upon, changed. Since, the whole inclined plane was at a certain height, so the slope had to be calculated theoretically after taking the measurements of the height.

Motion of a Rolling Disk Lab Report

2019-08-02 Height was measured when 𝑋 = 1𝑚. So, the distance from the left edge of the incline to the base of the triangle will be 230 − 150 = 80𝑚𝑚. Using trigonometry: sin 𝜃 =

𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 80 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 1000

𝜃 = arcsin(0.08) = 4.58° ≈ 4.6°.

Now, the angle is going to remain constant with the X values changing for analysis.

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn Figure 1. Apparatus1

2.2.

Experimental method

Since the slope is fixed, there’s nothing to be changed in the height of the inclined plane. Following is the procedure that needs to be followed to conduct the experiment: 1. Place the steel disk at the starting rest position. 2. Then release the disk and simultaneously start the stop watch. 3. When the disk reaches the end point, stop the stopwatch and record the time. Repeat the above steps 3 times for each values of X which are 0.2𝑚, 0.4𝑚, 0.6𝑚, 0.8𝑚 and 1𝑚. 2.3.

Disk Parameters • • • • •

Large disk diameter 300 𝑚𝑚

Axle diameter 20 𝑚𝑚 (effective) Disk thickness 20 𝑚𝑚

Total mass of disk and axles 11.6 𝑘𝑔

𝑘𝑔

Material: stainless steel. The density of stainless steel is 7700 𝑚3

1 O'Shea, D. (2019). Lab 2 - Motion of a Rolling Disk. School of Civil and Environmental Engineering, UNSW Sydney p. 2

Motion of a Rolling Disk Lab Report 2.4.

2019-08-02

Experimental observations

The first column in Table 1. Time recorded for rolling disk in shows the different distances taken for the disk to roll along the incline. Column 2-4 shows the data collected which is basically the time taken for the disk to roll the specific distances along the incline. Last column shows the average of the 3 trials. Table 1. Time recorded for rolling disk in experiment Distance (m)

Time 1 (s)

Time 2 (s)

Time 3 (s)

Average time (s)

0.2

8.05

7.71

7.97

7.91

0.4

10.73

10.46

10.38

10.52

0.6

12.79

12.95

13.02

12.92

0.8

14.72

15.04

15.20

14.99

1.0

16.26

16.50

16.55

16.44

3. Theoretical calculations

Considering the case as shown in Error! Reference source not found., the disk of radius 150 𝑚𝑚 and mass along with the axle 11.6 𝑘𝑔 rolls down an inclined plane of angle 4.6°. Using these values and Newtons laws of Motion, we will be calculating the theoretical results and further analyzing the experimental ones. 3.1.

Mass moment of inertia

Torque is the ability of a force to produce rotation. It is the product of the linear force and the perpendicular distance between the centre of gravity of the disk and the point of contact of the disk with the incline. This perpendicular distance is basically the radius of the small concentric circle. However, Torque can also be represented by the product of moment of inertia of the disk and its angular acceleration as well. So, 𝜏 = 𝐼𝑜 𝛼 = 𝐹 × 𝑟. So, to come up with the theoretical results, we will be needing the mass moment of inertia.

Since the disk used for this experiment is made up of 3 disks, on a concentric axis. Out of the three disks 2 are identical and small. So, the inertia of the composite disk is: 𝐼𝑡𝑜𝑡𝑎𝑙

𝑚2 𝑟22 𝑚1 𝑟12 𝑚2 𝑟22 𝑚1 𝑟12 2 = 𝑚1 𝑟1 + + + = 2 2 2 2

Volume of the large disk: 𝜋𝑟 2 ℎ = 𝜋 × 0.152 × 0.02

Mass of the large disk: 𝑉𝑜𝑙𝑢𝑚𝑒 × 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 = 𝜋 × 0.152 × 0.02 × 7700 = 10.89 𝑘𝑔 Mass of the each small disk:

11.6−10.89 2

= 0.355

p. 3

Motion of a Rolling Disk Lab Report

𝐼𝑡𝑜𝑡𝑎𝑙 = 𝑚1 𝑟12 +

𝑚2 𝑟22 2

2019-08-02

10.89 × 0.1502 = 0.123 𝐾𝑔 𝑚2 2 = 0.355 × 0.010 + 2

Now to calculate the linear force acting on the disk we will be using the free body diagram of the disk which is shown in Figure 2.

Figure 2. FBD of the rolling disc

There are three forces acting on the disk and they are: 1. Weight Force (𝑚𝑔)

2. Normal Force (𝑚𝑔 cos 𝜃) which is perpendicular to the incline

3. Forward Moving Force (𝑚𝑔 sin 𝜃) which is parallel to the incline

This forward moving force (𝑚𝑔 sin 𝜃) is basically the linear force that is rotating the disk. So, basically 𝐼𝑜 𝛼 = 𝐹 ∗ 𝑟

𝐼𝑜 𝛼 = 𝑚𝑔 sin 𝜃 ∗ 𝑟 3.2.

Calculation of time

As per Kinematics Equations of motion: 𝑋 = 𝑋𝑜 + 𝑢𝑡 + Here, • • • • •

𝑋 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑎𝑡 2 2

𝑋𝑜 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑢 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑡 = 𝑡𝑖𝑚𝑒

𝑎 = 𝑙𝑖𝑛𝑒𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

Since, initial distance and initial velocity is zero.

p. 4

Motion of a Rolling Disk Lab Report

𝑋= 3.3.

Calculation of velocity

𝑎𝑡 2 2

2019-08-02

2𝑋 →𝑡 =√ 𝑎

Change in the kinetic energy of the disk will be equal to the change in its potential energy in accordance with the law of conservation of energy. ∆𝐾𝐸 =

𝑚 (𝑣𝑓 − 𝑣𝑖 )2 𝐼𝑜 (𝜔𝑓 − 𝜔𝑖 )2 𝑚 𝑣𝑓 2 𝐼𝑜 𝜔𝑓 2 + = + 2 2 2 2 ∆𝑃𝐸 = 𝑚𝑔(ℎ𝑓 − ℎ𝑖 ) = 𝑚𝑔 sin 𝜃 ∗ 𝑋

𝑚 𝑣𝑓 2 𝐼𝑜 𝜔𝑓 2 = 𝑚𝑔 sin 𝜃 ∗ 𝑋 + 2 2 𝑣𝑓 2 𝑚 𝑣𝑓 2 𝐼𝑜 ( 𝑟 ) = 𝑚𝑔 sin 𝜃 ∗ 𝑋 + 2 2

∆𝐾𝐸 = ∆𝑃𝐸 →

𝑣𝑓 = √𝑋 ∗ 2 𝑚𝑔 sin 𝜃 . 3.4.

𝑟2 = 0.122√𝑋 𝐼𝑜

Calculation of acceleration

As per the results in section 3.1 we know that:

𝐼𝑜 𝛼 = 𝑚𝑔 sin 𝜃 ∗ 𝑟

So, the angular acceleration is: 𝛼=

𝑚𝑔 sin 𝜃 ∗ 𝑟 = 0.741 𝑟𝑎𝑑 𝑠−2 𝐼𝑜

4. Discussion and conclusion 4.1.

Comparison of experimental and theoretical

From the formulas obtained in the above section we can write distance travelled by the disk in terms of time taken and other known quantities. From section 3.2: 𝑡=√ From section 3.4: 𝛼=

2𝑋 𝑎

𝑚𝑔 sin 𝜃 ∗ 𝑟 𝐼𝑜

Product of angular acceleration and radius of the disk gives linear acceleration: p. 5

Motion of a Rolling Disk Lab Report

𝑎 = 𝛼𝑟 =

So, 𝑡=√

2019-08-02

𝑚𝑔 sin 𝜃 ∗ 𝑟2 𝐼𝑜

2 × 0.123 2𝐼𝑜 2𝑋𝐼𝑜 = 16.4√𝑋 = √𝑋√ = √𝑋√ 2 2 11.6 × 9.81 × sin(4.6) ∗ 0.012 𝑚𝑔 sin 𝜃 ∗ 𝑟 𝑚𝑔 sin 𝜃 ∗ 𝑟 𝑋=(

𝑡 2 ) 16.4

We’ll be using this formulae to calculate the theoretical timing. And then compare it to the experimental timing. Percentage error has been calculated using:

|𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 − 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑇𝑖𝑚𝑒 | × 100 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑇𝑖𝑚𝑒

Table 2. Time recorded for rolling disk (Experimental and Theoretical) Theoretical Time (s)

Experimental Time (s)

Percentage Error

0.2

7.33

7.91

7.9 %

0.4

10.37

10.52

1.5 %

0.6

12.70

12.92

1.7 %

0.8

14.67

14.99

2.2 %

1.0

16.40

16.44

0.2 %

Distance (m)

As per the data presented in Table 2. Time recorded for rolling disk (Experimental and Theoretical) There is a very low discrepancy between the experimental and theoretical values. This might be arising because of the following reasons: •

Friction at the contact surface: Since, the time taken is larger in experimental values, it might be because of the friction between the plane and the axle of the disk.

•

Rusty Surface: Surface of the rolling disk might be rusty that which must have increased the roughness. Since, it has been used by many students, so wearing and tearing of the disk might have effected its geometric conditions.

•

Inclined plane might not be aligned accurately: If the planes are not inclined properly, side of the disk will be sliding against the vertical planes and consequently increasing the friction.

•

Parallax error when taking the reading: Though the error margin rising through this is very small, but it might have contributed to the high discrepancy. p. 6

Motion of a Rolling Disk Lab Report

4.2.

2019-08-02

Conclusion

There is a very low discrepancy between the experimental and theoretical results. Both the data sets agree with the quadratic relation of time and distance for a body rolling down an incline.

Figure 3. Graphical comparison of the experimental and theoretical values

This quadratic relation has been presented as graph in Figure 3. Time, distance, acceleration and velocity have been calculated using the simplified kinematics equations and summation of mass moment of inertia. The small errors might be a result of frictional forces and parallax as mentioned above. From the results above it can be concluded that the angular acceleration was constant, and the velocity increased with the distance increasing. Hence, the relation between acceleration, distance, time, velocity and mass moment of inertia was successfully determined in this experiment neglecting the errors arising, due to friction. References •

O'Shea, D. (2019). Lab 2 - Motion of a Rolling Disk. School of Civil and Environmental Engineering, UNSW Sydney

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