Съдържание

Условие Part A: Levelling the base and ensuring the stand is exactly vertical. (1 т.)Part B: Effect of Spinning Speed (6,5 т.)Part C: Influence of Gyroscope Arm Length (2,1 т.)Part D: Influence of Gyroscope Disk weight (3,7 т.)Part E: Torque Induced by External Forces (3,5 т.)Part F: Nutation phenomenon (1,5 т.)Part G: Application of gyroscope in self balancing (1,7 т.)Решение

APhO 2024, Експеримент — Задача 1. Gyroscope experiment

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от APhO 2024

APhO 2024 — Gyroscope Experiment, English (Official) · 20 т.

Внимание: Бележка към темата

The official question and official marking scheme are reproduced with their printed inconsistencies retained. Instruction 3 on question page 2 has no text. The installation heading says “26 mm disk”; the apparatus table and later tasks say 26 cm. The marking scheme uses 20 mm/26 mm in A.1 and 20 mm in G.1. Table 1 labels current in A but prints mA entries. B.2 is labelled 0.8 points although its itemised marks total 0.7; B.3 asks for an exponential trendline in its marking criteria but displays a linear one. B.6(ii) uses 65 g instead of the 89 g specified for the requested 3 mm disk. B.6(iv) uses a 25 cm arm instead of the procedure's 15 cm, and prints rad⁻¹s⁻¹. Its geometry also prints L=L sin φ without distinguishing the two quantities. D.3 marks “faster” although its sample graph decreases with disk thickness. D.4 defines R as radius but substitutes 0.2 m, and again uses 25 cm. E.2's plotted ordinate is labelled Speed (rpm) although the sample data are in rad/s. Part F is headed 1.5 points in the question and 2.1 in the key; its printed questions sum to 1.5. The key's reduced Euler equations use minus signs where the question's equations use plus signs. No values or marking points have been silently replaced. The archive TXT companion only points to a Russian solution at https://pho.rs/p/4072/m; it is an archive pointer, not a second supplied solution text. That remote text was not available in this source packet.

Условие

Introduction

Figure 1 shows a gyroscope. A gyroscope is a device used to measure or maintain orientation and angular velocity. It consists of a spinning wheel mounted in a gimbal so that its axis can freely point in any direction. When rotating, the orientation of the axis is unaffected by tilting or rotation of the mounting, according to the conservation of angular momentum.

Gyroscope illustration labelled Gyroscope Frame, Spin Axis, Gimbal and Spinning wheel. — Figure 1. Gyroscope

The physics behind a gyroscope is based on the principle of conservation of angular momentum, which states that the angular momentum of a system remains constant if no external torques are applied to it. In a gyroscope, the spinning wheel creates a large angular momentum. When an external force attempts to turn the gyroscope's axis, the gyroscope resists this change in direction due to its angular momentum. This behavior is like a top (or whipping top) that spins around its axis, balancing due to gyroscopic effect, which stabilizes its tilt.

The behavior of gyroscopes is described by the equations of motion derived from Euler's equations for rigid body rotation. These equations account for the torques and rotational speeds and explain how and why the gyroscope behaves as it does under various conditions. This makes gyroscopes critical in applications like navigation systems for aircraft, spacecraft, and ships, where precise orientation information is crucial. Besides that, auto balancing vehicles, such as self-balancing scooters and advanced robotics incorporate gyroscopes and accelerometers to dynamically adjust and maintain the balance.

Experiment Instruction

Place the antistatic map on the table.
Screw the three star knobs to the bottom of the A shape retort stand and hand knob on the side as shown in Figure 1.

Photograph of the underside of the A shape retort stand with three star knobs and the side hand knob.

Place the A shape retort stand on the antistatic map which use to reduce the vibration and noise.
Insert the gyroscope main vertical stand into the retort stand.
Insert the aluminum tube through the center shaft and lock it tightly with star knob provided. Your setup should looks like Figure 2.

Photograph of the main vertical stand, centre shaft, aluminium tube and wiring on the retort stand.

Connect the wire jack at the bottom to the adjustable power adapter as shown in Figure 3.

Photograph of the wire jack at the base connected to the adjustable power adapter.

If you plan to change the polarity of the supply to the motor, you can always swap the wire at this connection jack using screw driver provided.

Installation of motor

Screw the motor into the aluminum tube and connect the DC jack of the motor to the DC jack on the vertical stand as shown in Figure 4.

Photograph of the motor installed on the aluminium tube with its DC jack connection.

The coupler can attach to the disk using the M3 bolts and nuts given as shown in Figure 5.

Both printed photographs of the disk coupler attached using M3 bolts and nuts.

The coupler then can install on the motor shaft and lock using the lock screw from side as shown in Figure 6.

Photograph of the coupler installed on the motor shaft and secured with the side lock screw.

Installation of 26 mm disk

Sandwich the 26 mm disk with 2 washers.
Lock it to the aluminum tube using a M6 screw provided as shown in Figure 7.

Photograph of the 26 mm disk, as named in this instruction, sandwiched between washers and attached using an M6 screw.

Equipments/Materials:

Equipment/materials
Gyroscope main vertical stand x1
Adjustable power adapter x1
DC motor with case and coupler x1
20cm spinning disk with thickness 2mm, 3mm, and 4mm. 26cm counter weight disk.
Goniometer (or angle ruler) x1
Stopwatch
Antistatic map
Aluminium tube
Retort stand
Bolts and nuts, Allen keys, screwdriver, wire tie, cable tie double sided tape, ruler, motor shaft coupler,

Complete equipment table with all photographs and labels, including stand, adapter, motor, disks, goniometer, stopwatch, mat, tube, retort stand and small tools.

To mount the disk to the DC motor, you need to connect and tighten it to the coupler.

Coupler and Allen key illustration, with red arrow labelled To tighten the shaft.

Mass of components

20 cm (diameter) x 2 mm (thickness) spinning disk → 65±1 g

20 cm (diameter) x 3 mm (thickness) spinning disk → 89±1 g

20 cm (diameter) x 4 mm (thickness) spinning disk → 131±1 g

26 cm (diameter) x 4 mm (thickness) spinning disk → 231±1 g

DC motor with casing → 94 g

The mass of the aluminium tube can be neglected throughout the whole experiment.

Precautious!!!

Before turning on the Adjustable power adapter, make sure that the knob is turned OFF. High voltage supply to the DC motor will burn the motor. Limit the driving voltage below 6 volts. Driving above 6 volts might cause the motor to overheat and burn. Candidates will need to do the soldering and replacement of the DC motor as the penalty of not operating properly.
Do not leave the setup continuously running for a long period of time (10 minutes and above). Leave the motor to rest after each experiment.
Hold the retort stand when the experiment starts or when the motor is running until it is stable. The experiment will experience strong vibration.
Note that the spinning disk can pose a safety hazard, do not touch the spinning disk with bare hands when it is spinning, switch off the power and wait for it to come to a complete stop.
The wire connection of the DC motor is thin, please handle the connection with extremely gently.
The 3D printed DC motor holder is fragile, do not use brute force to adjust these 3D printed part. You might break the part.

Part A: Levelling the base and ensuring the stand is exactly vertical. (1 т.)

You are provided with a retort stand with a leveling function. To conduct the experiment, you will need to level the gyroscope test kit until the vertical pole is perfectly vertical without any leveling device. You may use all the apparatus provided to achieve the purpose.

*Take note that leveling this set-up is crucial to all subsequent parts of the exam, ensure you do this carefully and maintain the position after leveling.

A.1. Sketch your setup to show how you could level the gyroscope test kit without a leveling device, label the parts in your diagram. Label all the parts with detailed information. [0,6 т.]

A.2. Label the force that interact in the setup. [0,2 т.]

A.3. Which of the following two factors will allow you to most precisely determine whether the stand is vertical?

(a) Equal length of the arm (b) Unequal length of the arm (c) Size of the disk (d) Thickness of the disk (e) Weight of the disk (f) Weight of the DC motor (g) Voltage applied to the DC motor [0,2 т.]

Part B: Effect of Spinning Speed (6,5 т.)

Objective: To investigate how the angular velocity of the gyroscope affects its stability and precession period.

Materials: Gyroscope test kit, adjustable power adapter, DC motor, goniometer, stopwatch, 20 cm (3 mm thick) spinning disk, ruler.

For this part of the experiment, the arm should not be balanced and allowed to hang freely.

Photograph of the freely hanging, unbalanced gyroscope arm with the 20 cm spinning disk.

Procedure:

Fix the 20 cm diameter spinning disk with 3mm thickness on the DC motor and mount it on one side of the gyroscope arm in perpendicular direction as shown in Figure 2.
Adjust the gyroscope arm so that the distance of the disk from the center is 15 cm.
Connect all the wiring connections as well as the adjustable power adapter.
Set the gyroscope to a low rotational speed using low voltage and allow it to stabilize.
Measure and record the precession period using the stopwatch.
Gradually increase the voltage of the adjustable power adapter in increments to increase the spinning speed, repeating the measurement at each driving voltage.

note: you may slightly increase the voltage to improve balance. Make sure that the aluminium tube is always lifted and not in contact with the holder.

B.1. Table down the values that you obtained with the calculated precession period (s). [1,5 т.]

B.2. Calculate the standard deviation and standard error of the precession period shows your steps with equations for one set of data used. [0,8 т.]

B.3. Plot the precession period against the driving voltage wirth error bars and trendline. [0,9 т.]

B.4. Determine the relation between precession period and the DC motor driving voltage using the above experimental results. [0,2 т.]

B.5. Given that the DC motor performance characteristic as shown in Table 1. Obtained the relation between precession velocity and DC motor speed.

Table 1

Voltage (V)	Current (A)	Speed (rpm)
3.7	176 mA	7200
4.8	185 mA	9700
6.0	205 mA	12600
7.4	230 mA	15600
9.6	245 mA	19800
12.0	298 mA	24500

note: Do not drive the voltage above 6V. This table is for reference and not for replication. [1 т.]

B.6. Given that the moment of inertia of a uniform ring is $I = mR^2$ as shown in Figure 2. Given that the mass is M and radius is R.

Uniform ring of mass M, radius R, axis AA', angle theta and printed I=mR². — Figure 2. Moment of inertia of a uniform ring.

(i) Derive the equation of the moment of inertia of a uniform disk with respect to the rotation axis AA'. [0.5]

(ii) Calculate the moment of inertia for the 20 cm diameter with 3 mm thickness spinning disk. [0.2]

(iii) Given that the precession rate is given by

\omega_p = \frac{mgr}{I \cdot \omega_s}

where

$m$ is the mass involved,

$r$ is the distance from pivot to center of mass of the system,

$I$ is the moment of inertia of the spinning disk (assume that the moment of intertia is solely contributed by the spinning disk).

$\omega_s$ is the spinning speed of the gyroscope.

Sketch a diagram and label all the external forces involved in this experiment. Derive the equation of precession rate of the setup from angular momentum. [1.0]

(iv) Calculate the theoretical precession rate based on the motor speed in Table 1 when the motor is driving at 6V. [0.4]

Part C: Influence of Gyroscope Arm Length (2,1 т.)

Objective: To study the effect of changing the length of the gyroscope's arm on its precession behavior.

Materials: Gyroscope test kit, adjustable power adapter, DC motor, goniometer, stopwatch, 20 cm (3 mm thick) spinning disk, ruler.

Procedure:

Based on the previous setup, loosen the star knob that is used to lock the gyroscope arm.
Adjust the length of the gyroscope arm so to change the length is now 12 cm from the center of the setup.
Tighten the star knob.
Set the gyroscope driving voltage to approximately 4.5V and allow it to stabilize.
Measure and record the rotation period using the stopwatch.
Repeat the experiment for different lengths between 12 cm to 22 cm.

*Error analysis is not necessary in this experiment.

note: Make sure that the aluminium tube is always lifted and not in contact with the holder.

C.1. Table down the values that you obtained with the calculated precession velocity (rad/s) [1,2 т.]

C.2. Plot the precession velocity against the measured lengths. [0,8 т.]

C.3. Choose the change of precession with respect to arm length.

(a) Arm length increase, precession rate increase

(b) Arm length increase, precession rate reduces.

Part D: Influence of Gyroscope Disk weight (3,7 т.)

Objective: To study the effect of changing the weight of the gyroscope's disk on its precession behavior.

Materials: Gyroscope test kit, adjustable power adapter, DC motor, goniometer, stopwatch, and 20 cm spinning disk (thickness of 2mm, 3mm, and 4mm).

For this part of the experiment, the arm should not be balanced and allowed to hang freely.

Procedure:

Based on the previous setup, loosen the star knob that is used to lock the gyroscope arm.
Adjust the length of the gyroscope arm so that the spinning disk is now back to 15 cm from the center of the setup.
Tighten the star knob.
Change the spinning disk to 2 mm thickness.
Repeat step 3 - 6 in Part B.
Repeat the experiment using 4 mm thick spinning disk.

note: error analysis is not needed.

D.1. Table down the values that you obtained with the calculated precession velocity (rad/s). [1,7 т.]

D.2. Plot the precession velocity against the driving voltage for all the thicknesses you measured including in Part B. [1,1 т.]

D.3. Select the correct response of precession with respect to disk weight.

(a) The gyroscope precession will not change as the disk weight increased.

(b) The gyroscope will precess faster as the disk weight increased.

D.4. Without referring to the given mass of the disks, based on your experimental measurements, calculate the mass of the disks. Assume that the moment inertial is only contributed by the disk. [0,8 т.]

Part E: Torque Induced by External Forces (3,5 т.)

Objective: To understand how external torques affect the gyroscope's behavior.

Materials: Gyroscope with motor, 20 cm (2 mm thick) spinning disk, stainless steel bolts and nuts, stopwatch, goniometer.

Procedure:

Install the 20 cm (2 mm thick) spinning disk.
Attach the 26 cm diameter counter weight disk on the other end of the gyroscope arm.
Loosen the star knob and adjust the arm length until the arm reach the balance horizontally.
Tighten the star knob so that the arm length does not change in this part of the experiment.
Apply 5 V to the DC motor and allow it to stabilize in both precession velocity and leverage angle.
Observe and ensure that precession does not happen.
Switch off the power adapter.
Add some stainless-steel bolts and nuts to the counter weight disk. Lock tightly through the holes created around the circumference of the disk.
Measure and record the rotation period using stopwatch and leverage angle using goniometer.
Measure the change of angle due to the change of number of bolts and nuts.
Repeat the measurements using different number of bolts and nuts.
Include error calculations in your results.

E.1. Table down the values that you obtained with the calculated precession velocity (rad/s) including error estimation. [1,3 т.]

E.2. Plot the precession velocity against the number of sets of bolts and nuts you measured. Include a linear trend line. [0,8 т.]

E.3. Plot the leverage angle against the number of sets of bolt and nut you measured. [0,8 т.]

E.4. Use experimental data to determine the mass of the bolt and nut set. [0,5 т.]

E.5. If the screws are now added to the spinning disk, it is notice that the precession is now in the opposite direction and the precession velocity will be different from previous results. What parameter that it affected? m, g, r, I, or $\omega_s$ ? If there is more than one, list down in the highest to the lowest effect. [0,1 т.]

Part F: Nutation phenomenon (1,5 т.)

Objective: To observe the characteristics of the impulse affect the nutation frequency and damping in a gyroscope. Nutation is a periodic variation in inclination.

Materials: Gyroscope test kit, adjustable power adapter, DC motor, goniometer, stopwatch, 20 cm spinning disk (thickness of 2mm) and 26 cm counter weight disk.

Procedure:

Removed all the bolts and nuts on the disk from previous experiment.
Loosen the star knob and adjust the arm length until the arm reach the balance horizontally.
Tighten the star knob.
Apply 4 V to the DC motor and allow it to stabilize.
Make sure that the disk is rotate in anticlockwise direction (disc facing person).
Apply an impulse by quick pressing the counter weight and release it.
Observed the changes. Record down the parameters that you think is important.
Change the length of the aluminium tube to create an unbalanced condition where the angle of the spinning disk to an angle that is higher than horizontal.
Repeat steps 5 and 6: while the disk is spinning, hold and lift the rotation arm on the side of the spinning disk to a higher level.
Release the hold and observe the motion of the disk.
Repeat steps 7 to 9 by having the spinning disk at a lower than horizontal.

F.1. Draw the nutation motion that you observed for angle lower than horizontal and higher than horizontal. [0,4 т.]

F.2. To understand the phenomenon, you observed. You may start with the general Euler's equation for rigid body dynamic

I_1\dot{\omega}_1 + (I_3 - I_2)\omega_2\omega_3 = \tau_1 \qquad \text{Eq-1}

I_2\dot{\omega}_2 + (I_1 - I_3)\omega_3\omega_1 = \tau_2 \qquad \text{Eq-2}

I_3\dot{\omega}_3 + (I_2 - I_1)\omega_1\omega_2 = \tau_3 \qquad \text{Eq-3}

where I $_1$ , I $_2$ , I $_3$ are the moments of inertia about the principal axes, $\omega_1$ , $\omega_2$ , $\omega_3$ are the components of angular velocity about these axes, and $\tau_1$ , $\tau_2$ , $\tau_3$ are components of the external torque.

(i) Assume that the disk is symmetric at the set voltage in the direction of axis 3. Consider a frame in which there is no apparent torque. Solve Eq. 1 to 3 in this frame to obtain the nutation frequency equation. [1.0]

(ii) State the part of your derivation in F.2. that shows nutation is simple harmonic motion. [0.1]

Part G: Application of gyroscope in self balancing (1,7 т.)

Introduction: A self-balancing gyroscope is a device that utilizes the principles of angular momentum to maintain orientation and balance automatically. It consists of a spinning wheel or rotor whose axis is free to assume any orientation. When rotating, the axis tends to remain pointing in the same direction due to the conservation of angular momentum. Applications of self-balancing gyroscopes are extensive and include stabilizing platforms in ships and aircraft, enhancing the stability of motorcycles and bicycles, and in robotics for balance and precise movement control.

Objective: To demonstrate and understand the principles of angular momentum and gyroscopic stabilization. By designing and constructing this prototype, you will explore how gyroscopes maintain orientation and balance under various conditions, gaining insights into their applications in real-world technologies such as navigation systems in aerospace, automotive stabilization, and robotic balancing mechanisms.

G.1 Try to build prototype/s using the parts that you have to demonstrate the ability of self-balancing gyroscope. Label all the parts that you used and state clearly the moving and stagnant part. Safety of your setup will be concerned. In this experiment you are allowed to remove the coupler, change the disc, as well as removing the four bolts locking the DC motor clamp to reconstruct a new device using any provided parts. Sketch your prototype with label. [1,7 т.]

Решение

Experiment Marking Scheme, English (Official)

Part A: Alignment of the setup (1 т.)

A.1. [0.6pt]

Alignment diagram with the tilted arm and labelled 26 cm disk.

Marking scheme:

Weight at one end of the gyroscope arm [0.1pt]

Weight at the longer end of the gyroscope arm [0.1pt]

Disk chosen (as printed)	Points
20 mm (diameter) x 2 mm (thickness)	0.1pt
20 mm (diameter) x 3 mm (thickness)	0.2pt
20 mm (diameter) x 4 mm (thickness)	0.3pt
26 mm (diameter) x 4 mm (thickness)	0.4pt

A.2. [0.2pt]

Complete force diagram with dotted precession circle, r, F, mg, phi and both dashed angle lines.

Marking scheme:

Gravitational force, $mg$ included [0.1pt]

External force, $F$ included [0.1pt]

A.3. [0.2pt]

(b) Unequal length of the arm

(e) Weight of the disk [0.2pt]

remark: zero score will be given if student selects more than 2 choices.

Part B: Effect of Spinning Speed (6,5 т.)

B.1. [1.5pt]

Arm length (cm)	Voltage (V)	10 cycles 1 (s)	2 (s)	3 (s)	4 (s)	5 (s)	Average 10 cycles (s)	Period (s)	Std. dev.	Std. error
15	4	34	35	35	34	34	34.4	3.44	0.0548	0.0245
15	4.4	35	36	36	36	36	35.8	3.58	0.0447	0.0200
15	4.8	38	37	37	37	37	37.2	3.72	0.0447	0.0200
15	5.2	40	40	41	40	40	40.2	4.02	0.0447	0.0200
15	5.6	41	42	43	41	41	41.6	4.16	0.0894	0.0400
15	6	43	42	42	42	42	42.2	4.22	0.0447	0.0200

Proper table to record the measurement values [0.1pt]

Record rotation period using multiple cycles: [0.1] if the number of multiple cycle taken is less than 3; [0.2] if less than 6; [0.3] if more or equal to 6. [0.3pt]

Repeating the experiment: [0.1] if repeated twice; [0.2] if three times; [0.3] if four times; [0.4] if five times or more. [0.4pt]

Calculate the average period of single cycle [0.1pt]

Proper label of unit V, s, and rad/s [0.3pt]

Calculate the precession velocity [0.1pt]

Evenly exploring a range of voltages [0.1pt]

B.2. [0.8pt]

SD=\sqrt{\frac{\sum_{i=1}^{N}(T_i-\overline T)^2}{N-1}},\qquad SE=\frac{SD}{\sqrt N}.

where $T_i$ is precession period and $T_{bar}$ is average of precession period.

Correct standard deviation equation [0.1pt]

Correct standard error equation [0.1pt]

Correct calculation [0.5pt]

B.3. [0.9pt]

Period versus voltage graph with complete axes, error bars, trendline y=0.4243x+1.7352 and R²=0.9702. — Precession Period vs Voltage

$y = 0.4243x + 1.7352$ , $R^2 = 0.9702$

Label axis correctly with unit (0.1pt for each axis) — [0.2pt]

Proper scale used (0.1pt for each axis) — [0.2pt]

Chart plotted correctly

[0.2] If proper exponential trendline drawn is within 20% of the answer — [0.2pt]

[0.1] If exponential trendline drawn

Error bar is included in the chart — [0.2pt]

Chart occupy more than 70% of the area. — [0.1pt]

B.4. [0.2pt]

$y = 0.4243x + 1.7352$ — Eq-1

where y is precession period (s), x is voltage (V)

Shown correct graph equation — [0.1pt]

Shown the workout of getting the equation — [0.1pt]

B.5. [1.0pt]

DC motor speed versus voltage graph with complete axes and y=2075x−143.66. — DC motor speed vs. Voltage

Obtained the linear equation - $z = 2075x - 143.66$

where z is rotor speed (rpm), x is voltage (V)

$x = (z + 143.66)\, /\, 2075$ — Eq-2

Replace Eq-2 into Eq-1

$y = 0.4243x + 1.7352 = 0.4243\ [(z + 143.66)\ /\ 2075] + 1.7352$

$y = 2.045e\text{-}4z + 1.76476$

$\to w = (2\pi/y) = 2\pi/(2.045e\text{-}4z + 1.76476)$ — Eq-3

where w is precession velocity.

Label axis correctly with unit (0.1pt for each axis) — [0.2pt]

Proper scale used (0.1pt for each axis) — [0.2pt]

Chart plotted correctly — [0.2pt]

Shown correct linear graph equation from graph — [0.1pt]

Shown the workout of getting Eq-2 — [0.1pt]

Replace Eq-2 into Eq-1 and obtained the new equation Eq-3 — [0.2pt]

B.6(i) [0.5pt]

From $I = mR^{2}$ Eq-4

For disk, $I = \sum_i m_i r_i^{2}$ Eq-5 [0.1pt]

\text{Thus, } dI = dmr^{2}

I = \int_0^R dm\, r^{2}

Eq-6 [0.1pt]

Relation of mass of ring and radius can be found through

\frac{dm}{M} = \frac{2\pi r\, dr}{\pi R^{2}}

dm = \frac{2r\, dr}{R^{2}} M

Eq-7 [0.1pt]

From $I = \int_0^R dm\, r^{2} = \int_0^R \frac{2r\, dr}{R^{2}} M \cdot r^{2}$ [0.1pt]

I = \frac{2M}{R^{2}} \int_0^R r^{3} dr

I = \frac{2M}{R^{2}} \left[ \frac{r^{4}}{4} \right]_0^R

I = \frac{2M}{R^{2}} \left[ \frac{R^{4}}{4} \right] = \frac{1}{2} MR^{2}

Eq-8 [0.1pt]

B.6(ii) [0.2pt]

I = \frac{1}{2} MR^{2} = \frac{1}{2} (0.065)(0.1)^{2} = 0.000325\ \mathrm{kg \cdot m^{2}}

Magnitude correct [0.1pt]

Unit correct [0.1pt]

B.6(iii) [1.0pt]

Complete precession geometry, dotted circle, r, L=I omega_s, mg, phi, delta theta and red angular momentum arrows.

*Correct diagram and proper label for mg, r, and L [0.3pt]

Line for $\tau = mgr\sin\varphi$ [0.1pt]

Line for $L = L\sin\varphi$ [0.1pt]

Precession angular velocity,

\omega_p = \frac{\Delta\theta}{\Delta t}

[0.1pt]

\Delta\theta \approx \frac{\Delta L}{L\sin\varphi}

[0.1pt]

\tau = \frac{\Delta L}{\Delta t} = mgr\sin\varphi

[0.1pt]

\omega_p = \frac{\Delta L}{\Delta t L\sin\varphi} = \frac{\Delta L}{\Delta t} \cdot \frac{1}{L\sin\varphi} = \frac{\tau}{L\sin\varphi} = \frac{mgr\sin\varphi}{L\sin\varphi} = \frac{mgr}{L} = \frac{mgr}{I \cdot \omega_s}

[0.2pt]

B.6(iv) [0.4pt]

\omega_p = \frac{mgr}{I \cdot \omega_s} = \frac{(0.089 + 0.094)\ \mathrm{kg} \cdot (9.81\ \mathrm{m/s^{2}}) \cdot (0.25\ \mathrm{m})}{(0.000325\ \mathrm{kg \cdot m^{2}})(12600\ \mathrm{rev/min})}

= \frac{0.5386\ \mathrm{kg \cdot m^{2}/s^{2}}}{0.4288\ \mathrm{kg \cdot m^{2}/s}} = 1.256\ \mathrm{rad^{-1}s^{-1}}

Replacement in equation correct [0.1pt]

Including mass of motor (0.089 + 0.094) [0.1pt]

Magnitude correct [0.1pt]

Unit correct [0.1pt]

Part C: Influence of Gyroscope Arm Length (2,1 т.)

C.1. [1.2pt]

L/cm	1	2	3	4	5	Average 10 cycles period (s)	Period (s)	Precession Rate (rad/s)
12	53.72	51.19	55.12	55.38	54.03	53.888	5.389	1.166
14	43.85	48.75	48.22	48.84	47.9	47.512	4.751	1.322
16	34.44	34.22	35.72	34.69	34.12	34.638	3.464	1.814
18	30.5	30.12	30.53	29	29.68	29.966	2.997	2.097
20	26.69	25.47	27.47	27.31	27.4	26.868	2.687	2.339
22	23.54	23.62	23.78	23.66	23.93	23.706	2.371	2.650

The first five numbered columns are headed "10 cycles period (s)".

*Proper table to record the measurement values [0.1pt]

Record rotation period using multiple cycles

[0.1] if the period of multiple cycle taken is less than 5

[0.2] if the period of multiple cycle taken is between 5 to 10

[0.3] if the period of multiple cycle taken is more than 10

(all within [0.3pt])

Number of data points collected

[0.1] If the number of data points collected is between 3 and 5

[0.2] If the number of data points collected is more than 5

(within [0.2pt])

Calculate the average period of single cycle [0.1pt]

Proper label of unit V, s, and rad/s [0.3pt]

Calculate the precession velocity [0.2pt]

C.2. [0.8pt]

Precession rate versus arm length graph with all six points, complete axes, y=0.1536x−0.7136 and R²=0.9869. — Precession Rate vs. Length

*Marking

Label axis correctly with unit (0.1pt for each axis) [0.2pt]

Proper scale used (0.1pt for each axis) [0.2pt]

Chart plotted correctly [0.2pt]

Proper linear trendline drawn [0.1pt]

Chart occupy more than 70% of the graph area [0.1pt]

C.3. [0.1pt]

(a) Arm length increase, precession rate increase.

Part D: Influence of Gyroscope Disk weight (3,7 т.)

D.1. [1.7pt]

Thickness (mm)	Voltage (V)	1	2	3	4	5	Average 10 cycles period	Period (s)	Precession rate (rad/s)
2	4	31.97	34.15	33.66	34.35	34.03	33.632	3.3632	1.8682
2	4.4	35.04	35.28	35.24	36.78	33.25	35.118	3.5118	1.7892
2	4.8	34.47	37.97	37.06	35.9	36.31	36.342	3.6342	1.7289
2	5.2	38.06	36.16	36.65	36	38.53	37.08	3.7080	1.6945
2	5.6	36.56	37.43	38.53	37.63	37.15	37.46	3.7460	1.6773
2	6	40.35	40.28	40.13	39.81	40.31	40.176	4.0176	1.5639
3	4	34	35	35	34	34	34.4	3.44	1.8265
3	4.4	35	36	36	36	36	35.8	3.58	1.7551
3	4.8	38	37	37	37	37	37.2	3.72	1.6890
3	5.2	40	40	41	40	40	40.2	4.02	1.5630
3	5.6	41	42	43	41	41	41.6	4.16	1.5104
3	6	43	42	42	42	42	42.2	4.22	1.4889
4	4	35.16	38.38	38.34	38.47	38.66	37.802	3.7802	1.6621
4	4.4	37.81	41.94	42.84	42.31	42.03	41.386	4.1386	1.5182
4	4.8	39.66	42.81	43.5	43	42.94	42.382	4.2382	1.4825
4	5.2	42.07	44.19	44.56	44.75	44.32	43.978	4.3978	1.4287
4	5.6	44.06	45.44	45.59	45.5	45.37	45.192	4.5192	1.3903
4	6	45.62	46.94	46.53	46.56	46.81	46.492	4.6492	1.3515

(The header of the column group over the five measurement columns reads "10 cycles period (s)".)

*Proper table to record the measurement values — [0.1pt]

Record rotation period using multiple cycles — [0.3pt]

[0.1] if the period of multiple cycle taken is less than 5

[0.2] if the period of multiple cycle taken is between 5 to 10

[0.3] if the period of multiple cycle taken is more than 10

Number of data points collected — [0.3pt]

[0.1] If the number of data points collected is between 3

[0.2] If the number of data points collected is more than 7

[0.3] If the number of data points collected is more than 10

Calculate the average period of single cycle — [0.1pt]

Proper label of unit V, s, and rad/s — [0.3pt]

Calculate the precession velocity — [0.3pt]

Table in all the value of 2mm, 3mm, and 4mm — [0.3pt]

D.2. [1.1pt]

Precession velocity versus voltage graph, all 2 mm, 3 mm and 4 mm series with complete legend and axes. — Precession Velocity vs. Voltage

*Marking

Marking item	Points
Label axis correctly with unit (0.1pt for each axis)	[0.2pt]
Proper scale used (0.1pt for each axis)	[0.2pt]
Chart plotted correctly	[0.3pt]
Trendline drawn	[0.3pt]
Chart occupy more than 70% of the graph area	[0.1pt]

D.3. [0.1pt] (b) The gyroscope will precess faster as the disk weight increased.

D.4. [0.8pt] From B.6(iii), the precession angular velocity is given by

\omega_p = \frac{(m+M)gr}{I\ \omega_s},

[0.1pt]

where $m$ is the mass of the disk and $M$ is the mass of the motor. From B.6(i), the moment of inertia of a uniform disk is

I = \frac{1}{2}mR^2,

[0.1pt]

where $R$ is the radius of the disk. Therefore,

\omega_p = \frac{(m+M)gr}{\frac{1}{2}mR^2\ \omega_s}

[0.1pt]

m = \frac{2Mgr}{R^2\omega_p\omega_s - 2gr}.

m_i = \frac{2(0.094)(9.81)(0.25)}{(0.2)^2(\omega_{p,i})\omega_{s,i} - 2(9.81)(0.25)}

[0.1pt]

The mass of the motor $M$ is given, $g$ is the gravitational acceleration, $\omega_s$ is provided in B.5, $r$ is the length of the arm fixed at 25cm, and $\omega_p$ is measured by the students.

Student calculate from $m_1$ to $m_n$ [0.2pt]

Calculate average $m$ [0.1pt]

Calculate standard deviation [0.1pt]

Part E: Torque Induced by External Forces (3,5 т.)

E.1. [1.3pt]

Number of sets of bolt and nut	Period 1 cycle (s) 1	Period 1 cycle (s) 2	Period 1 cycle (s) 3	Period (s)	Precession Velocity (rad/s)	Leverage Angle (rad)
4	46.34	47.9	46.79	47.01	0.1337	0.296705973
6	39.69	39	36.75	38.48	0.1633	0.261799388
8	31.65	29.96	32.16	31.26	0.2010	0.244346095
10	22.16	21.75	22.13	22.01	0.2854	0.366519143

*Proper table to record the measurement values [0.1pt]

Repeating record the data

[0.1] if the data is taken only once [0.2pt]

[0.2] if the data repeated 3 times or more

Calculate the average period of single cycle [0.1pt]

Calculate the standard deviation [0.1pt]

Calculate the standard error [0.1pt]

Calculate the tilted angle [0.1pt]

Proper label of unit V, s, and rad/s [0.3pt]

Calculate the precession velocity [0.2pt]

Even bolts and nuts are added [0.1pt]

E.2. [0.8pt] Precession velocity versus sets of nuts and bolts, trendline y=0.0269x+0.0063; the printed vertical axis reads Speed (rpm).

*Label axis correctly with unit (0.1pt for each axis) [0.2pt]

Proper scale used (0.1pt for each axis) [0.2pt]

Chart plotted correctly [0.1pt]

Linear trendline drawn [0.1pt]

Shows error bar [0.1pt]

Chart occupy more than 70% of the graph area [0.1pt]

E.3. [0.8pt] Leverage angle versus sets of nuts and bolts with y=0.0309x+0.0609 and complete axes.

*Label axis correctly with unit (0.1pt for each axis) [0.2pt]

Proper scale used (0.1pt for each axis) [0.2pt]

Chart plotted correctly [0.1pt]

Linear trendline drawn [0.1pt]

Shows error bar [0.1pt]

Chart occupy more than 70% of the graph area [0.1pt]

E.4. [0.5pt] From B.6(iii), the precession velocity is

\omega_p = \frac{Nmgr}{I\ \omega_s},

where $N$ is the number of nuts and bolts, $m$ is the mass of one set of nut and bolt, $g$ is the gravitational acceleration, $r$ is the length of the arm, $I$ is the moment of inertia of the disk, $\omega_s$ is the angular speed of the disk, and $\omega_p$ is the angular speed of the precession.

Number of nuts and bolts	m/kg
4	0.0055
6	0.0045
8	0.0041
10	0.0047
Average	0.0047
Std. dev.	0.0006

Calculated all the m values for different number of bolt and nut [0.2pt]

Calculated the average value of m [0.1pt]

Calculated the standard deviation [0.1pt]

Measure r from the experiment [0.1pt]

E.5. [0.1pt]

$I$ [0.1pt]

Part F: Nutation phenomenon (2,1 т.)

F.1. [0.4pt]

Both native looping nutation trajectories (a) and (b), including the axis-tip circles and direction arrows.

(a) When the angle is lower than the horizontal. [0.2 pt]
(b) When the angle is higher than the horizontal. [0.2 pt]

F.2(i) [1.0pt]

When rotation is at axis 3. $\omega_{3} \gg \omega_{2}$ and $\omega_{1}$ [0.1 pt]

When no external torque applied, $\tau_{1} = \tau_{2} = \tau_{3} = 0$ [0.1 pt]

At constant spinning speed, $\dot{\omega}_{3} = 0$ [0.1 pt]

I_{1}\dot{\omega}_{1} - (I_{3} - I_{2})\omega_{2}\omega_{3} = 0 \qquad \text{Eq-4} \quad [0.1\ \mathrm{pt}]

I_{2}\dot{\omega}_{2} - (I_{1} - I_{3})\omega_{3}\omega_{1} = 0 \qquad \text{Eq-5} \quad [0.1\ \mathrm{pt}]

For symmetric disk, $I_{1} = I_{2}$ , [0.1 pt]

I_{1}\dot{\omega}_{1} = (I_{3} - I_{1})\omega_{2}\omega_{3} \qquad \text{Eq-6}

I_{1}\dot{\omega}_{2} = -(I_{3} - I_{1})\omega_{3}\omega_{1} \qquad \text{Eq-7}

From Eq.6 $I_{1}\ddot{\omega}_{1} = (I_{3} - I_{1})\dot{\omega}_{2}\omega_{3}$ Eq-8 [0.1 pt]

From Eq.7 $\dot{\omega}_{2} = -\frac{(I_{3}-I_{1})}{I_{1}}\,\omega_{3}\omega_{1}$ Eq-9 [0.1 pt]

Substitute Eq-9 into Eq-8

I_{1}\ddot{\omega}_{1} = (I_{3} - I_{1})\left[-\frac{(I_{3}-I_{1})}{I_{1}}\,\omega_{3}\omega_{1}\right]\omega_{3}

I_{1}\ddot{\omega}_{1} = -\frac{(I_{3}-I_{1})^{2}}{I_{1}}\,\omega_{1}\omega_{3}^{2}

\ddot{\omega}_{1} + \left[\frac{(I_{3}-I_{1})}{I_{1}}\,\omega_{3}\right]^{2}\omega_{1} = 0 \qquad [0.1\ \mathrm{pt}]

\omega_{n} = \frac{(I_{3}-I_{1})}{I_{1}}\,\omega_{3} \qquad \text{Eq-10} \quad [0.1\ \mathrm{pt}]

F.2(ii) [0.1pt]

\ddot{\omega}_{1} + {\omega_{n}}^{2}\,\omega_{1} = 0 \qquad [0.1\ \mathrm{pt}]

Part G: Application of gyroscope in self balancing (1,7 т.)

G.1. [1.7pt]

First native prototype schematic and photograph: DC Motor Static, 26 cm disk Static, 20 mm disk Moving. The small paste icon is intrinsic to the source.

Correct setup [0.2pt]

Correct labeling of parts [0.3pt]

Correct labeling of static and moving parts [0.3pt]

Second native prototype schematic and photograph: 20 mm disk Moving, 26 cm disk Static, DC Motor Static, complete labels.

[0.2] Correct setup without 26 cm disk

[0.3] Correct setup with 26 cm disk [0.3pt]

Correct labeling of parts [0.3pt]

Correct labeling of static and moving parts [0.3pt]

Archive solution pointer

For solutions in Russian, see here:

E1. https://pho.rs/p/4072/m

If you use a browser plugin for translation, you should be able to find your way around.

Оригинал в Архива: exp.pdf · официални решения: exp_sol.pdf · допълнителни данни: exp_sol.txt