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General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Lab Manual

Gyroscope Ver.20170428

Lab Office (Int’l Campus)

Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) / 19

[International Campus Lab]

Gyroscope

Observe the motions of gyroscope.

1. Torque

Torque (or moment) is the tendency of a force to rotate a

body about an axis. We use the Greek letter 𝜏𝜏 for torque.

Fig. 1 The torque 𝝉𝝉�⃗ of the force 𝑭𝑭��⃗ about the point O is 𝝉𝝉�⃗ = 𝒓𝒓�⃗ × 𝑭𝑭��⃗ . In this figure, 𝒓𝒓�⃗ and 𝑭𝑭��⃗ are in the plane of the page and the torque vector 𝝉𝝉�⃗ points out of the page toward you.

Fig. 1 shows a force 𝑭𝑭��⃗ applied at a point 𝑃𝑃 described by a

position vector 𝒓𝒓�⃗ with respect to the chosen point 𝑂𝑂. Then

the magnitude of torque depends on 𝐹𝐹, 𝑟𝑟, and the angle 𝜙𝜙

between 𝑭𝑭��⃗ and 𝒓𝒓�⃗ .

𝜏𝜏 = 𝐹𝐹𝐹𝐹 = 𝑟𝑟𝐹𝐹 sin𝜙𝜙 = 𝐹𝐹tan𝑟𝑟 (1)

or 𝝉𝝉�⃗ = 𝒓𝒓�⃗ × 𝑭𝑭��⃗ (2)

2. Angular Momentum

The angular momentum 𝑳𝑳��⃗ in the rotational motion is the

analogue of (linear) momentum 𝒑𝒑��⃗ = 𝑚𝑚𝒗𝒗��⃗ in the translational

motion. Its relationship to momentum 𝒑𝒑��⃗ is exactly the same

as the relationship of torque to force, 𝝉𝝉�⃗ = 𝒓𝒓�⃗ × 𝑭𝑭��⃗ . We define

angular momentum 𝑳𝑳��⃗ as

𝑳𝑳��⃗ = 𝒓𝒓�⃗ × 𝒑𝒑��⃗ = 𝒓𝒓�⃗ × 𝑚𝑚𝒗𝒗��⃗ (3)

When a net force 𝑭𝑭��⃗ acts on a particle, its velocity and mo-

mentum change, so its angular momentum may also change.

To show this, we take the time derivative of Eq. (3):

𝑑𝑑𝑳𝑳��⃗𝑑𝑑𝑑𝑑 = �

𝑑𝑑𝒓𝒓�⃗𝑑𝑑𝑑𝑑 × 𝑚𝑚𝒗𝒗��⃗ � + �𝒓𝒓�⃗ × 𝑚𝑚

𝑑𝑑𝒗𝒗��⃗𝑑𝑑𝑑𝑑�

= (𝒗𝒗��⃗ × 𝑚𝑚𝒗𝒗��⃗ ) + (𝒓𝒓�⃗ × 𝑚𝑚𝒂𝒂��⃗ )

(4)

Objective

Theory

----------------------------- Reference --------------------------

Young & Freedman, University Physics (14th ed.), Pearson, 2016

10.1 Torque (p.327~330)

10.5 Angular Momentum (p.341~343)

10.7 Gyroscopes and Precession (p.346~349)

-----------------------------------------------------------------------------


Lab Manual




The first term is zero because it contains the vector product

of the vector 𝒗𝒗��⃗ = 𝑑𝑑𝒓𝒓�⃗ 𝑑𝑑𝑑𝑑⁄ with itself. In the second term we

replace 𝑚𝑚𝒂𝒂��⃗ with the net force 𝑭𝑭��⃗ , obtaining

𝑑𝑑𝑳𝑳��⃗𝑑𝑑𝑑𝑑 = 𝒓𝒓�⃗ × 𝑭𝑭��⃗ = 𝝉𝝉�⃗ (5)

We can also find the total angular momentum of a rigid body

rotating about the 𝑧𝑧-axis with angular speed 𝜔𝜔. Consider a

thin slice of the body lying in the 𝑥𝑥𝑥𝑥-plain (Fig. 2). Each parti-

cle in the slice moves in a circle centered at the origin, and at

each instant its velocity 𝒗𝒗𝚤𝚤��⃗ is perpendicular to its position

vector 𝒓𝒓𝚤𝚤��⃗ , as shown.

A particle with mass 𝑚𝑚𝑖𝑖 at a distance 𝑟𝑟𝑖𝑖 from 𝑂𝑂 has a

speed 𝑣𝑣𝑖𝑖 = 𝑟𝑟𝑖𝑖𝜔𝜔. Then the magnitude 𝐿𝐿𝑖𝑖 of its angular mo-

mentum is

𝐿𝐿𝑖𝑖 = 𝑚𝑚𝑖𝑖𝑣𝑣𝑖𝑖𝑟𝑟𝑖𝑖 = 𝑚𝑚𝑖𝑖(𝑟𝑟𝑖𝑖𝜔𝜔)𝑟𝑟𝑖𝑖 = 𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2𝜔𝜔 (6)

The total angular momentum of the slice of the body lying

the 𝑥𝑥𝑥𝑥-plain is the sum ∑𝐿𝐿𝑖𝑖

𝐿𝐿 = ∑𝐿𝐿𝑖𝑖 = �∑𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2�𝜔𝜔 = 𝐼𝐼𝜔𝜔 (7)

And for a rigid body rotating around an axis of symmetry, we

have the vector relationship

𝑳𝑳��⃗ = 𝐼𝐼𝝎𝝎��⃗ (8)

Fig. 2 In a rigid body rotating about 𝑧𝑧-axis at angular speed 𝜔𝜔, the angular momentum 𝐿𝐿�⃗ 𝑖𝑖 of a particle of mass 𝑚𝑚𝑖𝑖 is perpendicular to the plane of motion and has magnitude 𝐿𝐿𝑖𝑖 = 𝑚𝑚𝑖𝑖𝑣𝑣𝑖𝑖𝑟𝑟𝑖𝑖 = 𝑚𝑚𝑖𝑖(𝑟𝑟𝑖𝑖𝜔𝜔)𝑟𝑟𝑖𝑖 = 𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2𝜔𝜔.

3. Gyroscope

When the axis of rotation changes direction, some quite

unexpected physical phenomena can occur. For example,

consider a gyroscope that is supported at one end (Fig. 3).

If we hold it with flywheel axis horizontal and let go, the free

end of the axis simply drops owing to gravity if the flywheel

isn’t spinning.

But if the flywheel is spinning, what happens is quite differ-

ent. One possible motion is a steady circular motion of the

axis in a horizontal plane, combined with the spin motion of

the flywheel about the axis. This motion of the axis is called

precession.

To study this strange phenomenon of precession, we need

the relationship given by Eq. (5), 𝝉𝝉�⃗ = 𝑑𝑑𝑳𝑳��⃗ /𝑑𝑑𝑑𝑑. We take the

origin 𝑂𝑂 at the pivot and assume that the flywheel is sym-

metrical, with mass 𝑀𝑀 and moment of inertia 𝐼𝐼 about the

flywheel axis. The flywheel axis is initially along the 𝑥𝑥-axis.

The only external forces on the gyroscope are the normal

force 𝒏𝒏��⃗ acting at the pivot and the weight 𝒘𝒘��⃗ of the flywheel

that acts its center of mass, a distance 𝑟𝑟 from the pivot. 𝒏𝒏��⃗

has zero torque with respect to the pivot, and 𝒘𝒘��⃗ has a torque

𝝉𝝉�⃗ = 𝒓𝒓�⃗ × 𝒘𝒘��⃗ in the 𝑥𝑥-direction, as shown in Fig. 4(a).

Fig. 3 A gyroscope supported at one end. When the flywheel

spins, it and its axis float in the air while moving in a circle about the pivot. The horizontal circular motion of the flywheel and axis is called precession. The angular speed of precession is Ω.


Lab Manual




Initially, there is no rotation, and the initial angular momen-

tum 𝑳𝑳𝚤𝚤��⃗ is zero. From Eq. (5) the change 𝑑𝑑𝑳𝑳��⃗ in angular mo-

mentum in a short time interval 𝑑𝑑𝑑𝑑 following this is

𝑑𝑑𝑳𝑳��⃗ = 𝝉𝝉�⃗ 𝑑𝑑𝑑𝑑 (9)

This change is in the 𝑥𝑥-direction because 𝝉𝝉�⃗ is. As each

additional time interval 𝑑𝑑𝑑𝑑 elapses, the angular momentum

changes by additional increments 𝑑𝑑𝑳𝑳��⃗ in the 𝑥𝑥-direction be-

cause the direction of the torque is constant (Fig. 3(b)). The

steadily increasing horizontal angular momentum means that

the gyroscope rotates downward faster and faster around the

𝑥𝑥-axis until it hits the table.

Fig. 4 (a) If the flywheel in Fig 3 is initially not spinning, its ini-

tial angular momentum is zero (b) In each successive time interval 𝑑𝑑𝑑𝑑, the torque produces a change 𝑑𝑑𝑳𝑳��⃗ = 𝝉𝝉�⃗ 𝑑𝑑𝑑𝑑 in the angular momentum. The flywheel acquires an angular momentum 𝑳𝑳��⃗ in the same direction as 𝝉𝝉�⃗ , and the flywheel axis falls.

Fig. 5 (a) The flywheel is spinning initially with angular momen-

tum 𝑳𝑳ı��⃗ . The forces (not shown) are the same as those in Fig 4(a). (b) Because the initial angular momentum is not zero, each change 𝑑𝑑𝑳𝑳��⃗ = 𝝉𝝉�⃗ 𝑑𝑑𝑑𝑑 in angular momentum is perpendicular to 𝑳𝑳��⃗ . As the result, the magnitude of 𝑳𝑳��⃗ remains the same but its direction changes continuously.

If the flywheel is spinning initially, the initial angular momen-

tum 𝑳𝑳𝚤𝚤��⃗ is not zero (Fig. 5(a)). Since the flywheel rotates

around its symmetry axis, 𝑳𝑳𝚤𝚤��⃗ lies along the axis. But each

change in angular momentum 𝑑𝑑𝑳𝑳��⃗ is perpendicular to the axis

because the torque 𝝉𝝉�⃗ = 𝒓𝒓�⃗ × 𝒘𝒘��⃗ is perpendicular to the axis

(Fig. 5(b)).

This causes the direction of 𝑳𝑳��⃗ to change, but not its magni-

tude, the change 𝑑𝑑𝑳𝑳��⃗ are always in the horizontal 𝑥𝑥𝑥𝑥-plane,

so the angular momentum vector and the flywheel axis with

which it moves are always horizontal. In other words, the axis

doesn’t fall, it just precesses.

4. Precession Angular Speed

At the instant shown in Fig. 5(a), the gyroscope has angular

momentum 𝑳𝑳��⃗ . a short time interval 𝑑𝑑𝑑𝑑 later, the angular mo-

mentum is 𝑳𝑳��⃗ + 𝑑𝑑𝑳𝑳��⃗ ; the infinitesimal change in angular mo-

mentum is 𝑑𝑑𝑳𝑳��⃗ = 𝝉𝝉�⃗ 𝑑𝑑𝑑𝑑, which is perpendicular to 𝑳𝑳��⃗ . As Fig. 6

shows, this means that the flywheel axis of the gyroscope

has turned through a small angle 𝑑𝑑𝜙𝜙 = �𝑑𝑑𝑳𝑳��⃗ �/�𝑳𝑳��⃗ �. The rate at

which the axis moves, 𝑑𝑑𝜙𝜙 𝑑𝑑𝑑𝑑⁄ , is called the precession an-

gular speed; denoting this quantity by Ω, we find

Ω =𝑑𝑑𝜙𝜙𝑑𝑑𝑑𝑑 =

�𝑑𝑑𝑳𝑳��⃗ � �𝑳𝑳��⃗ ��𝑑𝑑𝑑𝑑 =

𝜏𝜏𝑧𝑧𝐿𝐿𝑧𝑧

=𝑤𝑤𝑟𝑟𝐼𝐼𝜔𝜔 (10)

Thus the precession angular speed Ω is inversely propor-

tional to the angular speed 𝜔𝜔 of spin about the axis. A rapidly

spinning gyroscope precesses slowly. If friction in its bearings

causes the flywheel to slow down, the precession angular

speed increases.

Fig. 6 Detailed view of past of Fig 5(b). In a time 𝑑𝑑𝑑𝑑, the angular momentum vector and the flywheel axis precess together through an angle 𝑑𝑑𝜙𝜙.


Lab Manual




We can use Eq, (10) when only the weight 𝒘𝒘��⃗ of the fly-

wheel acts force on the gyroscope. In this experiment, you

will use the gyroscope equipped with several sensors and

components on the axis of the flywheel. They all have torques

in the 𝑥𝑥-direction. Because we do not know their weights and

distances from the pivot, we cannot calculate net torque.

If the axis is exactly balanced in the horizontal position by

adjusting the position of weights on the counterweight arm,

we don’t have to consider the torque of any component (in-

cluding the flywheel) on the axis. Instead, to produce the turn-

ing torque, you will attach a small add-on weigh to the front of

the flywheel. If the add-on weight has mass 𝑀𝑀 and distance

𝑅𝑅 from the pivot, Eq. (10) becomes

Ω =𝜏𝜏𝐿𝐿 =

(𝑀𝑀𝘨𝘨)𝑅𝑅𝐼𝐼𝜔𝜔 (11)

𝐼𝐼 : Moment of inertia of the flywheel

𝜔𝜔 : Angular speed of the flywheel

𝑀𝑀 : Mass of the add-on weight

𝘨𝘨 : Acceleration due to gravity

𝑅𝑅 : Distance from the pivot to the add-on weight

Fig. 7 Nutation (a) released from rest (b) released with forward

speed (c) released with backward speed

Fig. 8 Rotating flywheel and free-body diagram

5. Nutation

As explained above, if a gyroscope is set at a tilt on a hori-

zontal surface and spun rapidly, its rotational axis starts pre-

cessing about the vertical. After a short interval, the gyro-

scope settles into a motion in which each point on its rotation

axis follows a circular path. Initially, however, there is no pre-

cession, and the gyroscope falls downward. This gives rise to

an imbalance in torques that starts the precession. In falling,

the gyroscope overshoots the level at which it would precess

steadily and then oscillates about this level, as shown in Fig.

7. This oscillation is called nutation.

6. Moment of Inertia

We can find the moment of inertia of the flywheel experi-

mentally.

In the figure 8, the weight with mass 𝑚𝑚 hangs from the

thread which is wrapped around the pulley attached on the

flywheel. The weight speeds up with linear acceleration 𝑎𝑎.

Applying Newton’s second law gives ∑𝐹𝐹 = 𝑚𝑚𝑎𝑎 = 𝑚𝑚𝘨𝘨− 𝑇𝑇

and solving for the tension in the thread gives 𝑇𝑇 = 𝑚𝑚(𝘨𝘨 − 𝑎𝑎).

Thus the torque 𝜏𝜏 caused by the weight hanging from the

thread is 𝜏𝜏 = 𝑟𝑟𝑇𝑇 = 𝑟𝑟𝑚𝑚(𝘨𝘨 − 𝑎𝑎), where 𝑟𝑟 is radius of the pulley.

The relationship of the linear acceleration 𝑎𝑎 to the angular

acceleration 𝛼𝛼 of the pulley (or the flywheel) is 𝑎𝑎 = 𝑟𝑟𝛼𝛼.

Since 𝜏𝜏 = 𝐼𝐼𝛼𝛼, the moment of inertia of the flywheel is

𝐼𝐼 =𝜏𝜏𝛼𝛼 =

𝑟𝑟𝑚𝑚(𝘨𝘨 − 𝑎𝑎)(𝑎𝑎 𝑟𝑟⁄ ) = 𝑚𝑚𝑟𝑟2 �

𝘨𝘨𝑎𝑎 − 1� (12)

𝑚𝑚 : Mass of the weight

𝑟𝑟 : Radius of the pulley

𝑎𝑎 : Acceleration of the weight


Lab Manual




1. List

Item(s) Qty. Description

PC / Software Data Analysis: Capstone Gyroscope: SensorLAB

1 set Records, displays and analyzes the data measured by various sensors.

Interface

1 Data acquisition interface designed for use with various sensors, including power supplies which provide up to 15 watts of power.

Gyroscope Apparatus

1 Allows studies of the motion of a gyroscope.

USB to RS-232 Cable DC Adapter (12V)

1 1

Connects the gyroscope apparatus to PC. Supplies DC power to the gyroscope apparatus.

Photogate (Pulley, Rod, and Cable included)

1 set Measures high-speed or short-duration events.

A-shaped Base (Small)

1 Provide stable support for experiment set-ups.

Support Rods (600mm)


Multi-clamps


Hanging Weight Add-on Weight

1 1

Mass: 50g for the moment of inertia experiment Mass: 100g for acting downward force

Equipment


Lab Manual




Item(s) Qty. Description

String

1 Spins the gyroscope flywheel.

Thread Scissors

Hang a weight.

Flywheel Part

Gives its mass and dimension so that you can compute the moment of inertia of the gyroscope flywheel.

Vernier Caliper

1 Measures external, internal diameter or depth of an object with a precision to 0.05mm.

Electronic Balance

Measures mass with a precision to 0.01g.

2. Details

(1) Gyroscope Apparatus

(2) Vernier Caliper

① 22 mm is to the immediate left of 0 on the vernier scale.

② Aligned (13th) line corresponds to 0.65 mm (= 0.05 × 13).

③ 22 + 0.65 = 22.65 mm


Lab Manual




Procedure

Caution

The gyroscope apparatus could easily malfunction or

produce incorrect results. It should be treated with extra

care.

1. Keep tight hold of the gyroscope axle while you pull the

thread to rotate the flywheel, or while you stop the fly-

wheel.

2. Subjecting the apparatus to strong impact could cause

the sensors to fail, or the axles to be bent.

Caution

3. Never touch the sensor modules.

Note

You will use two kinds of thread/string in this experiment.

Thread String

Use

For suspending a weight

from the flywheel pulley

(Experiment 1)

For spinning up the fly-

wheel of the gyroscope

(Experiment 2 ~ 5)

Where Lecture Table Each Lab Table

Shape

Wound on a spool

Color: Black or White

Thicker one with loops

Note Do NOT cut the string.

Do NOT untie endknots.


Lab Manual




Experiment 1. Moment of Inertia of Flywheel

(1) Set up equipment.

① Clamp the gyroscope axle.

Using a support rod and a multi-clamp, clamp the gyroscope

axle in the horizontal position.

② Install the photogate assembly.

Assemble the photogate assembly (the raised slot on the

photogate housing provides a seat for attaching a pulley us-

ing a pulley mounting post). And then mount it on the support

rod using a multi-clamp.

③ Arrange a weight to accelerate the flywheel.

Tie a length of thread to the hole of the pulley of the flywheel.

Tie a mass of 50g at the other end of the thread. And then

pass the thread over the photogate pulley.

④ Adjust the position of the photogate pulley.

Adjust the position of the photogate pulley so the thread

runs parallel to the table and the weight can fall out of the

table.

⑤ Adjust the length of the thread, if required.

The weight must stop falling before it reaches the ground.

⑥ Connect the Photogate to the interface.


Lab Manual




(2) Configure the Capston software.

① Add [Photogate with Pulley].

(Make sure you do not select [Photogate].)

② Create a timer.

Make sure that [Linear Speed] is checked, the spoke arc

length is 0.015m, and the spoke angle is 36°.

③ Create a graph display.

𝑥𝑥-axis: [Time(s)], 𝑥𝑥-axis: [Linear Speed(m/s)]

(3) Wind the thread around the pulley.

By rotating the flywheel, wind the thread around of the pulley,

until the weight is located near the photogate pulley.


Lab Manual




(4) Click [Record].

(5) Release the flywheel.

The torque of a tension force rotates the flywheel. Click

[Stop] before the weight reaches the ground.

(6) Find the acceleration 𝑎𝑎 of the system.

(7) Calculate 𝐼𝐼 of the flywheel.

𝐼𝐼 = 𝑚𝑚𝑟𝑟2 �𝘨𝘨𝑎𝑎 − 1� (12)

𝑚𝑚 ≒ 0.050 kg : Mass of the weight

𝑟𝑟 ≒ 0.029 m : Radius of the pulley

(8) Repeat the measurement.

Repeat the steps (3) to (7) more than three times.

𝑚𝑚 (kg) 𝑟𝑟 (m) 𝑎𝑎 𝐼𝐼

1st

2nd

3rd

…

average

(9) Compare the result with the theoretical value.

The flywheel part is prepared on lecture table. Measure the

mass, the inner radius, and the outer radius of the flywheel,

and calculate the theoretical value 𝐼𝐼flywheel of the flywheel.

𝑅𝑅1 = _______ (m)

𝑅𝑅2 = _______ (m)

𝑀𝑀 = _______ (kg)

Because the moment of inertia of the pulley is relatively

smaller than that of the flywheel, you can ignore it. If required,

use the pre-calculated value 𝐼𝐼pulley = 5.51 × 10−5 kg ⋅ m2.


Lab Manual




Experiment 2. Motion of Gyroscope, Part 1

(1) Set up equipment.

① Remove the photogate assembly.

② Remove the support rod and the multi-clamp clamping

the gyroscope axle.

③ Remove the thread connected to the weight.

(2) Balance the gyroscope axle in the horizontal position.

Adjust the position of the counterweights until the gyroscope

is balanced without an add-on mass.

(3) Observe the motion of the gyroscope without rotating.

Do not spin the flywheel. Tap the end of the gyroscope axle

in the horizontal direction, and observe the motion of the gy-

roscope.

(4) Observe the motion of the gyroscope with rotating.

Spin the flywheel. And then tap the end of the gyroscope

axle in the horizontal direction. Observe the motion of the

gyroscope.


Lab Manual




(5) Compare the results.

Q Describe the differences of two cases and explain why.

A

Experiment 3. Motion of Gyroscope, Part 2

(1) Spin the flywheel.

(2) Apply a steady force on the gyroscope axle.

With the flywheel rotating, pull the gyroscope axle downward

using a string (or push it downward using your finger). Ob-

serve the motion of the gyroscope.

Q Can you simply swing the gyroscope axle as you pull it? If

not, which direction is the axle going in? Explain why.

A

Q If you change the direction of the force, which direction is

the axle going in? Explain why.

A

Caution

When you stop the spinning flywheel, follow the instruc-

tions below.


Lab Manual




Experiment 4. Precession Angular Speed

(1) Set up the equipment.

Connect the gyroscope apparatus to the computer using

Serial–USB cable. Provide power to the apparatus using 12V

DC adapter. (Different type of DC adapter can be provided.)

(2) Run SensorLAB software.

① Connect or disconnect communication.

② Begin or stop recording.

③ Displays the sensor readings.

④ Plots graphs.

⑤ Select graph type.

⑥ Span graphs.

⑦ Converts the value of rotating angle to absolute value.

⑧ Change the sign of sensor values.

⑨ Adjust the ranges.

⑩ Shows data table.

(2-1) Click [Connect] to receive the sensors’ values.

[Received Data] displays the real-time values of the sensors.

Rotation : Rotating angle of the rotating axle ( ° )

(by the rotary motion sensor)

Tilt : Tile of the gyroscope axle ( ° )

(by the tilt sensor)

DiskRPM : RPM of the flywheel (revolutions per minute)

(by the RPM sensor)

Caution

Before you run SensorLAB, you should shut down PAS-

CO Capstone. If not, they suddenly crashes and you may

lose your experimental data.


Lab Manual




(2-2) Set [Absolute Value].

By [Absolute Value] icon, [Rotation] value will

[Continually increase regardless of the rotating direction]

↔ [increase or decrease according to the rotating direction]

Watch the [Rotation] values of [Received Data] and rotate

the rotating axle of the gyroscope clockwise or counterclock-

wise.

If [Rotation] value continually increases regardless of the

rotating direction, depress [Absolute Value] icon. Then it will

increase or decrease according to the rotating direction.

(2-3) Set [Flip Horizontal] and [Flip Vertical].

[Rotation] value initially increases when the rotating axle of

the gyroscope rotates counterclockwise. If you depress [Flip

Horizontal] button, [Rotation] increases when the axle rotates

clockwise.

[Tilt] value is initially positive when the arm with the flywheel

of the gyroscope axle is higher. If you want to change the

sign of [Tilt], depress [Flip Vertical].

(2-4) Select a graph type.

You can change graph types to display on the graph area.

Select any type of graph in [Graph Option].

The data collection will automatically end when the graph

point reaches the right end of the graph display.

It is not recommended to select the [Rotation-Tilt] graph

because the data collection finishes in a relatively short time.

(This graph type is for the 5th experiment, nutation.)

Select a flow mode graph type by depressing [Flow mode]

icon.

(2-5) Set the range of each axis.

Set the range of 𝑥𝑥-axis (time) at maximum (60 seconds).

The data collection will automatically end after 60 seconds.


Lab Manual




(3) Level the gyroscope.

Follow the steps below. Do not move the apparatus after

finishing leveling it.

(4) Balance the gyroscope axle.

The counterweight arm has three weights. Adjust the posi-

tions of these weights so that the gyroscope axle is exactly

balanced in the horizontal position.

(5) Spin the flywheel.

Spin up the flywheel to high speed using the string.

Note

In this experiment, you will use the gyroscope equipped

with several sensors and components on the gyroscope

axle. They all act downward force in the 𝑥𝑥-direction. Be-

cause we do not know their weights and distances from

the pivot, we cannot calculate net torque.

If the axle is exactly balanced in the horizontal position

by adjusting the position of counterweights, we don’t have

to consider the torque of any component (including the

flywheel) on the axle.

Instead, to produce the turning torque, you will attach an

add-on weight to the front of the flywheel. Then you can

easily calculate the torque by measuring the mass and

the position of the add-on weight.


Lab Manual




(6) Hang an add-on weight.

To produce a tipping torque, hang the 100g weight on the

𝑅𝑅 = 0.15 m hanger. (The hangers are 0.15, 0.17, 0.19, and

0.21 m away from the pivot.)

(7) Let the gyroscope precess.

If you spin the flywheel in the direction as shown above, the

gyroscope will precess counterclockwise.

In the general case, i.e. for arbitrary initial conditions, the

motion of the gyroscope is a superposition of precession and

nutation. Nutation is caused by a possible small deviation of

the vector of own angular momentum from the axis of sym-

metry. This deviation is absent only for carefully chosen spe-

cific initial conditions. It means you have to help the gyro-

scope start precessing with the gyroscope axle horizontal by

moving it along in the correct direction at the correct speed,

and then release gently.

If you touch the end of the axle, you can feel the speed of

the gyroscope. Help the gyroscope start precessing with the

gyroscope axle horizontal by moving it along in the correct

direction at the correct speed, and then release gently.

If you push it faster or slower than the correct speed, it will

move upward or downward.

When it precesses without nutation, begin data collection.

(8) Begin data collection.

Click [Start] to begin data collection. The display will start

plotting graph.

The data collection will automatically end after 60 seconds

(as you set in the step (2-5)).


Lab Manual




(9) Extract collected data.

Click [Datasheet] icon to extract data table

Click [Save] to save the data as text format file (*.txt).

(10) Open the data file.

You can use Microsoft Excel® to import data from a text into

a worksheet. To start the Text Import Wizard, on the Data (데

이터) tab, in the Get External Data (외부 데이터 가져오기)

group, click From Text (텍스트). Then, in the Import Text

File (텍스트 파일 가져오기) dialog box, double-click the text

file that you want to import.

In the step 1 of 3 of the Text Import Wizard, select Delimited

(구분 기호로 분리됨) for Original data type (원본 데이터

형식). In the step 2 of 3, check Comma (쉼표) and Space (공

백) for Delimiters (구분 기호).

(11) Analyze the data.

Cells in the column B contain the values of the rotating an-

gle of the rotating axle. The table below shows that the differ-

ence between B9 cell and B1061 cell is 360 degrees (1 com-

plete turn). Column A (time column) shows that it has taken

21.04 (= 21.12− 0.08) seconds for one complete turn.

The average RPM (rev/min) of the flywheel during this time

interval, is given by Average(D9:D1061).


Lab Manual




(12) Calculate the precession angular speed.

Verify equation (11).

Ω =𝑑𝑑𝜙𝜙𝑑𝑑𝑑𝑑 =

𝜏𝜏𝐿𝐿 =

𝑀𝑀𝘨𝘨𝑅𝑅𝐼𝐼𝜔𝜔 (11)

Experimental value of the precession angular speed is

Ω = 2𝜋𝜋/𝑇𝑇, where the period 𝑇𝑇 of the gyroscope is experi-

mentally obtained in the step (11).

Theoretical value of the precession angular speed is

Ω = 𝑀𝑀𝘨𝘨𝑅𝑅 𝐼𝐼𝜔𝜔⁄ , where 𝑀𝑀 is the mass of the add-on weight, 𝑅𝑅

is the distance from the pivot to the add-on weight, 𝐼𝐼 is the

moment of inertia of the flywheel which is obtained in the

experiment 1, and 𝜔𝜔 is the flywheel angular speed which is

obtained in the step (11). (1 rpm = 1 rev/min = 1/60 rev/s =

2𝜋𝜋 60⁄ rad/s)

(13) Repeat experiments.

Vary conditions and repeat experiment.

𝑀𝑀 𝑅𝑅 𝐼𝐼 𝜔𝜔 Ω =𝑀𝑀𝘨𝘨𝑅𝑅𝐼𝐼𝜔𝜔

Ω =2𝜋𝜋𝑇𝑇

1st

2nd

3rd

…

Experiment 5. Nutation

Change [Graph Option] to [Rotation-Tilt].

Spin up the wheel to high speed using the string. With the

add-on weight producing a tipping torque, hold the axle hori-

zontally, and then

① Gently tap it in the counter-precession direction

② Suddenly release it from rest

③ Gently tap it in the precession direction

The gyroscope undergoes nutation, as in patterns below.


Lab Manual




Your TA will inform you of the guidelines for writing the laboratory report during the lecture.

Please put your equipment in order as shown below.

□ Delete your data files and empty the trash can from the lab computer.

□ Turn off the Computer and the Interface.

□ Keep the Gyroscope Apparatus away from strong impact. Never touch the sensor units of it.

□ Unplug the Gyroscope DC Adapter.

□ Do not unplug the Serial-USB Cable from the computer.

□ Be careful not to knot or tangle the String.

□ Keep the Flywheel Parts, Spools of Thread, Scissors in the basket on the lecture table at the font of the laboratory.

Result & Discussion

End of LAB Checklist

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