# Gyroscope

It is a common experience that a bicycle can be parked only when we support it with a stand otherwise it will fall down. But the same bicycle when moving on the road can be balanced very well on its two wheels. Wheels of a moving bicycle have a very large spin angular momentum which helps in maintaining the balance of the bicycle. This is the basic principle of a gyroscope. In this demonstration we will see how a spinning cycle wheel is affected by external torque.

You need a cycle wheel with an axle, and a rope.

1. Attach a rope to the axle of the cycle wheel.
2. Hold the cycle wheel by the rope. See that the wheel topples down.
3. Hold the cycle wheel from its axle and give it a spin in clockwise direction.
4. When the wheel acquires a large angular velocity, leave the axle and hold the wheel by the rope attached to the axle.
5. See that the wheel instead of toppling this time starts prĂ©session around the rope in the anticlockwise direction when seen from above.
6. Now hold the cycle wheel from the axle, spin it in anticlockwise direction, and again hold it with the rope but this time see that the wheel prĂ©cess in a clockwise direction around the rope.
7. When the wheel is held by the rope, (i) why does it topple when it is not spinning and (ii) why does it start precession when it is spinning?

When we hold the cycle wheel by the rope attached to the axle, the tension in the rope and the weight of cycle wheel acting through the centre of mass of the wheel cause a torque which topples the wheel. A spinning wheel has a spin angular momentum $$\vec{L}$$, whose direction is given by the right hand thumb rule. If you spin it in clockwise direction, the spin angular momentum $$\vec{L}$$ is away from you and perpendicular to the plane of the wheel. If you spin it in anticlockwise direction, $$\vec{L}$$ is towards you. If the angular velocity is large, $$\vec{L}$$ is also very large.

Now, when a torque is applied by holding the rope, the torque acts in a direction perpendicular to $$\vec{L}$$. This causes an additional small angular momentum $$\mathrm{d}\vec{L}$$ in the direction of the torque. The net angular momentum is now the vector sum of $$\vec{L}$$ and $$\mathrm{d}\vec{L}$$ which is no more perpendicular to the plane of the wheel but is slightly tilted towards the applied torque. This causes the angular momentum $$\vec{L}$$ to follow the torque and the wheel starts precession about the rope.

Reversing the direction of angular momentum $$L$$ causes the wheel to precess in an opposite sense.

## Variant: Gyrsocope using Motor and CD

You need a toy motor, two CD, aluminium sheet, and a battery. The equipment consists of a toy motor powered by two AA batteries.

Making: Take two CD and an aluminium sheet of the size of CD. Place the aluminium sheet between two CD and join them with tape or adhesive. Fix this assembly to the axle of motor. Paste/draw Netwon's disc of seven colour on outer CD. This equipment can be used for three concepts as discussed below.

Newton Disc: When connected to battery, the motor start rotating and you get a low cost Newton's disc.

Magnetic Brake: Bring a magnet close to the rotating disc. The speed reduces. You can explain about eddy currents and magnetic brake.

Gyroscope: Suspend entire assembly with a thread fixed a point slightly away from the centre of mass of the assembly. When motor is switched on, the CD rotates providing angular momentum to the system. The torque due to weight about centre of mass causes angular momentum (direction) to change and the system precesses about vertical axis. Changing the polarity of battery change direction of rotation. Physical law is $$\vec{\tau}=\frac{\mathrm{d}\vec{L}}{\mathrm{d}t}$$. This set up is good for classroom demo as carrying bicycle wheel in classroom is little bit cumbersome.

## Variant: Vector nature of angular momentum

Angular momentum of an object is given by $$\vec{L}=\sum m_i \vec{r}_i \times\vec{p}$$. For a rapidly spinning disk about its axes, it is very nearly $$\vec{L}=I\vec{\omega}$$ where $$I$$ is the momentum of inertia about the spin axis and $$\vec{\omega}$$ is the angular velocity vector.

The angular momentum can be changed by the torque $$\vec{\tau}=\sum \vec{r}_i\times \vec{F}$$. The equation is given by $$\vec{\tau}=\frac{\mathrm{d}\vec{L}}{\mathrm{d}t}$$.

The following experiment brings out the vector character of the angular momentum and how angular momentums add according to the vector addition rules. This experiment is also a very popular one, done with a rapidly spinning bicycle wheel held by a rope fixed at some place on the axis, away from the center of mass. We have made a compact and portable unit in place of bulky bicycle wheel.

1. Look at the apparatus. See how the battery is connected to the motor both of which are fixed on the CD-Axel unit.
2. Hold the axel in your hand and connected the battery to the motor. The CD starts spinning at a high angular speed.
3. While holding the axel in one hand, stretch the string tied to the axel, in vertical position. Keeps the CD surface vertical using the hand holding the surface.
4. Now gently leave the axel, while you are firmly holding the string.

See how the spinning disk starts precessing about the vertical string acquiring orbital and spin motion. As seen from the above, is the orbital motion clockwise or anticlockwise?

Because of the rapidly spinning DC, the system has its angular momentum along the spin axis. The tension in the string exerts torque $$\vec{\tau}=\vec{R}\times\vec{F}$$ on the system and in a time cause a change in angular momentum $$\mathrm{d}\vec{L}=\vec{\tau}\mathrm{d}t$$. Check that direction of $$\vec{\tau}$$ and hence of $$\mathrm{d}\vec{L}$$ is horizontal and perpendicular to the spin axis. The changed angular momentum $$\vec{L}+\mathrm{d}\vec{L}$$ will be obtained by the vector addition rule and it will be tilted from the original direction of spin axis. The system therefore turns to align the spin axis with the new direction of angular momentum $$\vec{L}+\mathrm{d}\vec{L}$$. This continuous and the CD is set into orbit motion.

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