Kara Sears
Gyroscopes are devices that use angular momentum to demonstrate an interesting effect. When a gyroscope is spun, it has angular momentum and wants to keep spinning. If it gets off-balance, gravity will push down on the heavier side, but this slight nudge won't affect it too much—instead, it will slightly change the orientation of its spin. This is why gyroscopes wobble and can remain upright for a long time. The conservation of angular momentum is an important part of the gyroscope's behaviour. But does it break the law of angular momentum?
| Characteristics | Values |
|---|---|
| Angular momentum | Vector quantity with direction and magnitude |
| Conservation of angular momentum | Conserved everywhere and at all times |
| Precession | Caused by torque from gravity acting on spinning mass |
| Energy involved in precession | Originates from kinetic energy in spinning gyroscope |
| Angular velocity | Represented by \(\omega\) |
| Kinetic energy associated with angular velocity | Represented by \(\frac{1}{2}I_{\mathrm{axle}}\omega^2\) |
| Torque | Affects both direction and magnitude of angular momentum |
| Direction of angular momentum | Determined by right-hand rule |
| Net torque | Represented by \(\text{net }\tau=\frac{\Delta \mathbf{L}}{\Delta t}\) |
| Gyroscopes | Precess around a vertical axis due to horizontal torque |
| Non-spinning gyroscope | Acquires angular momentum in the direction of torque and rotates about a horizontal axis |
| Earth | Acts like a gigantic gyroscope, precessing once in about 26,000 years due to torque of Sun and Moon |
What You'll Learn
Torque and angular momentum
Angular momentum is a vector quantity, meaning it has a magnitude and a direction. The conservation of angular momentum is an important part of the gyroscope. The rotating wheel on the interior of a gyroscope has an angular momentum vector proportional to its angular velocity.
Torque affects both the direction and the magnitude of angular momentum. The direction of the angular momentum of a rotating object can be determined using the right-hand rule. The right-hand rule states that the direction of angular momentum is the same as the direction of the angular velocity.
The equation for torque is given by:
> [latex]\boldsymbol{\textbf{net }\tau\:=}[/latex][latex]\boldsymbol{\frac{\Delta\textbf{L}}{\Delta{t}}}.]
This equation shows that the direction of the change in angular momentum is the same as the direction of the torque that creates it. For example, consider a bicycle wheel with handles attached to it. If a person holding the wheel tries to rotate it, they might expect the wheel to rotate in the direction they push it. However, due to the torque created by the forces they exert, the wheel's angular momentum changes direction but not magnitude, resulting in the wheel moving towards the person instead of rotating as expected.
The same logic applies to gyroscopes. The torque produced by the forces acting on a spinning gyroscope is perpendicular to the angular momentum, changing the direction of the torque but not its magnitude. As a result, the gyroscope precesses around a vertical axis. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque and rotates about a horizontal axis, falling over.
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The gyroscope's stability
The gyroscope is a device that operates on the principle of the conservation of angular momentum. When a gyroscope is spinning, it has angular momentum and, as a result, resists any changes to its axis of rotation. This resistance to changes in the axis of rotation provides stability to the gyroscope.
The conservation of angular momentum is a vector law, meaning that it conserves the magnitudes in each respective dimension. Angular momentum is a vector quantity, and therefore, has both direction and magnitude. The direction of angular momentum is the same as the direction of angular velocity.
The stability of the gyroscope can be understood by considering the forces acting on it. The two primary forces acting on a spinning gyroscope are its weight and the supporting force from the stand. These forces create a horizontal torque on the gyroscope, which results in a change in angular momentum. However, this change in angular momentum does not affect the magnitude of the angular velocity of the spinning wheel.
In summary, the gyroscope's stability is a result of the conservation of angular momentum, the minimisation of torque on the interior rotating wheel, and its ability to self-correct and resist disturbances to its axis of rotation. These factors combine to provide the gyroscope with a remarkable degree of stability, making it a valuable tool in various applications, including guidance systems and astronautics.
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The role of kinetic energy
When a gyroscope is set in motion, it possesses kinetic energy due to its rotational motion. This kinetic energy is directly related to the angular velocity of the gyroscope and is given by the equation:
> K.E = 1/2 * I * ω^2
Where K.E is the kinetic energy, I is the moment of inertia of the gyroscope, and ω is its angular velocity. This kinetic energy is essential for the gyroscope's stability and its ability to resist changes in its axis of rotation.
The kinetic energy of the gyroscope is closely linked to its angular momentum. Angular momentum is a vector quantity, which means it has both magnitude and direction. The angular momentum of the gyroscope is directly proportional to its angular velocity and is given by the equation:
> L = I * ω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. The conservation of angular momentum is a key principle in the behaviour of gyroscopes.
When a spinning gyroscope experiences a torque, such as the torque from gravity when it is dropped, the kinetic energy of the system changes. This change in kinetic energy results in precession, where the gyroscope's axis of rotation starts to rotate around a vertical axis. The energy involved in this precession originates from the kinetic energy of the spinning gyroscope.
The kinetic energy and angular momentum of a gyroscope are interconnected. To change the angular momentum, a torque must be applied, which also affects the kinetic energy. This is because the torque can change the direction of the angular momentum without altering its magnitude, resulting in a change in the kinetic energy associated with the new direction.
In summary, the kinetic energy of a gyroscope is essential for its stability and behaviour. It is linked to the angular momentum, which is conserved due to the gyroscope's tendency to resist disturbances in its axis of rotation. The interplay between kinetic energy and angular momentum, along with external torques, results in the unique characteristics of gyroscopes, such as their ability to precess and maintain stability.
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The gyroscope's axis of rotation
The axis of rotation of a gyroscope is a central component in understanding its behaviour. Angular momentum, being a vector quantity, has a direction as well as a magnitude. The right-hand rule is used to determine the direction of angular momentum, where the thumb points in the direction of angular velocity, while the fingers curl in the direction of the disk's rotation.
The conservation of angular momentum is evident in the behaviour of the gyroscope. When a torque is applied to a spinning gyroscope, the direction of the torque changes, but not its magnitude. This results in the gyroscope precessing around a vertical axis, as the torque is always horizontal and perpendicular to the angular momentum.
The Earth itself acts as a giant gyroscope, with its angular momentum pointing towards the North Star, Polaris. The torque exerted by the Sun and the Moon on the Earth's non-spherical shape causes it to precess once every 26,000 years.
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The Earth as a gyroscope
A gyroscope is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rotation (spin axis) is free to assume any orientation by itself. When rotating, the orientation of this axis is unaffected by tilting or rotation of the mounting, according to the conservation of angular momentum.
A spinning celestial body can be considered a gyroscope. The Earth, for example, exhibits gyroscopic behaviour. Due to its rotation, the Earth can be considered a gyroscope with a period of about 26,000 years. This gyroscopic precession is known as the 'precession of the equinoxes'.
The Earth's gyroscopic motion is the reason we have seasons. As the Earth rotates around the sun, the axial direction stays fixed due to the gyroscopic motion, resulting in one hemisphere being closer to the sun than the other for half of the year, and vice versa for the other half. This leads to seasonal changes.
The moon's gravity exerts a torque on the Earth, causing its axis to precess, similar to a gyroscope subjected to an off-axis torque. As a result, the Earth's axis will point in different directions over time, with a full cycle lasting about 26,000 years.
The Earth's behaviour as a gyroscope has implications for our understanding of the planet's dynamics and its impact on various natural phenomena, such as the changing of seasons.
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Frequently asked questions
No, a gyroscope does not break the law of angular momentum. In fact, it operates on the principle of the conservation of angular momentum.
The gyroscope has angular momentum due to its spinning motion. When the axis of rotation is disturbed, the gyroscope readjusts itself to preserve the angular momentum, thus providing stability.
Torque affects both the direction and magnitude of angular momentum. The direction of the change in angular momentum (ΔL) is the same as the direction of the torque (τ) that creates it.
The right-hand rule is used to determine the direction of angular velocity, momentum, and torque. For a disk rotating counterclockwise, the direction of angular velocity (ω) and angular momentum (L) is defined by the direction in which the thumb of your right hand points when you curl your fingers in the direction of the disk's rotation.
The Earth's angular momentum is along its axis and points towards the North Star (Polaris). Due to the torque exerted by the Sun and the Moon on its non-spherical shape, the Earth slowly precesses once in about 26,000 years.
