DP IB Physics: SL
A. Space, time and motion
A.4 Rigid body mechanics
DP IB Physics: SLA. Space, time and motionA.4 Rigid body mechanicsLinking questions: |
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| a) | How does rotation apply to the motion of charged particles or satellites in orbit? |
| b) | How does conservation of angular momentum lead to the determination of the Bohr radius? |
| c) | How does a torque lead to simple harmonic motion? |
| d) | How are the laws of conservation and equations of motion in the context of rotational motion analogous to those governing linear motion? |
| e) | How can rotation lead to the generation of an electric current? |
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a) How does rotation apply to the motion of charged particles or satellites in orbit?
- Solution:
- The main way that rotation, whether of a charged particle or a celestial body, influences motion is by introducing forces that diverge from straightforward circular or straight-line pathways.
- The Earth’s rotation, a revolving celestial body, has a little but discernible impact on the course of satellites in orbit, leading them to gradually drift. The circular or spiral route that charged particles follow in a magnetic field is determined by how they rotate in response to the magnetic force.
- ⇒ The motion of a charged particle in an orbit:
- Helical Motion:
- A charged particle in a magnetic field feels a force that is perpendicular to both the magnetic field and its velocity.
- The particle moves in a circle due to this force, but if it also has a velocity component parallel to the field, the motion becomes a helix.
- Drift across the Field:
- The particle’s trajectory may progressively change in the direction of the magnetic field due to a drift caused by the combination of circular motion and the velocity component of the particle.
- Rotation:
- The particle’s charge and the magnetic field’s direction dictate the direction of rotation. According to the right-hand rule, positive charges spin in one direction while negative charges rotate in the other direction.
- Figure 1 Motion of a charged particle in an orbit
- ⇒ The motion of a satellite in an orbit:
- Geostationary Orbits:
- A satellite must circle above the equator in the same direction as Earth’s rotation and have the same orbital period (24 hours) as Earth in order to look stationary over a certain point on Earth. The satellite will remain in its location with respect to the ground thanks to this alignment.
- Stability of Circular Orbits:
- The rotation of the Earth adds to the overall dynamic equilibrium even if it has no direct effect on a satellite’s circular orbit. By balancing the force of gravity, the satellite maintains a steady orbit.
- Rotation of Satellites:
- Non-spherical satellites in particular have the ability to revolve on their own axis. The stability and orientation of the satellite may be impacted by this rotation, which is impacted by the gravitational pull of the planet they orbit.
- Figure 2 Satellite in orbit
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b) How does conservation of angular momentum lead to the determination of the Bohr radius?
- Solution:
- The Bohr radius in Bohr’s model is determined by the quantization of angular momentum and the conservation of angular momentum.
- Multiples of h/2π, where h is Planck’s constant, are used to quantify angular momentum. The computation of the potential orbital radii of electrons is made feasible by this quantization in conjunction with the classical requirement that the centripetal force (caused by the attraction of the nucleus) matches the Coulomb force.
- Figure 3 Conservation of angular momentum lead to determination of the Bohr radius
- ⇒ Coulomb force = Centripetal force:
- The electrical pull between the positive nucleus and the negative electron keeps the electron in orbit:
- [math]\frac{ke^2}{r^2} = \frac{m_e v^2}{r}[/math]
- – [math]k = \frac{1}{4\pi\varepsilon_0}[/math] (Coulomb^’ sconstant)
- – e is the elementary charge
- – R is the radius of the orbit
- ⇒ Solving to find the Bohr Radius:
- Use Bohr’s quantized angular momentum:
- [math]m_e v r = \hbar \\
v = \frac{\hbar}{m_e r}[/math] - Substitute into the force
- [math] \begin{gather}
\frac{ke^2}{r^2} = \frac{m_e}{r} \left( \frac{\hbar}{m_e r} \right)^2 \\
\frac{ke^2}{r^2} = \frac{\hbar^2}{m_e r^3} \\
r = \frac{\hbar^2}{k m_e e^2} \\
\text{This value is known as the Bohr radius, denoted by } a_0: \\
a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2} \\
a_0 \approx 5.29 \times 10^{-11} \, \text{m}
\end{gather} [/math] - ⇒ Role of angular momentum conservation:
- The fundamental explanation for why electrons can only occupy certain orbits is Bohr’s notion that angular momentum is quantized (only discrete values permitted) yet preserved.
- The Bohr radius — the minimum permitted orbital radius for an electron in hydrogen — is derived directly from this.
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c) How does a torque lead to simple harmonic motion?
- Solution:
- The secret to simple harmonic motion in rotating systems is a restoring torque that is proportionate to the angular displacement and in the opposite direction. The item oscillates back and forth around an equilibrium point as a result of this torque.
- Restoring Torque:
- In order to return the system to its equilibrium position in simple harmonic motion, a force or torque must be present.
- The displacement (or, in the case of rotational motion, the angular displacement) from the equilibrium determines the magnitude of this restoring force or torque.
- Angular Displacement:
- A torque serves as the restoring force for rotating systems, such as a pendulum or torsional pendulum. The system is rotated back towards its equilibrium position by this torque.
- Proportionality:
- The restoring torque must work in the opposite direction yet be proportionate to the angular displacement in order for simple harmonic motion to occur. This indicates that the restoring torque increases in strength as the item deviates from the equilibrium, drawing it back towards the Centre.
- Example:
- Pendulum
- Take a look at a basic pendulum. The gravitational component operating tangentially along the pendulum’s swing arc produces a restoring torque when the pendulum deviates from its vertical equilibrium.
- This torque serves to bring the pendulum back to its vertical position and is proportional to the sine of the angle of displacement (about proportional to the displacement itself for small angles).
- Figure 4 Simple pendulum
- Torsional Pendulum:
- A disc is fastened to a wire, which may be twisted, to create a torsional pendulum. The disc stores rotational energy when it is twisted, and the wire’s restoring torque is proportional to the twist angle.
- Figure 5 Torsional pendulum
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d) How are the laws of conservation and equations of motion in the context of rotational motion analogous to those governing linear motion?
- Solution:
- For rotational motion, the equations of motion and conservation laws are comparable to those for linear motion. Similar ideas and formulas apply in both situations, substituting rotating variables (such as torque and angular momentum) for linear ones (such as force and momentum).
- In particular, the equations of rotational motion are comparable to those of linear motion, and the conservation of angular momentum is comparable to that of linear momentum.
- ⇒ Conservation laws:
- Linear momentum:
- When there is no net external force operating on a system, linear momentum is preserved.
- Angular momentum:
- When there is no net external torque operating on a system, angular momentum is preserved.
The rotating equivalent of force is torque, which modifies angular momentum. - Analogies between Linear and rotational motion:
| Linear motion | Rotational motion |
|---|---|
| Displacement:[math]x[/math] | Angular displacement:[math]\theta[/math] |
| Velocity:[math]v = \frac{dx}{dt}[/math] | Angular velocity:[math]\omega = \frac{d\theta}{dt}[/math] |
| Acceleration:[math]a = \frac{dv}{dt}[/math] | Angular acceleration: [math]a = \frac{d\omega}{dt}[/math] |
| Mass: [math]m[/math] | Moment of inertia:[math]I[/math] |
| Force:[math]F[/math] | Torque: [math]τ[/math] |
| Newton’s Second law:[math]F = ma[/math] | Rotational form: [math]τ = Ia[/math] |
| Momentum:[math]p = mv[/math] | Angular momentum:[math]L = Iω[/math] |
| Impulse:[math]F∆t = ∆p[/math] | Angular impulse:[math]τ∆t = ∆L[/math] |
| Kinetic energy:[math]\frac{1}{2} mv^2[/math] | Rotational KE [math]\frac{1}{2} I \omega^2[/math] |
- Figure 6 Linear motion and rotational motion
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e) How can rotation lead to the generation of an electric current?
- Solution:
- An electric current can be produced by rotation, especially of a coil in a magnetic field, using a process known as electromagnetic induction.
- In order to produce an electromotive force (EMF) and, ultimately, an electric current, this process depends on the fluctuating magnetic flux as the coil turns.
- The Rotating Coil:
- – A magnetic field surrounds a coil of wire.
- – The coil is attached to a rotating source, such a motor or turbine.
- – The magnetic flux flowing through the coil varies as it turns.
- Figure 7 Rotary motion in a turbine convert into electric energy
- Changing Magnetic Flux:
- – The quantity of magnetic field lines that flow through a certain area is known as the magnetic flux.
- – The magnetic flux varies as the coil spins because of the change in the angle between the coil’s plane and the magnetic field.
- – The secret to creating an EMF is this fluctuating flux.
- Electromagnetic Induction:
- – A fluctuating magnetic flux causes an electromagnetic field (EMF) to be induced in a conductor, in this case the coil, in accordance with Faraday’s Law of Induction.
- – Electric current is driven through the coil by this EMF, which is a voltage.
- – The number of turns in the coil, the magnetic field’s intensity, and the rotational speed all affect how much current and EMF are induced.
- [math]\varepsilon = -\frac{d\Phi_B}{dt}[/math]
- – ε is the induces EMF (voltage)
- [math]\Phi_B = B A \cos{\theta}[/math] is the magnetic flux
- – B is the magnetic field strength
- – θ is the angle between the field and the area vector