DP IB Physics: SL

A. Space, time and motion

A.2 Forces and momentum

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  a) How do collisions between charge carriers and the atomic cores of a conductor result in thermal energy transfer?

• Solution:
• When charge carriers, such as electrons, collide with the atomic cores in a conductor, the energy of the collisions is transmitted to the atoms’ lattice, which intensifies their vibrations.
• Heat transmission results from this enhanced atomic vibration, which is a manifestation of thermal energy.
• Charge carriers gain kinetic energy:
• Electrons, or charge carriers, acquire kinetic energy from the electric field as an electric current pass through a conductor.
• Atomic core collisions:
• The positively charged atomic cores (ions) of the conductor’s lattice are continuously struck by these powerful charge carriers.
• Energy transfer:
• The atomic cores get a portion of the charge carriers’ kinetic energy during these encounters.
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• Figure 1 Thermal energy transfer
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  b) How can knowledge of electrical and magnetic forces allow the prediction of changes to the motion of charged particles?

• Solution:
• ⇒ Electric Forces on Charged Particles
• Electric forces are the interactions between charged particles, which can be either attractive or repulsive depending on whether the charges are opposite or the same.
• Opposite charges attract, while like charges repel. The strength of these forces depends on the magnitude of the charges and the distance between them.
• An electric force is exerted between any two charged objects.
• Objects with the same charge, both positive and both negative, will repel each other, and objects with opposite charges, one positive and one negative, will attract each other.
• [math]\bar{F} = q\bar{E}[/math]
• ⇒ Magnetic forces on moving charged particles:
• A magnetic field exerts a force on a moving charged particle, but only if the particle is in motion. This force is always perpendicular to both the particle’s velocity and the magnetic field, causing the particle to move in a curved path, often a circular path if the magnetic field is uniform.
• The magnitude of the force depends on the charge, velocity, magnetic field strength, and the angle between the velocity and the magnetic field.
• [math]\bar{F} = q\bar{v} \times \bar{B}[/math]
• No Force on Stationary Charges:
• Only moving charges are subject to the force of a magnetic field.
• Force Direction:
• The charge velocity and the magnetic field velocity are always perpendicular to the magnetic force. The right-hand rule determines this.
• Circular Motion:
• The charged particle travels in a circle due to the centripetal force of the magnetic field.
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• Figure 2 Attraction and repulsion forces between charges
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  c) How does the application of a restoring force act on a particle result in simple harmonic motion?

• Solution:
• Simple harmonic motion (SHM) is the result of a restoring force operating in the opposite direction and proportionate to the displacement from equilibrium.
• A particle is accelerated back towards equilibrium by the restoring force when it is moved out of equilibrium.
• The force diminishes as the particle gets closer to equilibrium, but because of inertia, it keeps moving past, repeating the process in the opposite direction.
• By Hooke’s Law
• [math]F = – kx[/math]
• – k is spring constant in SHM
• – This is restoring force
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• Figure 3 Harmonic motion
• By Newton’s second law
• [math]F = ma \\
  a = \frac{F}{m} \\
  F = -kx \\
  a = -\frac{k}{m}x[/math]
• Motion where a particle’s acceleration is proportional to its displacement from the equilibrium position and always points in that direction.
• [math]a = – ω^2 x[/math]
• – ω angular frequency from equilibrium
• – a acceleration
• – x displacement from equilibrium
• The negative sign shows the direction is opposite to displacement.
• Compare both equation
• [math]\omega = \sqrt{\frac{k}{m}}[/math]
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  d) How are concepts of equilibrium and conservation applied to understand matter and motion from the smallest atom to the whole universe?

• Solution:
• Equilibrium and conservation concepts are fundamental to understanding matter and motion, applicable from the smallest particles to the largest structures in the universe.
• Equilibrium describes a state of balance, where forces are balanced and there’s no net change, while conservation principles state that certain quantities (like matter, energy, or momentum) remain constant in a closed system.
• Conservation:
• Atoms can only be rearranged; they cannot be formed or destroyed. Similarly, the conservation of charge states that the total electric charge in a closed system stays constant.
• Equilibrium:
• When an object is at rest or travelling at a steady speed, its forces are balanced and it is in mechanical equilibrium.
• Equilibrium may also be reached by chemical processes when the forward and reverse reaction rates are equal.
• Electrostatic attraction between positive and negative charges balances the forces that would lead the electrons to fall into the nucleus, allowing atoms to exist in a state of dynamic equilibrium.
• By arguing that all matter is composed of minuscule particles (atoms or molecules) in perpetual, random motion, the kinetic theory of matter—also referred to as the kinetic molecular theory—explains how matter behaves.
• These particles have kinetic energy, and the theory uses their motion and organization to describe the characteristics of matter in its many states (solid, liquid, and gas).
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• Figure 4 Kinetic theory of matter
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  e) Why is no work done on a body moving along a circular trajectory?

• Solution:
• Because the centripetal force, which maintains a body’s circular motion, is always perpendicular to the body’s displacement, no work is done on a body moving on a circular trajectory.
• The force operating across a displacement is called work, and the dot product of vectors that are perpendicular to each other is zero.

  
  Figure 5 work done in a circular trajectory

• Centripetal Force:
• This force acts towards the center of the circle and is responsible for changing the direction of the body’s velocity, keeping it moving in a circular path.
• Perpendicularity:
• The centripetal force is always directed towards the center of the circle, while the body’s displacement is tangent to the circle at every point. Therefore, the angle between the force and displacement is 90 degrees.
• Work:
• Work done by a force is calculated as:
• [math]W = F · d · cos(θ),[/math]
• Where F is the force, d is the displacement, and θ is the angle between them.
• No Work Done:
• When a body moves on a circular trajectory, the work done is always zero since cos(90°) = 0 and the work done is the sum of force, displacement, and the cosine of the angle.
• Centripetal force, according to Physics Classroom, is unable to change an object’s overall mechanical energy.
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  f) In which way is conservation of momentum relevant to the workings of a nuclear power station?

• Solution:
• In nuclear power plant operations, conservation of momentum is essential, especially when it comes to heat transmission and nuclear fission events.
• The concept of conservation of momentum is crucial to comprehending how the released energy is used to produce electricity, even though it is not directly related to nuclear processes.
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• Figure 6 Working of the nuclear power station:
• Nuclear Fission and Momentum:
• When an atom splits, energy and neutrons are released.
• When these momentum-carrying neutrons strike other atoms, momentum is transferred and the atoms begin to move. The total heat transmission inside the reactor core is aided by this movement.
• Heat transport and Momentum:
• Convection and conduction work together to transport the heat produced in the reactor core to the water.
• A key component of this process is the movement of water molecules, which is fueled by the heat and momentum that are transmitted from the reactor core.
• Reactor Design and Momentum:
• The concepts of momentum conservation have an impact on the layout of a nuclear reactor’s fuel rods and the water flow through the reactor core. This keeps the reactor from overheating and guarantees effective heat transmission.
• Turbine Momentum and Operation:
• Hot water is utilized to power turbines as it leaves the reactor. The turbine blades rotate and produce energy as a result of the water’s momentum, which the reactor imparts to them.

• g) If experimental measurements contain uncertainties, how can laws be developed based on experimental evidence? (NOS)

• Solution:
• By carefully examining data, taking mistakes into account, and applying statistical techniques to make inferences within the bounds of uncertainty, scientific rules can be established based on experimental evidence despite the inherent uncertainty of experimental measurements.
• ⇒ Patterns and Consistency over Time:
• Repeated studies conducted under controlled settings can reveal distinct, persistent patterns, despite the uncertainty inherent in individual data. Scientists are able to create laws that explain dependable correlations in nature thanks to these patterns.
• Example:
• Several studies demonstrated that total momentum before and after collisions stayed relatively constant, even with measurement errors, leading to the development of the law of conservation of momentum.
• ⇒ Applying Statistical Techniques
• To account for uncertainties, scientists employ error analysis and statistics:
• – They compute confidence intervals, standard deviations, and averages.
• – This makes it easier to discern between genuine physical impacts and chance mistakes.
• ⇒ Increasing Precision:
• Improvements in tools and methods throughout time lower uncertainty and boost confidence in the outcomes.
• ⇒ Hypothesis and Forecasting
• Even if no one piece of evidence is flawless, a rule becomes stronger when it reliably explains findings and makes correct predictions about the future.
• – Reliable predictability, not absolute certainty, is the key.
• ⇒ Replication and Peer Review
• Reproducible evidence is the foundation of scientific laws. The validity of a rule is confirmed when several scientists repeat the same tests and obtain comparable findings.
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  h) What assumptions about the forces between molecules of gas allow for ideal gas behaviour? (NOS)

• Solution:
• The main presumption about intermolecular forces is that they are insignificant for a gas to behave optimally.
• Accordingly, it is believed that there are no forces of attraction or repulsiveness between gas molecules.

  
  Figure 7 Difference between ideal and non-ideal gas molecules

• No Intermolecular Forces:
• It is considered that ideal gases have no intermolecular forces, which means that their molecules are not attracted to or repulsed by one another.
• Intermolecular forces actually exist in real gas molecules, particularly at high pressures and low temperatures, thus this is oversimplified.
• Kinetic theory of gas:
• The foundation of the kinetic theory of gases, which offers a framework for comprehending gas behaviour, is this assumption.
• Ideal Gas Law:
• This presumption is necessary for the ideal gas law (PV = nRT).
• Real Gas Deviations:
• When intermolecular forces become substantial, real gases really behave differently from their ideal state. This occurs when molecules are moving slowly, like at low temperatures, or when they are near to one another, as under high pressures.