--> Stable isotopes tend to lie within a narrow band on the graph of neutrons vs protons (left chart)

--> Lighter elements are more stable when the number of protons = number of neutrons

--> For heavier elements, more neutrons are needed to counteract the repulsive forces between many protons, so the N/Z ratio increases (N > Z)

2) Nuclear Binding Energy Per Nucleon

--> A higher binding energy per nucleon means the nucleus is more tightly bound and stable

--> Heavier or lighter nuclei than iron-56 are less tightly bound, which is why light nuclei can fuse to become more stable and Heavy nuclei can split to become more stable

Radioactive waste is dangerous to living organisms because it emits ionizing radiation, such as alpha, beta, and gamma rays. This ionizing radiation has enough energy to:

1) break chemical bonds in DNA and other molecules inside cells, which can cause the cells to malfunction and die

2) alter DNA and lead to mutations which can result in cancer or be passed on if reproductive cells are affected

3) accumulate inside the body and irradiate tissues if inhaled, swallowed, or absorbed through the skin

4) remain active and hazardous for thousands of years

Potential energy is stored energy due to an object`s position and a negative potential energy means the object is in a state where it is naturally attracted to a lower energy state. To move the object to a higher potential energy state, you need to counteract the attractive force and do work against it.

Therefore, the induced current changes direction and the potential, which is directly proportional to the current would also change (Alternating Current), charging the phone wirelessly

E_{ktrans(lational)} = 1/2mv²

E_{krot(ational)} = 1/2 Iw²

E_{kvib(rational)}

emf: electromotive force in volts (V) or J C^-1

F/m = a = -w²x

a. The induced emf is proportional to the rate of change of magnetic flux, and because the magnetic field is changing as the magnet passes through the solenoid, therefore the magnetic flux will change and the emf will be induced.

b. Calculate the angular velocity of the magnet's oscillation.

F = ma and a = -ω²x

-kx = -mω²x

ω = sqrt(k/m)

ω = sqrt(20/0.2)

ω = 10

c. Determine an expressions for the induced emf in terms of B₀, ω, and other relevant quantities.

emf_induced = N x -Δ (BA cos (theta))/Δt

emf_induced = NB₀A (-si5n (ωt)) ω

d. Calculate the maximum induced current in the circuit

emf_peak = ωNB₀A

IR = ωNB₀A

I = 0.005 A

e. Explain what happens to the induced EMF if the amplitude of the magnet's oscillation doubles.

The induced EMF will increase as the amplitude of the magnet's oscillation doubles because a larger amplitude of oscillation means the magnet moves through a greater distance within the coil, resulting in a larger change in magnetic flux and thus a higher induced EMF.

The threshold frequency is the minimum frequency for photoelectric effect to take place. The threshold frequency can be found using the formula

The photoelectric effect is applied in the photovoltaic cells, which are found in solar panels.

Higher intensity, more electrons being emitted per second, meaning a larger current because current is the amount of electrons flowing through per second.

The work function ϕ is the minimum energy required to free an electron from the metal's surface and this is the electron's binding energy to the material.

The equation subtracts ϕ because you must use up part of the photon's energy to overcome the binding energy, E = hf is photon energy. If hf < ϕ, the minus sign makes K_max negative, which means no electrons are being emitted.

PET imaging is when a radioactive substance is injected to a patient. So, the factor that primarily influences the choice for the substance would be the isotope's half-life, shorter half-life, less time to stay in the body.

Alpha particles --> low penetration and high ionization ability (trapped in the skin)

Beta particles --> moderate penetration and moderate ionization ability

Gamma particles --> high penetration and low ionization ability

The magnitude of the potential energy here is the amount of energy needed to be supplied in order to separate the two masses to an infinite distance apart where gravitational interaction becomes zero. This is the gravitational binding energy, the energy debt needed to be overcome to break the gravitational bond.

The minus sign in the formula means the potential energy is negative when the two masses are bound together and by convention, zero potential energy is at r = infinity

Since it takes a POSITIVE energy input to move from the current position to r = infinity, the potential energy at any finite r must be LESS THAN ZERO.

A negative E_p indicates the system is bound and objects cannot be separated without doing work against gravity. Minus sign = the gravitational system is bound and you must add energy to reach ZERO potential at infinity.

The minus sign means the electron's energy is lower than zero, so it is not free. Therefore, you must add POSITIVE energy equal to the magnitude of that negative value to remove the electron from the atom or to ionize the atom. For example, to remove an electron from the first energy level, you need to add +13.6 eV of energy.

Because of this attraction from the SNF, the total mass of a nucleus is less than the sum of the individual masses of its separate nucleons and this mass defect Δm corresponds to the binding energy NBE = Δmc². This NBE is the energy you must supply to completely separate the nucleus into FREE protons and neutrons.

The potential energy of a bound nucleus is NEGATIVE compared to the zero level for free, infinitely separated nucleons. the minus sign means that the nucleus is in a bound state and its energy is lower than the total energy of the separated nucleons. The binding energy is the POSITIVE amount you need to add to overcome the SNF and bring the nucleus to ZERO potential energy (unbound state).

So, the minus sign clearly distinguish between BOUND systems with negative E_p from UNBOUND systems with zero or positive E_p.

In fission:

• A heavy nucleus splits into two (or more) smaller nuclei.

• The binding energy per nucleon of the fission products is greater than that of the original heavy nucleus.

• This means the nucleons in the products are held more tightly together.

The difference in total binding energy between the products and the original nucleus is released as kinetic energy of the split fragments, gamma rays, and energy by emitted neutrons.

A bound nucleus has negative potential energy compared to free nucleons at infinite separation (zero energy level). The minus sign means the system's energy is below zero, therefore the nucleus is in a bound state. To break it apart into free nucleons, you’d need to add positive energy equal to the magnitude of the binding energy.

In fission, when the products have more negative total potential energy, the change in energy is negative for the system, meaning energy is released to the surroundings.

The total energy of a bound system (like a nucleus) is negative relative to free nucleons at infinite separation.

• Energy of free nucleons (not bound) is set to 0.

• When nucleons bind, the system’s potential energy decreases and the bound state has less energy than the free particles.

• This lower energy is written as a negative value.

The minus sign means energy must be added (positive) to overcome the nuclear force and break the nucleus apart and a larger magnitude of negative binding energy means a more stable nucleus.

Four positions:

1. Above position

   • Magnetic flux is small and constant → No EMF.

   • Direction: Not applicable.

2. Below position

   • Magnetic flux is small and constant → No EMF.

   • Direction: Not applicable.

3. Left position

   • Magnetic flux increases when the switch is closed → EMF induced.

   • Direction: Field is going up, right-hand rule shows EMF going CCW.

4. Right position (same as left, symmetric)

   • Magnetic flux increases → EMF induced.

   • Direction: Symmetrical to the left position, so CCW.

Q2) 7.5 V

I_battery --> induces B inside the solenoid --> induces I --> Induces secondary B

EMF of the battery will create a change in flux, which induced a new or secondary EMF.

EMF_battery = -(dΦ/dt) = EMF_induced

AND the reaction force R when the lift is going upwards, the reaction force will change its direction and the initial direction of R going up will go down. Therefore, the 2 forces are acting on the same direction, so +.

(ii) Net torque about any axis is zero for rotational equilibrium

Δλ = λ (v/c)

larger wavelength shifts more

Destructive interference or the minimum would be when the crest meets the trough.

But frequency still changes because the closer the observer get to the source, the observer will hear more of the soundwave, which corresponds to the frequency.

1. Positions of maxima (where principal peaks occur):

   • The angles of principal maxima depend only on d and λ, not on N.

   • nλ = d sin θ

2. Intensity of principal maxima:

   • Maximum intensity scales as:

     I_max ∝ N²

   • If we double slits from 4 → 8, it should get brighter and not weaker

3. Intensity of secondary maxima:

   • Secondary maxima appear between the main peaks.

   • Their relative intensity compared to principal maxima decreases as N increases.

   • So when N increases, secondary maxima become weaker.

4. Width of principal maxima:

   • The central peaks become narrower (sharper) as N increases.

   • This is because constructive interference is “more selective” when there are more slits.

   • In fact, the angular width Δθ ∝ 1/N²

• With few slits (say 2 or 4), you don’t need the phase alignment to be perfect for the waves to still add up fairly strongly. This means that the intensity doesn’t drop very sharply when you move a little away from the central angle. → broader peaks.

• With many slits (say 8, 10, 100), the requirement for perfect alignment becomes stricter. If you move even slightly away from the constructive angle, the phase difference builds up quickly between the many slits, and destructive interference cancels the light much more strongly. → narrower peaks.

That’s why we say the system is “more selective”:

• It only “selects” angles that satisfy the constructive condition very precisely.

• Any small deviation leads to destructive interference, cutting the intensity sharply.

+++++

• When light enters a denser medium (glass), it slows down → bends towards the normal.

• When it exits back to air, it speeds up again → bends away from the normal.

• Since the two surfaces of the glass are parallel, the direction is restored, and the emergent ray is parallel to the incident ray, only shifted sideways.

So C is the answer

Because they share the same circular orbit, their speeds and hence the magnitudes of their radial velocities at these positions (purely along the line of sight) are equal. So the shifts are equal in size but opposite in direction, bringing the two spectral lines toward each other on the wavelength scale.

So D

Condition for constructive interference with initial phase difference

• Total phase difference at P must be an integer multiple of 2π

Δd = 8.2 − 7.6 = 0.6 m

Δϕ_total = π + 2π * Δd/λ = 2πn

Divide by π gives:

1+ 2Δd/λ = 2n

2Δd/λ = 2n - 1

λ = 2Δd / ( 2n-1 )

where 2n - 1 is always an odd integer.

SO, λ = 1.2 / odd integer

smallest odd integer is 1, substituting 1 to the equation gives λ = 1.2 m

Hence, the frequency heard will be lower than 1000 Hz and increasing as the train accelerates away.

SO:

• Before t = 10 s, frequency f is greater than 1000 Hz because the train is approaching.

• At t = 10 s, frequency exactly equals 1000 Hz (train stops).

• After t = 10 s, frequency f is less than 1000 Hz because the train is moving away.

++++++

• The train starts at some high speed approaching the station at t = 0 s and then slows down until it stops at t = 10 s.

• When the train is moving fast toward you, the frequency you hear is higher.

• As the train slows down, its speed decreases, so the Doppler shift effect reduces.

• This means the frequency you hear decreases over time as the train approaches because the train is getting slower even though it's still getting closer.

• At t=10 s, the train stops, so the frequency equals the emitted frequency (1000 Hz).

Only Graph D fits the equation

Explain how is this possible.

Hint: 1. What stays constant during refraction and what changes?

2. Think about topic E1 atomic physics and the cause of colour.

Colour in glass: WOULD STAY THE SAME

Even when the wavelength in glass changes, BECAUSE E = hf where if E changes, the colour would change as colour is the perception of the energy of light. THEREFORE, if f changes, then E changes and our perception of the energy would change, thus changing our perception of the colour.

However, in this case, the frequency stays constant, so the energy would stay the same and therefore, the colour would stay the same.

Insulating sphere where the charges are spread throughout and they're not free to move.

A sphere is HIGHLY symmetric, so all field strength would cancel out on all directions, which gives zero field strength until the radius is changed.

Galilean does not have Lorentz factor because it is not moving at 0.2c or above.

Time is constant in Galilean, but here it is not constant, it is dilated.

In Lorentz transformation, the t-coordinate is the instantaneous time seen on the clock.

Two types of q: t-coordinate, time dilation