Physics NCERT Exemplar Solutions Class 12th Chapter Thirteen Nuclei

Consider a radioactive nucleus A which decays to a stable nucleus C through the following sequence:

                                                                                   A -> B -> C

Here B is an intermediate nuclei which is also radioactive. Considering that there are N₀ atoms of A initially, plot the graph showing the variation of number of atoms of A and B versus time.

A piece of wood from the ruins of an ancient building was found to have a ¹⁴ C activity of 12 disintegrations per minute per gram of its carbon content. The ¹⁴ C activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of  ¹⁴ C is 5760 yr.

Are the nucleons fundamental particles, or do they consist of still smaller parts? One way to find out is to probe a nucleon just as Rutherford probed an atom. What should be the kinetic energy of an electron for it to be able to probe a nucleon? Assume the diameter of a nucleon to be approximately 10^-15

A nuclide 1 is said to be the mirror isobar of nuclide 2 if Z₁=N₂ and Z₂ = N₁ (a) What nuclide is a mirror isobar of ₁₁²³Na ? (b) Which nuclide out of the two mirror isobars have greater binding energy and why?

In pair annihilation, an electron and a positron destroy each other to produce gamma radiations. How is the momentum conserved?

Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is

³⁸sulphur $\stackrel{\mathit{h}\mathit{a}\mathit{l}\mathit{f}\mathit{}\mathit{l}\mathit{i}\mathit{f}\mathit{e}}{\to }$ ³⁸Cl $\stackrel{\mathit{h}\mathit{a}\mathit{l}\mathit{f}\mathit{}\mathit{l}\mathit{i}\mathit{f}\mathit{e}}{\to }$ 38Ar(stable)

Assume that we start 1000 ³⁸S nuclei at time t=0. The number of ³⁸Cl is of count zero at t=0 and will be again zero at t= $\mathit{\infty }$  at what value of t, would the number of counts be a a maximum?

Deuteron is a bound state of a neutron and a proton with a binding energy B = 2.2 MeV. A γ-ray of energy E is aimed at a deuteron nucleus to try to break it into a (neutron + proton) such that the n and p move in the direction of the incident γ-ray. If  E = B, show that this cannot happen. Hence, calculate how much bigger than B must be E be for such a process to happen.

The deuteron is bound by nuclear forces just as H-atom is made up of p and e bound by electrostatic forces. If we consider the force between neutron and proton in a deuteron as given in the form a coulomb potential but with an effective charge e’.

F= ke’²/r estimate the value of e’/e given that the the binding energy of a duetron is 2.2MeV.

Before the neutrino hypothesis, the beta decay process was thought to be the transition.
n —> p + e
If this was true, show that if the neutron was at rest, the proton and electron would emerge with fixed energies and calculate them. Experimentally, the electron energy was found to have a large range.

The activity R of a unknown radioactive nuclide is measurement at hourly intervals the result found are tabulated as follows.

Plot the graph of R verses t and calculate half life from the graph.

Plot the graph of In(R/Ro) verses t and obtain the value of half life from the graph.

Nuclei with magic no. of proton Z=2,8,20,28,50,52 and magic no. of neutrons N=2,8,20,28,50,82 and 126 are found to be stable. (i) Verify this by calculating the proton separation energy Sp for. ¹²⁰Sn ( Z = 50 ) and ¹²¹Sb( Z = 51 ). The proton separation energy for a nuclide is the minimum energy required to separated the least tightly bound proton form a nucleus of that nuclide. It is given by Sp = ( M_z-1 , _N + M_H−M_z ,_N )c² 𝑆𝑝 = ( 𝑀𝑧 - 1 , 𝑁 + 𝑀_H – 𝑀_Z , 𝑁 ) 𝑐 2 . given ¹¹⁹𝑆𝑛 = 118.9058 𝑢 , . ¹²⁰𝑆𝑛 = 199.902199  , . ¹²¹𝑆𝑏 = 120.903824 𝑢 ,¹𝐻 = 1.0078252 𝑢 (ii) what does the existence of magic number indicate?

Suppose we consider a large number of containers each containing initially 10000 atoms of a radioactive material with a half life of 1 year. After 1 year,
(a) All the containers will have 5000 atoms of the material
(b) All the containers will contain the same number of atoms of the material but that number will only be approximately 5000
(c) The containers will in general have different numbers of the atoms of the material but their average will be close to 5000
(d) None of the containers can have more than 5000 atoms

The gravitational force between a H atom and a another particle of mass m will be given newton’s law

F=GMm/r² where r in km

(a) M=m_proton+m_electron

(b) M=m_proton+m_electron-

(c) M is not relate to the mass of the hydrogen atom

(d) M=m_proton+m_electron-(|V|)= magnitude of the potential energy of electron in H atom.

When a nucleus in an atom undergoes a radioactive decay, the electronic energy levels of the atom
(a) Do not change for any type of radioactivity
(b) Change for α and β-radioactivity but not for γ-radioactivity
(c) Change for α -radioactivity but not for others
(d) Change for β-radioactivity but not for others

M_e and M_s denote the atomic masses of the parent and he daughter nuclei respectively in radioactive decay. The Q value for a beta^-1 decay is Q₁ and that for beta+ decay is Q₂.if m_e denotes the mass of an electron, then which of the following statements is correct?

(a) Q₁=(M_x-M_y)c² and Q₂=[M_x-M_y-2m_e]c²

(b) Q₁=(M_x-M_y)c² and Q₂=[M_x-M_y]c²

(c) Q₁=(M_x-M_y-2m_e)c² and Q₂=[M_x-M_y-2C_e]c²

(d) Q₁=(M_x-M_y +2m_e)c² and Q₂=[M_x-M_y+2m_e]c²

Tritium is an isotope of hydrogen whose nucleus triton contains 2 neutrons and 1 proton. Free neutrons decay into p + e + n . If one of the neutrons in Triton decays, it would transform into He³ nucleus. This does not happen. This is because
(a) Triton energy is less than that of a He³ nucleus
(b) The electron created in the beta decay process cannot remain in the nucleus
(c) Both the neutrons in triton have to decay simultaneously resulting in a nucleus with 3 protons, which is not a He³ nucleus.
(d) Free neutrons decay due to external perturbations which is absent in triton nucleus

Heavy stable nuclei have more neutrons than protons. This is because of the fact that .

(a) Neutrons are heavier than protons
(b) Electrostatic force between protons are repulsive
(c) Neutrons decay into protons through beta decay
(d) Nuclear forces between neutrons are weaker than that between protons

In a nuclear reactor, moderators slow down the neutrons which come out in a fission process. The moderator used have light nuclei. Heavy nuclei will not serve the purpose, because
(a) They will break up
(b) Elastic collision of neutrons with heavy nuclei will not slow them down
(c) The net weight of the reactor would be unbearably high
(d) Substances with heavy nuclei do not occur in liquid or gaseous state at room temperature

Fusion processes, like combining two deuterons to form a He nucleus are impossible at ordinary temperatures and pressure. The reasons for this can be traced to the fact 
(a) Nuclear forces have short range
(b) Nuclei are positively charged
(c) The original nuclei must be completely ionized before fusion can take place
(d) The original nuclei must first break up before combining with each other

Samples of two radioactive nuclides A and B are taken. λ_Aand λ_B are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?
(a) Initial rate of decay of A is twice the initial rate of decay of B and λ_A = λ_B
(b) Initial rate of decay of A is twice the initial rate of decay of B and λ_A > λ_B
(c) Initial rate of decay of B is twice the initial rate of decay of A and  λ_A > λ_B
(d) Initial rate of decay of B is same as the rate of decay of A at t = 2h and  λ_A < λ_B

The variation of decay rate of two radioactive samples A and B with time is shown in figure.

Which of the following statements are true?

(a) Decay constant of A is greater thanB, hence A is always decays faster than B

(b) Decay constant of B is greater than that of A but its decay rate is smaller than that of A

(c) Decay constant of A is greater than that og B but it is does not always faster than B

(d) Decay constant of B is smaller than that of A but still decay rate becomes equal to that of A at later instant

Two projectiles are thrown with same initial velocity making an angle of $45°$ and $30°$ with the horizontal respectively. The ration of their respectively ranges will be:

In a Vernier Calipers, 10 divisions of Vernier scale is equal to the 9 divisions of main scale. Then both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of zero of the main scale and 4^th Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to 1mm. While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between 30 and 31 divisions of main scale reading and 6^th Vernier scale division exactly coincides with the main scale reading. The diameter of the spherical body will be:

A ball of mass 0.15 kg hits the wall with its initial speed of 12 ms^-1and bounces back without changing its initial speed. If the force applied by the wall on the ball during the contact is 100 N, calculate the time during of the contact of ball with the wall.

A body mass 8kg and another of mass 2kg are moving with equal kinetic energy. Thee ratio of their respective momenta will be:

Two uniformly charged spherical conductors A and B of radii 5mm and 10mm are separated by a distance of 2cm. If the spheres are connected by a conducting wire, then in equilibrium condition, the ratio of the magnitudes of the electric fields at the surface of the sphere A and B will be:

The oscillating magnetic field in a plane electromagnetic wave is given by $B_y=5\times 10^{-6}sin100\pi \left(5x-4\times 10^{8}t\right)T.$ The amplitude of electric field will be:

Light travels in two media M₁ and M₂ with speeds 1.5 × 108 ms1 and 2.0 × 108 ms1 respectively. The critical angle between them is:

A body is projected vertically upward from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be:

The maximum and minimum voltage of an amplitude modulated signal are 60V and 20V respectively. The percentage modulation index will be:

A nucleus of mass M at rest splits into two parts having masses $\frac{M\text{'}}{3}$ and $\frac{2M\text{'}}{2}\left(M\text{'}<M\right),$ The ratio of de Broglie wavelength of two parts will be:

An ice cube of dimensions 60cm × 50cm × 20cm is placed in an insulation box of wall thickness 1cm. The box keeping the ice cube at $0°C$ 898

of temperature is brought to a room of temperature $40°C.$ The rate of melting of ice is approximately:

(Latent heat of fusion of ice is 3.4 × 105 J kg1 and thermal conducting of insulation wall is 0.05 $W{m}^{-1}°C^{-1}$ )

A gas has n degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be:

A transverse wave s represented by y = $2sin\left(\omega t-kx\right)cm.$ The value of wavelength (in cm) for which the wave velocity becomes equal to the maximum particle velocity. Will be:

A battery of 6V is connected to the circuit as shown below. The current I drawn from the battery is:

A source of potential difference V is connected to the combination of two identical capacitors as shown in the figure. When key ‘K’ is closed, the total energy stored across the combination is E₁. Now key ‘K’ is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now E₂. The ratio E₁/E₂ will be:

Two concentric circular loops of radii r₁ = 30cm and r₂ = 50cm are placed in X – Y plane as shown in the figure. A current I = 7A is flowing through them in the direction as shown in figure. The net magnetic momentum of this system of two circular loops is approximately:

A velocity selector consists of electric field $\overrightarrow{E}=E\widehat{k}$ and magnetic field $\overrightarrow{B}=B\widehat{j}$ with B = 12 mT. The value of E required for and electron of energy 728 eV moving along the positive x-axis to pass undeflected is:

(Given. Mass of electron = 9.1 × 1031kg)

Two masses M₁ and M₂ are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass M₂ is twice that of M₁. The acceleration of the system is a₁. When the mass M₂ is thrice that of M₁ the acceleration of the system is a₂. The ratio $\frac{a_1}{a_2}$ will be:

Mass numbers of two nuclei are in the ratio of 4 : 3. Their nuclear densities will be in the ratio of

The area of cross section of the rope used to lift a load by a crane is 2.5 × 10^-4m². The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be:

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the centre will be________m.

Given, the length of the rod is $10\sqrt[]{3}m.$

In the given figure, the face AC of the equilateral prism is immersed in a liquid of refractive index ‘n’. For incident angle $60°$ at the side AC, the refracteel light beam just grazes along face AC. The refractive index of the liquid $n=\frac{\sqrt[]{x}}{4}.$

The value of x is_______.

(Given refractive index of glass = 1.5)

Two lighter nuclei combine to form a comparatively heavier nucleus by the relation given below: ${}_1{}^{2}X{+}_1^{2}X=\frac{4}{2}Y$

The binding energies per nucleon for are 1.1 Me V and 7.6

Me V respectively. The energy released in this process is________ Me V.

A uniformly heavy rod of mass 20kg, cross sectional  area 0.4 m² an length 20m is hanging from a fixe support. Neglecting the lateral contraction. The elongation in the rod due to its own weight is x × 10^-9m. The value of x i________.

The typical transfer characteristics of a transistor in CE configuration is shown in figure. A load resistor of $2k\Omega$ is connected in the collector branch of the circuit used. The input resistance of the transistor is $0.50k\Omega .$ The voltage gain of the transistor is________.

Three point charges of magnitude $5\mu C,0.16\mu C,and0.3\mu C$ are located at the vertices A, B, C of a right angled triangle whose sides are AB = 3cm, BC = $3\sqrt[]{2}$ and CA = 3 and point A is the right angle corner. Charge at point A experiences __________ N of electrostatic force due to the other two charges.

In a coil of resistance $8\Omega ,$ the magnetic flux due to an external magnetic field varies with time as $\varphi =\frac{2}{3}\left(9-t^2\right).$ The value of total heat produced in the coil, till the flux becomes zero, will be_______J.

A potentiometer wire of length 300 cm is connected in series with a resistance 780 $\Omega$ and a standard cell of emf 4V. A constant current flows through potentiometer wire. The length of the null point for cell of emf 20 mV is found to be 60 cm. The resistance of the potentiometer wire is ________ $\Omega$ .

As per given figures, two spring constants k and 2k are connected to mass m. If period of oscillation in figure (a) is 3s, then the period of oscillation in figure (b) will be $\sqrt[]{x}s.$ The value of x is________.

In a meter bridge experiment, for measuring unknown resistance ‘S’, the null point is obtained at a distance 30cm from the left side as shown at point D. If R is 5.6 $k\Omega$ , then the value of unknown resistance ‘S’ will be ___________ $\Omega .$

The one division of main scale of Vernier calipers reads 1mm and 10 divisions of Vernier scale is equal to the 9 divisions on main scale, When the two jaws of the instrument touch each other, the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between 4.1 cm and 4.2cm and 6^th Vernier division coincides with a main scale division. The diameter of the bob will be ________× 10^-2 cm.

Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the two beams are p/2 and p/3 at points A and B respectively. The difference between the resultant intensities at the two points is xI. The value of x will be________.

To light, a 50 W, 100 V lamp is connected, in series with a capacitor of a capacitance $\frac{50}{\pi \sqrt[]{x}}\mu F,$ with 200V, 50Hz AC source. The value of x will be________.

A 1 m long copper wire carries a current of 1 A. If the cross section of the wire is 2.0mm² and the resistivity of copper is 1.7 × 10^-8 $\Omega m,$  the force experienced by moving electron in the wire is _______× 10^-23N.

A long cylindrical volume contains a uniformly distributed charge of density $\rho C{m}^{-3}.$  The electric field inside the cylindrical volume at a distance $x=\frac{2{\epsilon }_0}{\rho }m$  from its axis is_________Vm^-1.

A mass 0.9kg, attached to a horizontal spring, executes SMH with an amplitude A₁. When this mass passes through its mean position, then a smaller mass of 124 g is placed over it both masses move together with amplitude A₂. If the ratio $\frac{A_1}{A_2}is\frac{\alpha }{\alpha -1},$  then the value of a will be______.

A square aluminium (shear modulus is 25 × 10⁹ Nm^-2) slab of side 60 cm and thickness 15 cm is subjected to a shearing force (on its narrow face) of 18.0 × 10⁴ N. The lower edge is riveted to the floor. The displacement of the upper edge is ________   $\mu m.$

A pulley of radius 1.5m is rotated about its axis by a force F = (12t – 3t²)N applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is 4.5 kg m², the number of rotation made by the pulley before its direction of motion is reversed, will be $\frac{K}{\pi }.$  The value of K is ________.

A ball of mass m is thrown vertically upwards. Another ball of mass 2 is thrown at an angle q with vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is $\frac{1}{x}.$  The value of x is _______.