## ^Calculating work from F – x graph

^Calculating work from F – x graph is read as work done by a force ‘Fx’ in moving an object form point A to point B is equal to area under F – x graph bounded with the displacement – axis under position limits of the point A & B. Conventionally upward areas are +ve & downward –ve.

## ^Friction enables us to walk

^Friction enables us to walk

In order to run or walk forward we push (P) the ground backward at some angle to vertical.

The reaction of this push can be resolved in components. The vertical component balances our weight, where as the horizontal component push enables us to walk forward.

For no slip, FL ≥ P sinθ

or  μ Pcosθ ≥ Psinθ or  μ ≥ tanθ

On a smooth surface we fail to exert enough horizontal push on the ground as a result receive insufficient reaction i.e. the friction between our feet and the ground is greatly reduced, which causes us to slip. We can avoid slip by taking small steps.

## ^Cars on curved bridges

^Cars on curved bridges

Three identical cars A, B and C are moving at the same uniform, speed on three bridges. The car  A goes  on  a  plane  bridge, B on a  bridge convex  upward  and  C goes  on   a  bridge concave upward . Let  NA, NB and NC be  the normal  forces exerted  by  the cars  on the bridges  when they  are at  the  middle  of  bridges.

## ^Moderator

^Moderator

The neutrons produced in fission of 235U nuclei have average KE about 2 MeV. Such neutrons are called fast neutrons. These fast neutrons have more tendency to escape instead of triggering another fission reaction. Slow neutrons are more efficient in inducing fission in 92U235 nuclei than fast neutrons. By the use of a moderator, the fast neutrons are slowed to thermal velocities i.e. velocities » 2200 m/s & energies » 0.0235 eV, it is same as that of atoms and molecules at room temperatures, such slow moving neutrons are called thermal neutrons. Light target are better moderators. The commonly used moderator are water, heavy water (D2O), graphite and beryllium. About 25 collisions with deutrons (present in heavy water) or 100 collisions with carbon or beryllium are sufficient to slow down a neutron from 2 MeV to thermal energies.

A good moderator must have:

1. low atomic weight
2. should collide elastically with neutrons.
3. should not absorb the neutrons

## ^Comparison of α, β & ϒ  rays Here IP = Ionizing power & PP = Penetrating power

Also All the three type of rays namely α, β & ϒ affect photographic plate  produce flourescence & artificial radioactivity.

## ^γ-decay

^γ-decay

Most radioisotopes, after an alpha decay or a beta decay, leave the daughter nucleus in an excited state, these excited nuclei make a transition to a state of lower energy by emitting a photon. These photons are charge less, mass less & high energy electromagnetic waves (of the order of million electron volt) & are called the gamma rays.

ZXA (unstable nuclei) → ZXA (stable nuclei) + γ

## ^Linear momentum conservation

^Linear momentum conservation will be a constant , in other words the linear momentum of a system is constant in time if net external force acting on it is zero. LCLM is equivalent to NTL. Consider a two particle system. Let the total linear momentum of the system is not changing with the time i.e.  i.e., the two forces i.e. action & reaction are equal & opposite. i.e. for a two particle system the linear momentum of a system will not change with the time only if forces acting on the system are equal & opposite so that net force on the system is zero.

This justifies that LCLM is equivalent to NTL.

## ^β-decay

^β-decay

In the beta-minus decay, a neutron inside the nucleus transforms into a proton with the emission of an electron and anti-neutrino are emitted. Note, the spins of the neutron, proton and electron are all 1/2. In the beta-plus decay, a proton inside the nucleus  transforms into a neutron with the emission of a positron and neutrino are emitted. ## ^α-decay

^α-decay

Consider the following decay As a nucleus decays due to internal force of repulsion, there is no net external force on it, hence in any nuclear reaction linear momentum must be conserved.

Before disintegration, the nucleus can be assumed to be at rest, so the total momentum was zero. After disintegration let it be mava & mD vD for  alpha particle & daughter nuclei respectively. To conserve linear momentum the total vector momentum must still be zero i.e.  mava + mDvD = 0 or mava = -mDvD

i.e. momentum of a particle must be equal & opposite to that of daughter’s nucleus.

In magnitude, mava = mDvD  As mass of alpha particle is much lighter than thorium, thus the lighter α particle carries off most of the energy in the form its KE (about 98% of the total KE).

1.    A heavy unstable nucleus (e.g. Uranium, polonium, radium, thorium, actinium, etc.) disintegrates itself naturally, spontaneously & randomly without being forced by any external agent to do so until it acquires stability.

2.    The disintegration is independent of all physical and chemical conditions and so it can neither be accelerated nor retarded.

3.    The disintegration is random. It is purely a matter of chance for any atom to disintegrate first. It is not possible to predict whether a particular nucleus will decay in a given time interval.

4.    The activity (or rate of disintegration, A or R) of a radioactive sample at any instant is directly proportional to the number of undecayed nuclei present in the sample at that instant. Here λ = disintegration constant or decay constant. & N0 = no. of the atoms present initially i.e. at t = 0.

From above result we can say

• The number of active nuclei in a radioactive sample decreases exponentially with time.
• The disintegration is fast in the beginning but becomes slower and with the passage of time.
• Irrespective of its nature a radioactive sample will take infinitively long time to disintegrate complete.
• The larger the value of decay constant l the higher is the rate of disintegration.

5. Half life (T): 6. Fraction ‘f’ of substance left undecayed after ‘n’ half lives is given by: 7. Mean life (τ): 8. Decay constant (λ) is the reciprocal of time for which 9. λ = 0 for a stable element (e.g. Pb).

10. (a) 1 Bacqueral (Bq) = 1 d.p.s.

(b) 1 Curie (rd) = 3.7 x 1010 d.p.s.

(c) 1 Rutherford (Rd) = 106 d.p.s.

Here d.p.s. = disintegrations per second.

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