Sign of work depends on sign of cosθ. As cosθ can be 0, +ve or – ve (recall –1 ≤ cosθ ≤ +1), hence the work done by a force also can be 0, + ve or – ve depending upon the angle between the force and the displacement.
Work done by a force acting at acute angles to displacement of a system is +ve.
Work done by a force acting at obtuse angles to displacement of a system is – ve.
+ve work increase the KE of the system.
– ve work decrease the KE of the system.
/wp-content/uploads/2021/04/logo-2.png00kp-web-admin/wp-content/uploads/2021/04/logo-2.pngkp-web-admin2022-01-18 14:57:272022-01-18 14:57:27^Sign of work
The y component of the gravitational pull mg cosθ presses the plane & is balanced by normal reaction from the plane, thus N = mg cosθ.
The x component of the gravitational pull mg sinθ (say, F) tends to move the block down the plane, but can do so, only if it exceeds the limiting friction (fL = msmg cosθ) on the block.
As the inclination θ is increased mgsinθ increases while the limiting frictional force (mS mg cosθ) decreases. At one stage, mgsinθ & mS mg cosθ become equal & balance each other then the body placed of the inclined plane at the verge of motion. If the block at this stage is given some velocity it will keep sliding down at constant velocity.
This angle θ is called angle of repose & abbreviated by
symbol β. At θ = β,
or mg sin θ = mS mg cosθ
or tanβ = μS
Also we can make following conclusions
If θ < β, then F < fL & block remains at rest. Friction on block is static & equals mg sinθ.
If θ = β, then F = fL & block remains at rest. Friction on block is static & equals mg sinθ or msmg cosθ.
If θ > β, then friction acting on the block is kinetic mK mg cosθ. This is less than mg sinθ & the block slides down the incline with a uniform acceleration. Using kinematic equations for UAM it can be checked that a block starting from rest will strike the ground with speed in time This is independent of mass of block.
If the inclined plane is smooth (i.e. μ = 0), then the acceleration of a body sliding down a of inclination is, a = g sin θ
Retardation by friction on a rough horizontal surface (i.e. θ = 00) is, a = μ g
The force required to move the block up along the inclined plane with const. acceleration a is, F = mgsinθ + μmgcosθ + ma
/wp-content/uploads/2021/04/logo-2.png00kp-web-admin/wp-content/uploads/2021/04/logo-2.pngkp-web-admin2022-01-17 19:10:052022-01-17 19:10:05^Motion on a rough inclined plane
When seen from inertial frame two types of forces act is a circular motion, one that changes speed & the other that changes direction.
A force that acts tangential of velocity changes the speed only, called tangential force. Its
magnitude is given by
A force acting normal to velocity towards turning centre, along the radius & changes direction is called centripetal force or radial force.
It is not a different type of force. It is actually the resultant of the forces acting on a system & directed towards the center of the circle. For planet revolving around sun gravitational force is centripetal. For a string whirled in a horizontal circle tension in the string is centripetal force. For oscillating pendulum resultant of tension in the string & normal component of weight is centripetal.
The electrostatic force of attraction between electrons & nucleus changes the direction of electrons revolving around the nucleus thus we can write
FCP = Felctrostatic
Gravitational force of attraction between the moon & the earth changes the direction of moon thus we can
write, FCP = Fgravitational
Magnitude of centripetal force on a mass moving with velocity v in a circular path of radius r at constant speed (i.e. uniform circular motion) is
/wp-content/uploads/2021/04/logo-2.png00kp-web-admin/wp-content/uploads/2021/04/logo-2.pngkp-web-admin2022-01-15 18:44:102022-01-15 18:44:10^Forces in circular motion
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.
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.
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 foralpha 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