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Rutherford atom model doesn't explain stability of an atom properly. Following are the two drawbacks of Rutherford’s model:
(i) Centripetal acceleration is experienced by an electron when it moves in a circular orbit.According to electromagnetic theory, an accelerated electron 
must radiate energy in the form of electromagnetic waves. Therefore, if the accelerated electron lose energy by radiation, the energy of the electron continuously decreases and it must spiral down into the nucleus.Thus, the atom cannot be stable. But, it is well known that most of the atoms are stable.
 (ii) According to classical electromagnetic theory, the accelerating electron must radiate energy at a frequency proportional to the angular velocity of the electron. Therefore, as the electron spiral towards the nucleus, the angular velocity tends to become infinity and hence the frequency of the emitted energy will tend to infinity. This will result in a continuous spectrum with all possible wavelengths. But experiments reveal only line spectra of fixed wavelength from atoms. 
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Rutherford's model was compared with the solar system in which planets revolve around the sun in fixed circular paths.....but there is an important difference between the two .firstly whereas the electron and the nuclei are the microscopic particles,sun and planets were big bodies and hence newtonian law of motion do not apply to the motion of microscopic particles..(there is a separate branch called quantum mechanics for microscopic particles)
secondly sun and planets do not carry any charge like electrons and nuclei 
according to Maxwell's electromagnetic theory whenever a charged particle like electron is revolving in a field of force like that of nucleus it losses energy continuously in the form of radiations...because when a particle is revolving it undergoes acceleration due to change in direction even if the speed remains constant..thus the orbit of the revolving electron will keep on becoming smaller and smaller following a spiral path and ultimately the electron will fall into the nucleus (calculations show that it will take 10^-8 second to spiral into nucleus) other words atom should collapse...however this does not happen and the atom is quite stable thus rutherford model could not explain the stability of atom.