A familiar example is pushing a child on a swing. The child-swing system, like a pendulum, has a natural resonance frequency that depends on the weight of the child and the length of the chains. Optimal swinging height is achieved when the frequency of pushing closely matches the natural frequency of the swing. Pushing faster or slower provides a less efficient transfer of energy.
The natural precession frequency of a spin system is also known as the Larmor frequency.
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It should be noted that the B1 field does not have to applied exactly at the resonance frequency for some tipping of M to occur. It just has to be relatively "close" to the Larmor frequency (i.e, within 3-5%). Similarly, a grandmother pushing her child on a swing does not have to push at exactly the natural frequency to get motion to occur. It is just that the motion is maximal/optimal when applied precisely at that frequency.
Rigden JS. Quantum states and precession: the two discoveries of NMR. Rev Mod Physics 1986; 58(2):433-448.
Who was Larmor and how did he discover his famous frequency?