Why did Sita let Ravana abduct her when she could lift playfully Shiva's bow ?

There are 3 issues mentioned in the question.

1. Sri Rama broke Shiva’s bow, which was playfully lifted by Sita.
2. Ravana could not lift Shiva’s bow.
3. Sita lifts that great bow

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a) Sri Rama did not break Shiva Dhanush, as commonly believed. It is a myth. Sri Rama broke a very powerful Dhanush of Varuna.

In Ayodhya Kanda, Sita while explaining Anasuya about her marriage narrated that Janaka himself received a Great bow from VARUNA, the god of Water/rain-god but not from Lord Shiva.

महा यज्ने तदा तस्य वरुणेन महात्मना |

दत्तम् धनुर् वरम् प्रीत्या तूणी च अक्षय्य सायकौ || (Ayodhya Kanda 118th Sarga 39th Sloka)

"In ancient days, Janaka on the occasion of a great sacrifice, received with affection from Varuna (वरुणेन महात्मना) the rain-god, an excellent bow with two quivers that should never lack arrows."

असंचाल्यम् मनुष्यैः च यत्नेन अपि च गौरवात् |

तन् न शक्ता नमयितुम् स्वप्नेषु अपि नर अधिपाः || (Ayodhya Kanda 118th Sarga 40th Sloka)

"That bow was so heavy in weight that no man could lift it up nor any of the kings were bale to bend it even in their dreams."

The story of King Janaka himself receiving a Great bow from Varuna, the god of Water/rain-god, might have been altered with the story King Janaka's ancestor Nimireceiving the Great bow of Lord Shiva by the Vaishnavaites, who for the purpose of establishing the Superiority of Lord Vishnu over Lord Shiva, might have incorporated the story of Sri Rama breaking the bow of Lord Shiva in Bala Kanda.

Hence, Sri Rama broken the Great bow of VARUNA only, but not that of Lord Shiva.

It is to remembered that Varuna was very powerful God in Vedic era, which continued upto Mahabharata period.

Even Arjuna received his Gandiva of Varuna, through Agni only, and after the war it was returned to Varuna.

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b) Seetha further narrates the story of her marriage to Anasuya. Then Seetha relates that story to Anasuya in detail.

पति सम्योग सुलभम् वयो दृष्ट्वा तु मे पिता |

चिन्ताम् अभ्यगमद् दीनो वित्त नाशाद् इव अधनः || २-११८-३४

"Seeing my age to be such, when union with a husband can be easily had, my father (King Janaka) fell a prey to anxiety like an indigent man, feeling miserable by the loss of all his possessions."

तस्य बुद्धिर् इयम् जाता चिन्तयानस्य सम्ततम् |

स्वयम् वरम् तनूजायाः करिष्यामि इति धीमतः || २-११८-३८

"After reflecting thus deeply, the thought came to him, 'I shall inaugurate a Svayamvara, a process of self-choosing marriage, for my daughter."

तद् धनुः प्राप्य मे पित्रा व्याहृतम् सत्य वादिना |

समवाये नर इन्द्राणाम् पूर्वम् आमन्त्र्य पार्थिवान् || २-११८-४१

"My truthful father called all the princes first and informed them in a meeting about the bow to be lifted."

तच् च दृष्ट्वा धनुः श्रेष्ठम् गौरवाद् गिरि सम्निभम् |

अभिवाद्य नृपा जग्मुर् अशक्ताः तस्य तोलने || २-११८-४३

"Seeing that excellent bow, resembling a mountain in weight, and being unable to lift it up, the princes offered salutation to it and went away."

सुदीर्घस्य तु कालस्य राघवो अयम् महा द्युतिः |

विश्वामित्रेण सहितो यज्नम् द्रष्टुम् समागतः || २-११८-४४

लक्ष्मणेन सह भ्रात्रा रामः सत्य पराक्रमः |

"After a very long time, this Rama born in Raghu dynasty with a great splendour possessing a true valour, along with his brother Lakshmana together with a sage Visvamitra, came to witness a sacrifice."

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So there was no mention about Ravana in this narration. So Ravana did not attend to Sita's Swayamvara. And, hence the question of his inability to lift Shiva’s bow does not arise.

c) Finally, Ramayana does not mention about Sita lifting the Great bow. It is an interpolated story, created by film directors of Telugu Film industry.