Learn Spanish (Latin America): Veo el péndulo cayendo hacia nosotros.

When using perception verbs like ver, using the gerund (cayendo) emphasizes that you are witnessing an action currently in progress. If you used the infinitive (caer), it would frame the action as a whole, completed event from start to finish. Here, the focus is on the continuous, terrifying motion of the pendulum approaching.

Could I replace el péndulo with a pronoun, and if so, where would it go?

Yes, you can replace it with the direct object pronoun lo. It must go before the conjugated verb: Lo veo cayendo hacia nosotros. You might be tempted to attach it to the end of the gerund (veo cayéndolo), but that is incorrect here. The pronoun lo is the object of what you see (ver), not the object of the falling (caer), so it cannot attach to the gerund.

Why is the gerund spelled cayendo instead of caiendo?

To form the gerund of caer, you add -iendo to the stem ca-. However, Spanish spelling rules dictate that an unaccented 'i' between two vowels must change to a 'y'. Therefore, caiendo becomes cayendo. You have likely seen this same spelling rule with verbs like leer (leyendo) and oír (oyendo).