Imagine there is a string, not a massless one but a heavy one, which is attached to the ceiling where the angle between the ceiling and the tangent at the end of the string is $\theta$. (First visualize it, or draw it somewhere). Let at any point the tension in the string be T. Now obviously there are components of forces acting, as T$\sin\theta$ is balancing its weight. But if I nudge the string from that point with a very lightly, I actually do not experience a resistive force (I've tried it several times). But I know T$\cos\theta$ is acting towards me.
So my question is, why do not I experience a resistive force?
Also, when we derive the expression for centripetal force, we do not take components of Tension but of its weight, because it "sounds" meaningless.
Is it logical to take components of T?
Answer
Look up the Catenary Curve properties and notice that the weight of the cable causes a reaction in both $x$ and $y$ axis as you noted. The vertical components of the reactions sum up to the weight of the cable, and the horizontal components are such as the tension being tangent to the shape of the cable.
Now what you "feel" when you move the support is the inertial force of moving the cable which I suppose it too small for you to feel. If you try with a heavy steel cable, and you accelerate in the same order of magnitude as gravity then you will feel inertial resistance.
Notice that you also feel an effective stiffness as you pull on the cable as the sag decreases and the incident angle creates higher horizontal reactions. In addition you might have elastic deformation also decreasing the above stiffness by $1/k_\mathrm{eff} = 1/k_\mathrm{elastic} + 1/k_\mathrm{geometric}$.
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