So I was trying to figure out the free-body diagram for a pendulum, and I stumbled upon a video that explains it pretty well (see screenshot).
What I don't understand is how we can have a centripetal acceleration. Shouldn't the tension of the rope be equal to the radial component (or in this case, $x$ component) of the gravitational force? I assumed we have
$$
T=-G_{g_x}.
$$
However, that seems odd too, because sure we have a centripetal force if our motion is circular… So could someone explain what's going on?
Best Answer
Just because the motion of the bob is along a circle doesn't mean that its acceleration has to be tangent to it - which is, I think, the root of your confusion. The bob has both a tangential acceleration to change the magnitude of its velocity vector (because it is speeding up or slowing down according to whether it is going down the circle or up the circle currently) and also a radial (centripetal acceleration) that changes the direction of the velocity vector.
So, no, the tension should not equal the radial component of gravity because the bob does have net acceleration in this direction.