A question I encountered is whether a pendulum undergoing SHM on earth will oscillate on the ISS. Apparently the answer is it won't due to no effective gravity. The argument is there will be no tension force as all of it is on free fall. How is this different to on earth, where the entire setup is acted upon by g.
If you go the the International Space Station (ISS) with a pendulum and set it up, it will NOT oscillate. I will just float there. This is true even though there is powerful gravity present in the ISS. After all, it is only a few hundred kilometers above the surface of the earth and so the gravitational strength there is almost as strong as on the surface of the earth.
The ISS and everything in it including your pendulum, is in low orbit around the earth. Its acceleration towards the earth is close to $9.8\,m/s^2$ which is the value on the earth's surface. There is no other significant force acting on the ISS (or our pendulum) so it's in free fall.
On the earth's surface, there is a second force acting on our pendulum. This is exerted by whatever is holding up our pendulum against gravity. So the pendulum does work.
You could make a pendulum work in a spacecraft in space far away from all stars, planets, etc. You would hook up the pendulum to something in the spacecraft and then fire your rocket motors causing the spacecraft to accelerate. The accelerating spacecraft would pull on the pendulum string and you could then push the pendulum bob sideways and it would oscillate. No gravity needed.
No! A pedulum wouldn't oscillate inside the ISS. It's simply going to float much like every other object inside the space station. Objects oscillate only if there's a restoring force. For a pendulum, it is gravity and NOT the rope tension that provides the required restoring force. The tension is only there to guide the pendulum in a circle and not to make it oscillate. You may as well consider that a mass attached to a horizontal spring would oscillate even though there is no "tension". It is the weight of the pendulum that drives it back and forth while reversing action at every amplitude. This is the reason a pendulum oscillates. In microgravity however, a pendulum wouldn't oscillate since there is no gravity to drag it back and forth. But a pendulum can still do other things inside the ISS like move in a circle when you nudge it forward. There is no "gravity" (so to say) to stop it from climbing up and circling the pivot. The pendulum would continue moving with a uniform speed around the circle, with the rope tension providing the required central force. See Will a Pendulum Oscillate in Space