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My friend phrased the original question like this:

If I tied a rope to myself and wrapped it all the way around the earth and then pulled (assuming the rope was already taut), which direction would I go and how long would it take for me to feel the tension force from the pull?

I then started to think about how long would it take for something really far away, say $300,000,000 \ \text{meters}$, to feel the tension from the pull of a rope? My initial thought was instantly but I feel like that would break some laws of physics.

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how long would it take for something really far away, say 300,000,000 meters, to feel the tension from the pull of a rope?

When you first pull a rope you produce a mechanical wave in the rope. This wave travels at a characteristic speed called the speed of sound. The speed of sound differs for different materials.

The speed of sound in a steel cable is around 5000 m/s. So it would take about 60,000 seconds or a little less than 17 hours. Other rope/cable materials would have a slower speed of sound and take longer.

My initial thought was instantly but I feel like that would break some laws of physics.

Clearly 17 hours is not instantaneous. During the time that the wave is in transit there will be no change in the tension on the distant end. This is regardless of the amplitude or force of the pull. The pull may even be fast or strong enough to break the rope without causing any disturbance on the far end until the wave has time to propagate. Thus, the rope will break well before the laws of physics break.

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    $\begingroup$ I guess I didn’t really think about a rope pull being a mechanical wave. That clears things up for me. Thanks! $\endgroup$ – Trey Jun 11 at 13:03
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Notice, the rope consists of the atoms & the tension force (pull) from one end of rope to the other is transmitted via the atoms along the length. Tension force from end of rope is transmitted by one atom to the next & so on until it reaches other end. This mechanism takes certain time to transmit pull from one end to the other. The time taken to transmit pull force from one end to the other will depend on the length of rope i.e. the greater the length of rope, the longer is the time taken to transmit pull from one end to the other.

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