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Gymnast climbing up or down rope - acceleration's direction? Watch

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    I understand that if you use the standard xy-plane, then the acceleration upwards is positive and downwards is negative.

    But because I'm thinking about this problem differently that confuses me. If a gymnast is accelerating up a rope (they climb up faster and faster), then Tension acts up and their weight acts down, but surely the force due to acceleration acts up?

    So T + F = mg (where F = ma)??

    But apparently not...apparently T = mg + ma
    Why?!

    Also...the tension on the rope is apparently greater (ignoring it's mass) if the gymnast accelerates up rather than down. Is this because the tension acts on a smaller and smaller section of rope as they accelerate closer to the top end of the rope (which btw is stuck to the ceiling)? Or is there a different reason why there can be such a significant increase in Tension as the gymnast climbs up the rope?
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    If you're hanging onto a rope and you want to accelerate upyards, do you pull on the rope with more or less force than if yo want to remain stationay?
    What's the difference in your pull force going to do to the tension in the rope above you?
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    Are you familiar with free-body diagrams? They are important for understanding problems like this. I won't draw one, but you should. Remember each force has an equal and opposite reaction, so you can't really talk about the direction of a force without saying which body its acting on.

    Case 1: Equilibrium

    The rope is in tension, and supports exactly the weight of the gymnast:

    T = mg

    Case 2: Gymnast accelerates up

    In order for the gymnast to accelerate upwards, the tension has to supply an additional force to the gymnast in addition to that needed to support his/her weight:

    T = ma + mg

    The acceleration upwards is positive, so there is an additional positive component to the tension.

    Case 3: Gymnast accelerates downwards

    The equation is, I believe, the same as before:

    T = ma + mg

    But now the acceleration is negative. Can you see what happens when a = -g? The rope will become slack, and after than the equation breaks down, because a rope can't support a negative tension (in a simple model).
 
 
 
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