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In this post we will explore the physics of a swing, and answer the question: does the rope supporting the swing act on the girl at top of her swing?
The first step is to calculate how high up she is. To do that, let’s use a little trigonometry! We know that swinging once across an arc takes time t seconds. So if we call d meters (the distance between anchor point and where she swings) as x then from basic trigonometry we can find out what angle θ it would take for someone to go all way around in x seconds by solving tan(θ)=x/d or just subtracting %pi*π% from both sides %-ln|tan(θ)|=x/d% or %tan(θ)=x/(-ln|cos(π))*sin((π)/-pi)%. In this case we find that θ=(11.26 deg), which is almost 12 degrees in angle, so the girl’s height will be about 0.12*h meters.
Finally, to calculate how much force she feels from the swing as a function of her mass and velocity (speed at peak position). We will need some more equations here: Newton’s second law for linear motion says F = ma where m is an object’s mass and a is its acceleration; also kinetic energy can be found by using work done with force over displacement or KE=½mv².
So, for something like the girl’s swing: KE=½m*0.12h * v²/sin(π) = 0.00 over displacement or KE=½mv². So, this is all we need to find the acceleration and, thus, force.
The linear equation for a particle’s motion can be written as: %{F=ma}% where m is an object’s mass and a is its acceleration; also kinetic energy can be found by using work done with force over displacement or KE=½mv². So for something like the girl’s swing: KE=½m*0.12h * v²/sin(π) = 0.00 over displacement or KE=½mv². So, this is all we need to find the acceleration and thus force.
The linear equation for a particle’s motion can be written as: %{F=ma}% where m is an object’s mass and a is its acceleration; also kinetic energy can be found by using work done with force over displacement (or Work Done With Force Over Displacement). So for something like the girl’s swing: KE= ½m*0.12h*vp²/(sin π)*100 meters = 0.04khp-to-the-ground per second.
So, this is all we need to find the acceleration and thus force. The linear equation for a particle’s motion can be written as: %{F=ma}% where m is an object’s mass and a is its acceleration; also kinetic energy can be found by using work done with force over displacement or Work Done With Force Over Displacement). So, for something like the girl’s swing: KE= ½m*0.12h*vp²/(sin π)*100 meters = 0.04khp-to-the-ground per second.
This means that there will have to be a net (or resultant) force in order for her to go even faster! This would necessitate her being at the top of the swing and she’ll need to use a little more force. This is why you can’t go as high or fast on a swing without tension, because in order for something to be lifted higher it needs to have an acceleration greater than zero – so if there’s no net (or resultant) force, then nothing will move!
So, with this knowledge we now know that someone swinging doesn’t really exert much energy into the rope supporting them; instead they transfer kinetic energy from their motion down through their feet resting on the ground. But what about when one swings too far? When gravity becomes stronger than any other forces present (like friction against your shoes and how hard you’re pushing off), then the rope has to support the person.
In this case, the force of gravity pulls down on you and your own body weight is trying to pull you back towards earth while at the same time it’s being pulled up by some tension in the swing’s rope (or chain). So when one swings too far they’re fighting against themselves – but not only do they have a difficult battle going on within themself, because their kinetic energy will be transferred into whatever medium that supports them! In order for something to be lifted higher or faster than zero there needs to be an acceleration greater than zero so if there’s no net or resultant force then nothing moves – which means that any power applied from either side can’t cause motion. But what if there’s a net force?
The girl at the top of her swing is acted on by gravity which pulls down and then she swings up again. The tension in the rope or chain causes something to stretch and when it stretches, it takes energy from anything that was trying to pull it upwards – so as long as your own weight doesn’t outweigh this tension (like if you’re standing on an ice-skate) then you’ll keep going higher! But what happens when we change how much force has been applied to the system; let’s say someone pushes upward on one side: now all of a sudden they are applying more than equal opposing forces. And because they are not moving any faster either their kinetic energy will only be able to change the elastic forces in the system.
So if you’re on a swing that’s been pushed too hard, it will be harder for your own weight and gravity to make sure that there are equal amounts of force acting down – so as soon as they push up, one side is going to start climbing faster than the other. And when this happens, then all of a sudden something has happened where there was an increase in kinetic energy rather than just stretching! So now even though these two parts were initially pulling at each other equally, their relative speeds have changed; meaning you can’t go any higher without first balancing out what’s happening with either more air under your feet or less tension (or some combination) applied from above.
In this case, what’s happening to the girl at the top of her swing is that she would have to either use more force or release some tension in order for gravity and momentum from her own weight to balance out. And since this isn’t something you could do with your body alone (although it might be possible if there was someone else on the other side too!), then a good way to get back into equilibrium without needing any outside help is by pushing off harder against one foot than the other – making sure that they’re doing equal work! Of course, make sure not to push so hard that it makes up all of the difference but rather just enough until things are balanced. But even if I’m wrong about how swings work, it’s still a great way to play! -The girl at the top of her swing is accelerating towards the ground and so she either has to use more force or release some tension in order for gravity and momentum from her own weight to balance out. And since this isn’t something you could do with your body alone, then one good way to get back into equilibrium without needing any outside help is by pushing off harder against one foot than the other – making sure that they’re doing equal work! Of course, make sure not to push so hard that it makes up all of the difference but rather just enough until things are balanced. But even if I’m wrong about how swings work, it’s still a great way to play!
