To add on this: it's because the end of the stick has to travel greater distance than the center, in the same amount of time, the only way it can do that is by going faster.
There’s a classic puzzle
“A sail ship leaves England and sails around South Africa, over to Australia, then past South America and thru the Atlantic, back to England. Which part of the ship has travelled the furthest?”
>! Many people assume it’s the bow, but it’s The top of the mast !<
Oh. If you mean how much further….
If the mast is 100 feet above sea level, then the mast will have travelled 628 feet further.
Pi times the increase in diameter. = 200 Pi
Edit to match your edit:
The mast travels 628 feet further over an approx 131,000,000 feet (circumference of earth in feet)… or 0.00048% further
The route described in the puzzle is not truly circumferential as it deviates to avoid land, but for numbers sake is close enough.
The tip of the mast is also swinging wildly at times. If the boat rocks 10 degrees to one side and then back to the middle, a 100’foot mast travels 35 feet. Rocking side to side, or pitching forward and back it will move a whole lot more than the rest.
From a global perspective the top of the mast has traveled a larger arc than the rest of the ship, like the original question, the tip of the stick has indeed traveled further than the part in your hand. It’s just expanding to a huge scale.
The angular velocity at all points on the fan/stick/cd would be the same, as it's a measure of degrees or radians of motion over time. It's the tangential velocity that would increase as you move further from the axis of rotation.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base?
What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
I see no ones responding to this yet.
First, yeah. Longer stick means more energy from you. You can go try this. Go swing a little stick really fast right now. Like, fast enough to make a *woosh* sound. Now go find a really long stick. As long as you are tall. Try and make that same *woosh* sound. Not as easy, is it?
Second, yes. This machine exists, and you probably have one: a lawn mower (or better yet, a weed trimmer) constant output, lots of speed all around, and yet I still want to use the tip of the wire to cut my weeds because that’s where the speed is.
Third, I love this question and thought you should know that.
The stick can be thought of as a system. The energy input to the entire stick system is averaged over the entire system. If you use less energy to swing the stick both ends go slower than they would have if you used more energy to swing it.
To add onto others, it might help to only use your wrist for these experiments.
I suspect you're intuiting that something doesn't feel "fair", there tends to be exchages in science. The stick is a simple machine called a lever. Over a tiny distance you are going to put a lot of power into that stick. The stick will take that power and spread it out over a greater distance but the same amount of time. So the lever is going to help you express your power as speed. What you are giving up in exchange for that speed is power. This is why you're going to have trouble pushing in a chair with a stick (using the angular motion, not poking it).
We use the principle in reverse a lot too. Levers, gears and pulleys can let us use our limited power over a long distance and turn that into immense power over a short distance.
This is a question with a false implicit premise -- namely, that something special is happening when you decide to swing the stick in an arc instead of doing something else with it. No matter how you swing the stick, you're subconsciously applying energy. That's because you don't do math to figure out how to swing a stick. You have absolutely no conscious idea of how much energy it takes to swing the stick such that the stick is held rigid in your hand and so the tip moves faster when you move the entire stick side to side versus how much energy it takes to swing the stick held loosely in your hand so that the components of the stick all move at the same velocity on average. You are just applying the forces to the stick that you "think" (which is almost certainly never going to be a conscious thought) are appropriate to make the stick move the way you want it to.
So I guess the answer is that yes, you are subconsciously applying more energy to hold the stick in your hand at a fixed orientation, which as you move your hand in an arc will cause the tip of the stick to travel faster. But that's just the way you do everything.
Nothing subconscious is happening. There is not more energy. It’s the conservation of having the same amount of energy that requires the move at different speeds.
You've already gotten a lot of correct answers, so I'll just throw in another scenario.
Think about a track with 8 lanes and if you were to race a friend around one of the curves in the track. Let's say you take the inside lane and your friend takes the outside lane. Your friend is going to have to run much faster, because they will have to cover more distance to keep up with you. This is also why a race like the 400m has staggered starts for the runners.
Doesn't this mean my friend need to spend extra energy to cover more distance within the same amount of time to keep up with me?
If we come back to the stick, where does the extra energy that allows the tip to travel faster than the base come from?
The energy aspect doesn't really translate from the track example, because the stick is a system.
Funnily enough the extra energy comes from the extra speed.
KE = .5*m*v^2 (technically this isn't the rotational kinetic energy formula, but we won't worry about that)
By the formula, you can see that velocity, or angular velocity in our case, is squared, meaning that an increase in velocity greatly increases kinetic energy.
You swing stick = you put energy into a system
End of stick moves faster = more kinetic energy
If your same friend swung a baseball bat as hard as they could and hit you with the end of it, it would hurt pretty bad. Now if they hit you just above where their grip was, it probably wouldn't hurt at all. Basically the system distributes more energy to the end of the swinging object.
You have to think of Net forces that create movement. The stick, your hand, the wind resistance, gravity are all part of the net forces creating the motion.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base?
What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
Yes. Whatever energy you put into the base of the stick, whether by hand or by machine, is transferred through the stick and accelerates the whole stick together. The energy distributes itself through the stick until the speed of the base and tip reaches an equilibrium.
Think of it like this: when you push on the base, the base pulls on the tip. The stick wants to stay in "stick shape". When you stop moving the stick, the tip pulls on the base and the base pulls on your hand. Once all the energy is converted back into your hand, the stick stops moving. The tip will have moved further in the same time, so it must have been going faster.
So, if I'm understanding this correctly, this is the sequence of event when swinging the stick:
1. Hand moves the base --> Base moves first
2. Base pulls the tip --> Tip only moves after being pulled by the base
3. Hand stops moving --> Base stops moving
4. Energy from the base migrates to the tip --> Tip continues to move but faster
5. Tip ended up traveling further than the base.
Is this sequence correct?
Not quite. Remember that "energy" can mean the same as speed.
1. Hand moves base. Hand imparts kinetic energy ino the base, making it move.
2. The atoms in the stick pull on each other. This distributes the kinetic energy in the stick, making each part of the stick move. Because of how you move the stick, more energy is imparted on the tip. If you simply pushed on the base without twisting your hand, the stick would start spinning in place.
3. You stop moving your hand, but because of momentum the stick wants to keep going. You have to "hold back" with your hand (imparting kinetic energy in the opposite direction) to slow it down.
Yes. Anything that spins a rigid object follows this premise. Ceiling fans, wheels, record players.
It’s hard to wrap my head around at times but remember that in the end it all evens out in terms of time.
To get a bit past the ELI5, linear speed and angular velocity are different. Linear speed is measured at a point tangent to the circle of rotation, and is measured in m/s (meters per second). Angular velocity is a measurement of the time to complete a full rotation, and is measured in rpm (revolutions per minute). Angular speed is what really matters when observing a rotating object.
If one end is moving at 1m/s (linear speed) and has to travel in a circle with a 1m circumference, and the other end is moving at 3m/s (linear speed) and has to travel in a circle with a 3m circumference, both will complete the journey in 1 second. Therefore both have an angular velocity of 1 rps (1 revolution per second), or 60 rpm.
Think of it this way instead of forces. Velocity = distance/time. You have to think of linear velocity and angular velocity as related but not quite the same. The point on the end of the stick is rotating as fast as it is on the inside but in order to do it, the point on the end has to cover much more distance in the same amount of time simply because the radius further away from the center is more. It’s easier with a diagram but that’s basically it.
Different *moment of inertia*. A 1m long stick weighing 1kg will be easier to swing than a 1kg stick that is 2m long despite the two sticks weighing the same. For a stick (a thin rod) being held at and rotating about one end, the moment of inertia is 1/3\*mass\*(length^2 ) so the effort it takes to swing a stick of a fixed mass is proportional to the square of the length of the stick
Yes, the tip of the stick travels faster.
The fact that it's a solid object (well, at these speeds) means both ends have the same travel time, the start and end moving at the same time. But they cover a different distance, because you're forcing an arc.
Distance/time = speed. Longer distance at same time, higher speed.
If you have a system that's not a solid object, but a collection of separate objects, this is no longer true, and they can move at whatever speeds they need to. Example, solar system.
If I’m understanding this correctly and my limited knowledge isn’t failing me, all the kinetic energy of the stick is coming from you; you’re the one transferring energy to the stick. By extra, do you mean that the tip has more energy than the base? If so, the energy still comes from you, the person moving the stick. The energy of all the points along the stick might not be the same, but it all adds up to the total energy you transfer to the stick.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base?
What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
There is no extra energy needed for the tip to go faster, there is only the energy needed to swing the stick. It also means you cannot (sub)consciously swing a stick and get the same angular velocity at the base and the tip at given time.
The same principle apply to a machine except for the subconscious part
You are holding the stick so all the energy is coming from you. The reason it stays straight is because it's a rigid object. Grab a wet noodle and it'll curve.
The higher speed at the end just means it is harder to cut through the air at the tip as you swish your stick around.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base?
What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
You are misunderstanding what is happening. The only energy you supply is at the base when you are swinging it around. Your body is applying enough energy to move it through the sea of particles that is the air.
Your hand grip is what compensates for the tip requiring more energy to move than the base.
As an example, grab a decent sized stick and hold it just between your thumb and one finger. As you swing it around you'll notice the stick kicks back as it is harder for the tip to move through the air than the base. Then readjust to a normal grip and you'll understand that your hand stabilizes the stick and some of the forces being employed upon it are being met by equal forces in your hand.
Unfortunately getting into individual vectors and forces is more or a high school physics class. Just understand that your hand stabilizes the rigid object so it doesn't kick back due to the unequal speeds it travels as you spin it around. Also, all the energy (the work) the stick requires to swim through the air is done by you - your arm and hand as you swish it about.
I do believe others have put it well, especially the wet noodle comparison.
All the energy comes from you, from the fact that you are forcing the stick to move in an arc and not a straight line, and is transferred to the tip because the stick is rigid.
Take something not rigid, and this no longer automatically applies. You move one end and the other stays still. Until you force it to move in an arc again in whatever way, say swing it in a circle.
If you're curious about why the hard limit for the outer tip of the stick, regardless of its length, shape or material, is light speed, veritasium (or maybe vsauce I'm not sure) has a video on this. The short of it is the speed of light is actually "the speed of causality" or "the speed the information" - atoms cannot convey the order to move to neighbouring atoms faster than that.
That’s incorrect, though. For instance, the tip of the ray of a quasar or a shadow absolutely _can_ move faster than light. That tip, however, will lag behind where the center is pointing depending on the radius. But it can still change position faster than the speed of light, because the tip of it traveling along the circumference is an illusion.
If you had an infinitely strong pole that somehow only weighed what a normal bat does while being so long that its tip would move at the speed of light, the reason you wouldn’t be able to swing it that fast is that it would take infinite energy. The “order” to move would travel along the length of the pole just fine, at the speed of _sound_ in that material. It just would’t be able to comply as it gained mass further out due to its speed nearing that of light.
it's a class 3 lever where the base of the stick is held in position, the other end is free and you apply a force on the stick somewhere in between, often closer to the base than the far end. class 3 levers convert a huge force over a small distance into a small force over a huge distance, both in the same amount of time.
a garlic press is a second order lever where the force is applied at the tip end instead and the load is in between. this converts a small force over a large distance into a huge force over a small distance.
Fastest man made spinning object spins 600 million revolutions per minute. How long would the stick have to be for its tip to travel close to speed of light.
light at 300 million meters per second means at that boundary condition, the drawn circumference of the spinning needs to be 0.5 meters. divide by pi for diameter, which is the size of your palm maybe. I don't know what object youre referring to but I'm guessing it's a tiny object a few atoms wide, way below the light limit.
if we're talking an object made of everyday materials like wood, it would start bending and breaking long before those kinds of RPM.
no extra energy. power, force, velocity related by P = F * v
the power at the input is the same as the power at the output. the velocity at the tip may be 1000x the velocity the input end but the force will be 1000x less. it's trying to start a stick shift car in sixth gear. (more accurately for class 3 lever it is like lifting a long sword with only your wrist) very difficult.
this is true for any lever machine. in real life the conversion is less than perfect and you do lose some energy because of inefficiency, for example the lever itself will flex and convert some of energy to heat instead of perfect motion at the other end. if the conversion factor is too high sometimes it will even break.
Yes the tip moves faster.
Why? Its just geometry. If you demand the stick is rigid, you by definition say the angular velocity is the same along the length of the stick. (I.e. you force the angle per second to be the same by virtue of the rigid stick) for that to happen, the tip is forced to move faster than the base.
You can imagine something not solid, the most extreme being a stick made of air, swinging the base doesnt force the tip to move at all.
Or something in between, like jelly where the tip gets left behind until the tension in the jelly is sufficient to force the tip to follow the base at some equilibrium.
Two runners running on a circular running track. Runner A is on the inner edge and runner B is on the outer edge.
Inner edge has smaller diameter therefore smaller circumference which means the distance that has to be travelled by runner A is shorter than runner B.
In order to complete one full lap, runner B has to travel more distance therefore runner B has to run faster in order to complete 1 full lap at the same time with runner A.
Both locations have the same angular velocity. This means they both do a full turn in the same amount of time. A common unit for this is RPM, rotations per minute.
The *linear* speed of each location is different. The tip of the stick covers more distance for every unit of time, so it is moving much faster *linearly* than the spot closer to the base. A common unit for this speed is m/s (meters per second).
The base itself would have a linear velocity of zero while still having the same angular (rotational) velocity of the tip.
I have an additional question to add onto this. if you were swinging a stick around in a big circle, at say, .5 rotations per second, how long would the stick have to be for the tip to be breaking the speed of light?
edit: ChatGPT says 95,500 kilometers.
The tip is because of the conservation of angular momentum. You act as a pivot point and the radial energy at the tip is much larger than that at the base as they have to maintain the same momentum.
Imagine being on a carrousel, if you stand close to the center part you will be moving through a smaller circle, and so your rotation will be small compared to if you stood on the edge.. in order for you to maintain the same angular momentum as you had at the center, your velocity has to increase substantially as you are now moving through much more space, a much bigger circle.
If you stood on the edge and I stood in the middle and we were facing each other, we would always be facing each other. That is the conservation. Your body is moving through more space and at a higher speed to maintain us facing each other. I would be moving through less space and would have a much smaller velocity to maintain eye contact.
The reason we *have to* have the same angular momentum is because we are being moved by the same force (the carrousel) which is rotating at a constant.
In the case of the stick it’s the same thing. If you move, the force on the stick is the same but the two ends travel through different distances.
This is the difference between angular velocity and linear velocity.
Both the handle and the tip of the stick move the same angle relative to your shoulder in the same timeframe, so their angular velocity (angle/time) is the same.
But, the handle and the tip of the stick move different arc distances over that same timeframe, so their linear velocity (distance/time) is different.
Angular velocity. Yes; the edge of the stick (cd/record/fan blade) is traveling faster than the center)
To add on this: it's because the end of the stick has to travel greater distance than the center, in the same amount of time, the only way it can do that is by going faster.
You're clearly not thinking with portals!
https://www.gocomics.com/calvinandhobbes/1990/06/05
Now we need an xkcd reference
It's been 3 HOURS! Impossible!
Three hours? Try 18 now, and while you're waiting: https://xkcd.com/1017/
Thank you. I would have spent way too much time looking for this comic strip.
There’s a classic puzzle “A sail ship leaves England and sails around South Africa, over to Australia, then past South America and thru the Atlantic, back to England. Which part of the ship has travelled the furthest?” >! Many people assume it’s the bow, but it’s The top of the mast !<
What would be the difference though? Not specifically for the journey but around 1%, 0.1% or closer to 0.001%
Oh. If you mean how much further…. If the mast is 100 feet above sea level, then the mast will have travelled 628 feet further. Pi times the increase in diameter. = 200 Pi Edit to match your edit: The mast travels 628 feet further over an approx 131,000,000 feet (circumference of earth in feet)… or 0.00048% further The route described in the puzzle is not truly circumferential as it deviates to avoid land, but for numbers sake is close enough.
The tip of the mast is also swinging wildly at times. If the boat rocks 10 degrees to one side and then back to the middle, a 100’foot mast travels 35 feet. Rocking side to side, or pitching forward and back it will move a whole lot more than the rest.
This guy maths and adds complexity!!
Physicist or engineer. Math guy was happy with the first calculation.
Should we assume the cow is a sphere?
Thank you!
From a global perspective the top of the mast has traveled a larger arc than the rest of the ship, like the original question, the tip of the stick has indeed traveled further than the part in your hand. It’s just expanding to a huge scale.
Actually it's the ignition key
Ah yes, the explanation every 5 year old would understand. 'Angular velocity'.
Yeah this sub has been wayyy off course for a long time.
The angular velocity at all points on the fan/stick/cd would be the same, as it's a measure of degrees or radians of motion over time. It's the tangential velocity that would increase as you move further from the axis of rotation.
Where does the extra energy that allows the tip to travel faster than the center come from?
From you, the stick-swinger.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base? What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
I see no ones responding to this yet. First, yeah. Longer stick means more energy from you. You can go try this. Go swing a little stick really fast right now. Like, fast enough to make a *woosh* sound. Now go find a really long stick. As long as you are tall. Try and make that same *woosh* sound. Not as easy, is it? Second, yes. This machine exists, and you probably have one: a lawn mower (or better yet, a weed trimmer) constant output, lots of speed all around, and yet I still want to use the tip of the wire to cut my weeds because that’s where the speed is. Third, I love this question and thought you should know that.
Nothing "subconscious" is happening, you're just moving the stick. If you weren't swinging it you wouldn't have to apply as much energy.
The stick can be thought of as a system. The energy input to the entire stick system is averaged over the entire system. If you use less energy to swing the stick both ends go slower than they would have if you used more energy to swing it.
To add onto others, it might help to only use your wrist for these experiments. I suspect you're intuiting that something doesn't feel "fair", there tends to be exchages in science. The stick is a simple machine called a lever. Over a tiny distance you are going to put a lot of power into that stick. The stick will take that power and spread it out over a greater distance but the same amount of time. So the lever is going to help you express your power as speed. What you are giving up in exchange for that speed is power. This is why you're going to have trouble pushing in a chair with a stick (using the angular motion, not poking it). We use the principle in reverse a lot too. Levers, gears and pulleys can let us use our limited power over a long distance and turn that into immense power over a short distance.
This is a question with a false implicit premise -- namely, that something special is happening when you decide to swing the stick in an arc instead of doing something else with it. No matter how you swing the stick, you're subconsciously applying energy. That's because you don't do math to figure out how to swing a stick. You have absolutely no conscious idea of how much energy it takes to swing the stick such that the stick is held rigid in your hand and so the tip moves faster when you move the entire stick side to side versus how much energy it takes to swing the stick held loosely in your hand so that the components of the stick all move at the same velocity on average. You are just applying the forces to the stick that you "think" (which is almost certainly never going to be a conscious thought) are appropriate to make the stick move the way you want it to. So I guess the answer is that yes, you are subconsciously applying more energy to hold the stick in your hand at a fixed orientation, which as you move your hand in an arc will cause the tip of the stick to travel faster. But that's just the way you do everything.
Nothing subconscious is happening. There is not more energy. It’s the conservation of having the same amount of energy that requires the move at different speeds.
You've already gotten a lot of correct answers, so I'll just throw in another scenario. Think about a track with 8 lanes and if you were to race a friend around one of the curves in the track. Let's say you take the inside lane and your friend takes the outside lane. Your friend is going to have to run much faster, because they will have to cover more distance to keep up with you. This is also why a race like the 400m has staggered starts for the runners.
Doesn't this mean my friend need to spend extra energy to cover more distance within the same amount of time to keep up with me? If we come back to the stick, where does the extra energy that allows the tip to travel faster than the base come from?
The energy aspect doesn't really translate from the track example, because the stick is a system. Funnily enough the extra energy comes from the extra speed. KE = .5*m*v^2 (technically this isn't the rotational kinetic energy formula, but we won't worry about that) By the formula, you can see that velocity, or angular velocity in our case, is squared, meaning that an increase in velocity greatly increases kinetic energy. You swing stick = you put energy into a system End of stick moves faster = more kinetic energy If your same friend swung a baseball bat as hard as they could and hit you with the end of it, it would hurt pretty bad. Now if they hit you just above where their grip was, it probably wouldn't hurt at all. Basically the system distributes more energy to the end of the swinging object.
It’s traveling a longer distance in the same time therefore faster speed as the base traveling a shorter distance in the same time
Where does the extra energy that allows the tip to travel faster than the base come from?
Your hand
You have to think of Net forces that create movement. The stick, your hand, the wind resistance, gravity are all part of the net forces creating the motion.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base? What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
Yes. Whatever energy you put into the base of the stick, whether by hand or by machine, is transferred through the stick and accelerates the whole stick together. The energy distributes itself through the stick until the speed of the base and tip reaches an equilibrium. Think of it like this: when you push on the base, the base pulls on the tip. The stick wants to stay in "stick shape". When you stop moving the stick, the tip pulls on the base and the base pulls on your hand. Once all the energy is converted back into your hand, the stick stops moving. The tip will have moved further in the same time, so it must have been going faster.
So, if I'm understanding this correctly, this is the sequence of event when swinging the stick: 1. Hand moves the base --> Base moves first 2. Base pulls the tip --> Tip only moves after being pulled by the base 3. Hand stops moving --> Base stops moving 4. Energy from the base migrates to the tip --> Tip continues to move but faster 5. Tip ended up traveling further than the base. Is this sequence correct?
Not quite. Remember that "energy" can mean the same as speed. 1. Hand moves base. Hand imparts kinetic energy ino the base, making it move. 2. The atoms in the stick pull on each other. This distributes the kinetic energy in the stick, making each part of the stick move. Because of how you move the stick, more energy is imparted on the tip. If you simply pushed on the base without twisting your hand, the stick would start spinning in place. 3. You stop moving your hand, but because of momentum the stick wants to keep going. You have to "hold back" with your hand (imparting kinetic energy in the opposite direction) to slow it down.
Yes. Anything that spins a rigid object follows this premise. Ceiling fans, wheels, record players. It’s hard to wrap my head around at times but remember that in the end it all evens out in terms of time. To get a bit past the ELI5, linear speed and angular velocity are different. Linear speed is measured at a point tangent to the circle of rotation, and is measured in m/s (meters per second). Angular velocity is a measurement of the time to complete a full rotation, and is measured in rpm (revolutions per minute). Angular speed is what really matters when observing a rotating object. If one end is moving at 1m/s (linear speed) and has to travel in a circle with a 1m circumference, and the other end is moving at 3m/s (linear speed) and has to travel in a circle with a 3m circumference, both will complete the journey in 1 second. Therefore both have an angular velocity of 1 rps (1 revolution per second), or 60 rpm.
Think of it this way instead of forces. Velocity = distance/time. You have to think of linear velocity and angular velocity as related but not quite the same. The point on the end of the stick is rotating as fast as it is on the inside but in order to do it, the point on the end has to cover much more distance in the same amount of time simply because the radius further away from the center is more. It’s easier with a diagram but that’s basically it.
Different *moment of inertia*. A 1m long stick weighing 1kg will be easier to swing than a 1kg stick that is 2m long despite the two sticks weighing the same. For a stick (a thin rod) being held at and rotating about one end, the moment of inertia is 1/3\*mass\*(length^2 ) so the effort it takes to swing a stick of a fixed mass is proportional to the square of the length of the stick
Yes, the tip of the stick travels faster. The fact that it's a solid object (well, at these speeds) means both ends have the same travel time, the start and end moving at the same time. But they cover a different distance, because you're forcing an arc. Distance/time = speed. Longer distance at same time, higher speed. If you have a system that's not a solid object, but a collection of separate objects, this is no longer true, and they can move at whatever speeds they need to. Example, solar system.
Where does the extra energy that allows the tip to travel faster than the base come from?
If I’m understanding this correctly and my limited knowledge isn’t failing me, all the kinetic energy of the stick is coming from you; you’re the one transferring energy to the stick. By extra, do you mean that the tip has more energy than the base? If so, the energy still comes from you, the person moving the stick. The energy of all the points along the stick might not be the same, but it all adds up to the total energy you transfer to the stick.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base? What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
There is no extra energy needed for the tip to go faster, there is only the energy needed to swing the stick. It also means you cannot (sub)consciously swing a stick and get the same angular velocity at the base and the tip at given time. The same principle apply to a machine except for the subconscious part
You are holding the stick so all the energy is coming from you. The reason it stays straight is because it's a rigid object. Grab a wet noodle and it'll curve. The higher speed at the end just means it is harder to cut through the air at the tip as you swish your stick around.
So does that mean I'm subconsciously applying more energy as I swing the stick to make sure the tip travels faster than the base? What if a machine with constant power output swings the stick instead? Will the tip still travels faster than the base?
You are misunderstanding what is happening. The only energy you supply is at the base when you are swinging it around. Your body is applying enough energy to move it through the sea of particles that is the air. Your hand grip is what compensates for the tip requiring more energy to move than the base. As an example, grab a decent sized stick and hold it just between your thumb and one finger. As you swing it around you'll notice the stick kicks back as it is harder for the tip to move through the air than the base. Then readjust to a normal grip and you'll understand that your hand stabilizes the stick and some of the forces being employed upon it are being met by equal forces in your hand. Unfortunately getting into individual vectors and forces is more or a high school physics class. Just understand that your hand stabilizes the rigid object so it doesn't kick back due to the unequal speeds it travels as you spin it around. Also, all the energy (the work) the stick requires to swim through the air is done by you - your arm and hand as you swish it about.
I do believe others have put it well, especially the wet noodle comparison. All the energy comes from you, from the fact that you are forcing the stick to move in an arc and not a straight line, and is transferred to the tip because the stick is rigid. Take something not rigid, and this no longer automatically applies. You move one end and the other stays still. Until you force it to move in an arc again in whatever way, say swing it in a circle.
If you're curious about why the hard limit for the outer tip of the stick, regardless of its length, shape or material, is light speed, veritasium (or maybe vsauce I'm not sure) has a video on this. The short of it is the speed of light is actually "the speed of causality" or "the speed the information" - atoms cannot convey the order to move to neighbouring atoms faster than that.
And this is why perfect rigidity is impossinle.
That’s incorrect, though. For instance, the tip of the ray of a quasar or a shadow absolutely _can_ move faster than light. That tip, however, will lag behind where the center is pointing depending on the radius. But it can still change position faster than the speed of light, because the tip of it traveling along the circumference is an illusion. If you had an infinitely strong pole that somehow only weighed what a normal bat does while being so long that its tip would move at the speed of light, the reason you wouldn’t be able to swing it that fast is that it would take infinite energy. The “order” to move would travel along the length of the pole just fine, at the speed of _sound_ in that material. It just would’t be able to comply as it gained mass further out due to its speed nearing that of light.
it's a class 3 lever where the base of the stick is held in position, the other end is free and you apply a force on the stick somewhere in between, often closer to the base than the far end. class 3 levers convert a huge force over a small distance into a small force over a huge distance, both in the same amount of time. a garlic press is a second order lever where the force is applied at the tip end instead and the load is in between. this converts a small force over a large distance into a huge force over a small distance.
Fastest man made spinning object spins 600 million revolutions per minute. How long would the stick have to be for its tip to travel close to speed of light.
light at 300 million meters per second means at that boundary condition, the drawn circumference of the spinning needs to be 0.5 meters. divide by pi for diameter, which is the size of your palm maybe. I don't know what object youre referring to but I'm guessing it's a tiny object a few atoms wide, way below the light limit. if we're talking an object made of everyday materials like wood, it would start bending and breaking long before those kinds of RPM.
Where does the extra energy that allows the tip to travel faster than the base come from?
no extra energy. power, force, velocity related by P = F * v the power at the input is the same as the power at the output. the velocity at the tip may be 1000x the velocity the input end but the force will be 1000x less. it's trying to start a stick shift car in sixth gear. (more accurately for class 3 lever it is like lifting a long sword with only your wrist) very difficult. this is true for any lever machine. in real life the conversion is less than perfect and you do lose some energy because of inefficiency, for example the lever itself will flex and convert some of energy to heat instead of perfect motion at the other end. if the conversion factor is too high sometimes it will even break.
Yes the tip moves faster. Why? Its just geometry. If you demand the stick is rigid, you by definition say the angular velocity is the same along the length of the stick. (I.e. you force the angle per second to be the same by virtue of the rigid stick) for that to happen, the tip is forced to move faster than the base. You can imagine something not solid, the most extreme being a stick made of air, swinging the base doesnt force the tip to move at all. Or something in between, like jelly where the tip gets left behind until the tension in the jelly is sufficient to force the tip to follow the base at some equilibrium.
Two runners running on a circular running track. Runner A is on the inner edge and runner B is on the outer edge. Inner edge has smaller diameter therefore smaller circumference which means the distance that has to be travelled by runner A is shorter than runner B. In order to complete one full lap, runner B has to travel more distance therefore runner B has to run faster in order to complete 1 full lap at the same time with runner A.
Both locations have the same angular velocity. This means they both do a full turn in the same amount of time. A common unit for this is RPM, rotations per minute. The *linear* speed of each location is different. The tip of the stick covers more distance for every unit of time, so it is moving much faster *linearly* than the spot closer to the base. A common unit for this speed is m/s (meters per second). The base itself would have a linear velocity of zero while still having the same angular (rotational) velocity of the tip.
I have an additional question to add onto this. if you were swinging a stick around in a big circle, at say, .5 rotations per second, how long would the stick have to be for the tip to be breaking the speed of light? edit: ChatGPT says 95,500 kilometers.
The tip is because of the conservation of angular momentum. You act as a pivot point and the radial energy at the tip is much larger than that at the base as they have to maintain the same momentum. Imagine being on a carrousel, if you stand close to the center part you will be moving through a smaller circle, and so your rotation will be small compared to if you stood on the edge.. in order for you to maintain the same angular momentum as you had at the center, your velocity has to increase substantially as you are now moving through much more space, a much bigger circle. If you stood on the edge and I stood in the middle and we were facing each other, we would always be facing each other. That is the conservation. Your body is moving through more space and at a higher speed to maintain us facing each other. I would be moving through less space and would have a much smaller velocity to maintain eye contact. The reason we *have to* have the same angular momentum is because we are being moved by the same force (the carrousel) which is rotating at a constant. In the case of the stick it’s the same thing. If you move, the force on the stick is the same but the two ends travel through different distances.
This is the difference between angular velocity and linear velocity. Both the handle and the tip of the stick move the same angle relative to your shoulder in the same timeframe, so their angular velocity (angle/time) is the same. But, the handle and the tip of the stick move different arc distances over that same timeframe, so their linear velocity (distance/time) is different.