Why does a pendulum only swing back and forth in a straight line?

[SusanC1105]

bsbtstrrbt

 

[diazona]

What have you done so far? If you don't show us any work you've been able to do, we can't tell you what you might be misunderstanding.

 

[SusanC1105]

erererger

 

[RoyalCat]

Well I know the answers to just about all of them, because we have discussed them in class, but I'm just unclear about some of the detailed physics explanations behind why the answer are correct. Like #1 - I know that the speed on the way up is the same as the way down, I just don't know how to explain why...

Please post your answers to each of the questions, and how you reached them, so we can help you understand your mistakes.

That is incorrect.

The velocity of the rock on the way up at height $$x$$ is the same as the velocity of the rock on the way down at height $$x$$
You can prove this using kinematics ($$v_f^2=v_0^2+2ad$$), or using preservation of energy. But that is irrelevant for this problem, as it deals with just the upwards motion.

The first segment you're asked about is from 0 meters above the ground, to 5 meters above the ground. What is the rock's initial velocity (You can calculate this using the two approaches I cited above) and what is its acceleration? You can easily construct an expression describing the time it takes for it to traverse the 5 meters.

Now consider the second segment, which is from 15 meters above the ground, to 20 meters above the ground. What is the rock's velocity at the start of this motion and what is its acceleration? As with the previous segment, you can construct an expression for how long it takes to traverse the 5 meters, or any distance, really.

I suggest you solve this parametrically, as any exam questions are very likely to have follow-up questions which you will find your parametric solution very useful for solving.EDIT: The answer in my spoiler is irrelevant. I misread the question. Your reasoning is still incorrect, though.

Spoiler

The velocity of the stone is not the same the way up as it is the way down. The acceleration is constant, that much is true, but the last 5 seconds of its flight are the 5 seconds before it hits the floor (20 meters below the throwing point) and first 5 seconds of its flight are when its first cast upwards. They are VERY different.
The formula that has all the answers in this case is $$v_f^2=v_0^2+2ad$$
Since we know the velocity is symmetrical with respect to the maximum height (The velocity of the stone 5 seconds before reaching the maximum height, is the same as its velocity 5 seconds after reaching the maximum height. The same can be said about its height), we'll make the argument as though our stone started from the maximum height, some $$x$$ meters above the throwing point, which is 20 meters above the ground.

Going by the above formula, when is the stone's velocity greater, in the last 5 seconds of its flight (When it clears the ADDITIONAL 20 meters of the height of the throwing point), or in the first 5 seconds of its flight?

 

[SusanC1105]

yjttytytjty

 

[Amar.alchemy]

Well I know the answers to just about all of them, because we have discussed them in class, but I'm just unclear about some of the detailed physics explanations behind why the answer are correct. Like #1 - I know that the speed on the way up is the same as the way down, I just don't know how to explain why...

I think its better to explain it in terms of "the direction of acceleration"

 

[RoyalCat]

Well see, this is why I'm here then! :tongue2:

Here are my answers:

2. The apparent weight is lighter than the actual weight because of Newtons second law. The true weight and the normal force are acting on the person. So the apparent weight= mg +ma

Your final answer is correct, but your reasoning is murky. Make a free body diagram and apply Newton's second law of motion. Your apparent weight is defined by the normal force you exert on the scale (And the scale exerts on you).
$$mg-N=ma$$ where $$a$$ is the acceleration of the elevator.
From here you can isolate N and compare it with its value when the elevator isn't accelerating.

3.Pulling your arm back will reduce the force because you extend the impulse over a greater period of time, which reduces the impact of the force.

This is correct. On a test, I would phrase my answer like this, though:
$$J\equiv \frac{dP}{dt} \Delta t$$
In order to catch the ball, we must reduce its momentum to 0.
Pulling our arm back would mean we increase the time it takes for the ball to undergo the change in momentum. In doing so, we are reducing the average force it exerts on us. Mathematically speaking, $$J$$ is a known quantity, so if we increase $$\Delta t$$, then the average force must decrease accordingly.

4. I'm not too sure on this one, but I think the ball will come out at a 90 degree angle because it is still influenced by the rotational momentum, but doesn't have the string tension there to keep it on course anymore, so it goes off in a new direction?

This is incorrect. Remember Newton's first law of motion, and remember that a change in velocity requires that a force acts on the mass in question. And since velocity is a vector quantity (Well, momentum to be more precise), a change in its direction is as much a change as is a change to its magnitude. Try and relate what the tension does in this motion. What it changes.

5. A. potential energy is added to the pendulum prior to release, the pendulum wants to get back to equilibrium and it will take energy to do so.
B. kinetic energy is the greatest just as it swings through the equilibrium point on its way down, because it is assisted by gravity?
C. The potential energy is highest when the pendulum is at the top of each swing, I don't know how to explain why though.

This is correct, but again your reasoning is murky at best. There are only conservative forces acting on the system. Therefore, $$E_k+U_g=constant$$
At what point is the kinetic energy at its maximum (Hint: The potential energy would have to be at its minimum), and at what point is the kinetic energy at its minimum? (Remember that it depends on the velocity of the mass)

6. The frictional force is necessary to negotiate a curve, and gravitational forces are not acting in the right direction to be able to help the car through the curve.

This is incorrect.
In order to negotiate a curve, there has to be such a force that the car doesn't slip out of the curve, or into it, right?
You can take a curve to the limit where it's a circular arc. This is a very big hint.
Look at the forces acting on it, and demand that it be in equilibrium so that it doesn't slide up or down the incline.
That means that it stays in motion throughout the curve.

7. The sled can not climb a hill higher than its starting point because it only started with a certain amount of potential energy, and since some is lost to frictional forces, there will not be enough to climb a taller hill.

8. There is an advantage with longer contact time, this increased the impulse and puts a greater force on the bat. The bat speed is what creates the force, which is the other part of the impulse equation. J = FdeltaT. Increasing the bat speed will also increase the impulse, making the ball go further.

9.The airbag does the same thing as moving your hand back when catching the baseball, extends the impulse over a greater period of time, reducing the impact you feel.

10. I had to guess at this one, do you swing the bag of oranges back and forth, constantly changing your center of gravity, causing you to move in the direction the oranges are going?

11. this is a transverse wave because the force you are exerting to create these waves is perpendicular to the direction of the waves.

12. It is possible for 2 waves traveling in the same direction to create a smaller wave, as long as the 2 waves are out of phase with each other, creating interference and thus a smaller wave is produced.

13. Increased tension on a guitar string will increase the frequency and decrease the wavelength of the standing waves. I'm sure there is an equation that proves this, but I'm not sure what it is.

14. I'm pretty clueless on this one.

15. Object A will be easier to set into rotational motion, but I don't understand the physics behind this.

16. Gravity is acting on the person but what is the other force? I don't think its friction because the person is at rest. Could it be the floor acting on the person or a normal force or something?

17. The velocity the ball is rolling at is not an important factor when calculating the time it takes to hit the floor, because this is only dependent on gravity, and the height that the ball falls from.

18. You will see the lighting before you hear the thunder because light waves travel faster than sound waves. I don't really know what physics explanation he is looking for other than that?

19. Sound waves cannot travel through a vacuum because the waves need air particles to move in, and without them, there would be no sound waves.

20. The bird increases the potential energy of the clam when it takes it from the ground, when the bird drops the clam, gravity is acting to bring the clam back to earth. Then there is a collision between the clam and the rock where they both exert a force on each other, but energy is always conserved.

I feel like I have an idea what the answers are, I just can't recall the equations or laws that explain them.

A bit busy now, will reply to the rest later.
Replies in bold.

 

[SusanC1105]

dtyjtjdtj

 

[shahin1234]

Did you ever fugure out number 4,7,10, 12, 15 or 16, i got the rest if you need any?

 

[SusanC1105]

thrhrhrth

 

[shahin1234]

Is there any that you need

 

[SusanC1105]

srthsrthh

 

[shahin1234]

for 5 i used the PE=mgh and explained that height is the only thing changing. and KE= .5mv^2, the other ones I am kinda iffy about. 4,10,15 are the worst.

 

[queenofbabes]

Well see, this is why I'm here then! :tongue2:

Here are my answers:

2. The apparent weight is lighter than the actual weight because of Newtons second law. The true weight and the normal force are acting on the person. So the apparent weight= mg +ma

3.Pulling your arm back will reduce the force because you extend the impulse over a greater period of time, which reduces the impact of the force.

4. I'm not too sure on this one, but I think the ball will come out at a 90 degree angle because it is still influenced by the rotational momentum, but doesn't have the string tension there to keep it on course anymore, so it goes off in a new direction?

5. A. potential energy is added to the pendulum prior to release, the pendulum wants to get back to equilibrium and it will take energy to do so.
B. kinetic energy is the greatest just as it swings through the equilibrium point on its way down, because it is assisted by gravity?
C. The potential energy is highest when the pendulum is at the top of each swing, I don't know how to explain why though.

6. The frictional force is necessary to negotiate a curve, and gravitational forces are not acting in the right direction to be able to help the car through the curve.

7. The sled can not climb a hill higher than its starting point because it only started with a certain amount of potential energy, and since some is lost to frictional forces, there will not be enough to climb a taller hill.

You're forgetting the push mentioned in the question, which will give it some extra energy, so it might be enough to climb a taller hill, depending on how tall it is

8. There is an advantage with longer contact time, this increased the impulse and puts a greater force on the bat. The bat speed is what creates the force, which is the other part of the impulse equation. J = FdeltaT. Increasing the bat speed will also increase the impulse, making the ball go further.

Careful with terminology. "bat speed is what creates the force" makes little sense. Think more along the lines of force being the rate of change of momentum. Now, what's momentum?

9.The airbag does the same thing as moving your hand back when catching the baseball, extends the impulse over a greater period of time, reducing the impact you feel.

Terminology! You're reducing the force experienced by your body. "Impact" isn't a physical quantity

10. I had to guess at this one, do you swing the bag of oranges back and forth, constantly changing your center of gravity, causing you to move in the direction the oranges are going?

If you're swinging it back and forth, then you'll just be oscillating about where you were standing without going anywhere. And besides, even if you do, it doesn't have anything to do with your center of gravity. No, they want you to throw the oranges somewhere. I'll leave you to try to reason it out.

11. this is a transverse wave because the force you are exerting to create these waves is perpendicular to the direction of the waves.

Correct, but I'm not sure I like the phrasing...

12. It is possible for 2 waves traveling in the same direction to create a smaller wave, as long as the 2 waves are out of phase with each other, creating interference and thus a smaller wave is produced.

You might want to specify that it is destructive interference

13. Increased tension on a guitar string will increase the frequency and decrease the wavelength of the standing waves. I'm sure there is an equation that proves this, but I'm not sure what it is.

I'm pretty sure the implicit assumption is that the length is unchanged. If the length is unchanged, the wavelength is unchanged. But of course, the frequency increases. The equation you want is v = f*lamba. (In your case, v increases)

14. I'm pretty clueless on this one.

Google/wiki Doppler effect

15. Object A will be easier to set into rotational motion, but I don't understand the physics behind this.

Have you come across moments of inertia? If not, consider the forces required to set a mass into circular motion at different radii.

16. Gravity is acting on the person but what is the other force? I don't think its friction because the person is at rest. Could it be the floor acting on the person or a normal force or something?

Yes, it is the normal force, exerted by the floor on the person.

17. The velocity the ball is rolling at is not an important factor when calculating the time it takes to hit the floor, because this is only dependent on gravity, and the height that the ball falls from.

Correct

18. You will see the lighting before you hear the thunder because light waves travel faster than sound waves. I don't really know what physics explanation he is looking for other than that?

I think that's all there is to it, really.

19. Sound waves cannot travel through a vacuum because the waves need air particles to move in, and without them, there would be no sound waves.

Yup.

20. The bird increases the potential energy of the clam when it takes it from the ground, when the bird drops the clam, gravity is acting to bring the clam back to earth. Then there is a collision between the clam and the rock where they both exert a force on each other, but energy is always conserved.

Describe the energy conversions more fully. PE->KE when dropped.

I feel like I have an idea what the answers are, I just can't recall the equations or laws that explain them.

Attempting to pick up where RoyalCat left off. Answers in bold.

Seems like you're still confused over 4. Once the string breaks, what are the forces acting on it? Apply Newton's First Law. That's all there is to it.

 

[SusanC1105]

rsthsrth

 

[SusanC1105]

srthhrs

 

[queenofbabes]

Well, what are your thoughts on the responses I've provided?

 

[RoyalCat]

Well see, this is why I'm here then! :tongue2:

Here are my answers:

2. The apparent weight is lighter than the actual weight because of Newtons second law. The true weight and the normal force are acting on the person. So the apparent weight= mg +ma

3.Pulling your arm back will reduce the force because you extend the impulse over a greater period of time, which reduces the impact of the force.

4. I'm not too sure on this one, but I think the ball will come out at a 90 degree angle because it is still influenced by the rotational momentum, but doesn't have the string tension there to keep it on course anymore, so it goes off in a new direction?

A body persists its state of rest or of uniform motion unless acted upon by an external unbalanced force.

That is one phrasing for Newton's first law of motion. Uniform motion is motion in a straight line at a constant velocity. Though once the string breaks, you still have gravity to consider.

There's no "momentum of the rotation" acting on the ball as soon as the string can provide no tension. That's just an illusion. Perform a simple experiment if you'd like. Twirl a keychain around your finger, and have it slip off at some point, it won't be moving in loops, but it'll just go off in a straight line (Well, projectile motion due to gravity, to be precise)

5. A. potential energy is added to the pendulum prior to release, the pendulum wants to get back to equilibrium and it will take energy to do so.
B. kinetic energy is the greatest just as it swings through the equilibrium point on its way down, because it is assisted by gravity?
C. The potential energy is highest when the pendulum is at the top of each swing, I don't know how to explain why though.

6. The frictional force is necessary to negotiate a curve, and gravitational forces are not acting in the right direction to be able to help the car through the curve.

Again, I'll mention that any curve can be approximated by a circular arc at some limit (Or at least so I've been told, sorry I can't cite a good proof of this)
For circular motion, what is the net force in the radial direction?
Look at a free-body diagram of the car, I suggest you draw it from a cross section of the elevated curve, and look at all the forces you've got acting on it. Remember that it is undergoing circular motion (Which is what's helping it get through the curve) and that it is neither slipping up the curve, nor down. If you still need more help, I'll attach a diagram clarifying my point.
A good question to ask right about now is whether you're familiar with fictitious forces?

7. The sled can not climb a hill higher than its starting point because it only started with a certain amount of potential energy, and since some is lost to frictional forces, there will not be enough to climb a taller hill.

I'm fairly certain you are to assume smooth surfaces unless told otherwise.
What is the sled's initial energy? Is it greater than/equal/less than the potential energy at the hump ever so slightly over the height of its initial height?
Remember it didn't just slide off the hill, it was PUSHED.

8. There is an advantage with longer contact time, this increased the impulse and puts a greater force on the bat. The bat speed is what creates the force, which is the other part of the impulse equation. J = FdeltaT. Increasing the bat speed will also increase the impulse, making the ball go further.

Again, you are correct, but your reasoning is bogged down by poor terminology.
$$J\equiv \frac{dP}{dt} \Delta t$$
That's our starting point.
When you hit the ball with the bat, you are changing the ball's momentum. You can think of it as transferring the bat's momentum to the ball.
The longer the two remain in contact, the bigger the change in momentum will be for the ball. And that's what you're trying to achieve. (Greater momentum=greater velocity)
The faster you swing the bat, the more initial momentum you're introducing to the system. There's more momentum available to "transfer" to the ball.
Note that your mathematical argument regarding bat speed falls apart, there is no "velocity of the bat" factor in the definition of impulse as you've presented it.

9.The airbag does the same thing as moving your hand back when catching the baseball, extends the impulse over a greater period of time, reducing the impact you feel.

Exactly. The longer you spread the impulse over, the lower the average force.

10. I had to guess at this one, do you swing the bag of oranges back and forth, constantly changing your center of gravity, causing you to move in the direction the oranges are going?

This would be pointless, since you'd just oscillate around your center of gravity, you want progress!
Think about conservation of momentum and about throwing the oranges.

11. this is a transverse wave because the force you are exerting to create these waves is perpendicular to the direction of the waves.

Can't help you there, haven't studied waves yet.

12. It is possible for 2 waves traveling in the same direction to create a smaller wave, as long as the 2 waves are out of phase with each other, creating interference and thus a smaller wave is produced.

Well, interference is something I'm familiar with. And yes, this is a correct explanation. If the two waves are out phase enough, the resultant wave will be of a lower amplitude, even 0!

13. Increased tension on a guitar string will increase the frequency and decrease the wavelength of the standing waves. I'm sure there is an equation that proves this, but I'm not sure what it is.

Again, can't help you here. :X

14. I'm pretty clueless on this one.

Think Doppler effect, look over your class notes, it should be in there somewhere. It's pretty simple on an intuitive level, too, though.
Without getting into the precise math of it all, this link from NASA should show you what you need to know.
http://www.grc.nasa.gov/WWW/K-12/airplane/doppler.html

15. Object A will be easier to set into rotational motion, but I don't understand the physics behind this.

This is a question of moments of inertia. If you've delved deeply into rotational motion, this should be a concept you're familiar with. If not, consider the following:
For an object undergoing rotational motion under an angular velocity $$\omega$$, its momentary LINEAR velocity is $$\omega r$$ where $$r$$ is the distance to the axis of rotation (Proof of this is available through VERY simple circle geometry)
To change this angular velocity, that is to say, to rotate the object, you would need to change its linear velocity. Changing the linear velocity of an object with mass (Changing its momentum) requires force.
For two objects of the same mass, undergoing the same angular acceleration, but at different distances from the axis of rotation then, you can see that greater force is required for the object that is further away. So it is easier to rotate the object that's close.

16. Gravity is acting on the person but what is the other force? I don't think its friction because the person is at rest. Could it be the floor acting on the person or a normal force or something?

It IS the normal force. The person is being pulled down by gravity, pushed into the floor. And yet, he remains at rest. The force responsible for this is the normal force.
Now be careful of the pitfall this question has in store. The gravitational pull and the normal are not an action-reaction pair.
The reaction force to the Earth pulling the man is the man pulling the earth, and the reaction force to the normal force, is a normal force of equal magnitude going in the opposite direction, into the ground. Action-reaction pairs each act on different masses.

17. The velocity the ball is rolling at is not an important factor when calculating the time it takes to hit the floor, because this is only dependent on gravity, and the height that the ball falls from.

Yep, that's correct. Mathematically speaking, once it leaves the edge of the table, its initial velocity in the $$y$$ direction is 0, so only the force of gravity determines how long it takes for it to fall.

18. You will see the lighting before you hear the thunder because light waves travel faster than sound waves. I don't really know what physics explanation he is looking for other than that?

At this level you can just cite empirical fact, I think. Light waves travel much faster in air than do sound waves.

19. Sound waves cannot travel through a vacuum because the waves need air particles to move in, and without them, there would be no sound waves.

This is correct, but I'd just change "air particles" to "medium." Since sound waves can propagate in many different materials.

20. The bird increases the potential energy of the clam when it takes it from the ground, when the bird drops the clam, gravity is acting to bring the clam back to earth. Then there is a collision between the clam and the rock where they both exert a force on each other, but energy is always conserved.

You've missed a couple of things.
The bird increases the potential energy of the clam by lifting it above the plane of reference (The ground).
When it is dropped, gravity performs work on it, converting the potential height energy into kinetic energy.
Upon impact, the clam and the rock exert equal forces on each-other. Momentum is conserved in the collision, but a lot of energy is lost to internal disorder (Heat and deformation).
Remember, unless you're told otherwise, a collision is not completely elastic.

I feel like I have an idea what the answers are, I just can't recall the equations or laws that explain them.

Replies in bold.