Thread: Exercises in Physics

Since I am currently re-studying basic physics to get a firmer grip, I thought I might post some of the exercises I come across - just in case anyone here would like to have a go by him/herself. These are not homework questions for which I seek the answer, they are simply exercise examples for whoever might be interested.

So here is the first one now - let me warn you, it looks deceptively easy but requires some interesting trigonometry ( some of which I had forgotten, so this little exercise took me a good hour to figure out, and an entire A4 sheet of paper !!! ) :

"You fire a ball with an initial speed v at an angle above the surface
of an incline, which is itself inclined at an angle above the
horizontal. (a) Find the distance, measured along the
incline, from the launch point to the point when the ball strikes the
incline. (b) What angle gives the maximum range, measured
along the incline? Ignore air resistance."

Cheers and good luck
Feel free to post your solution if you like !

EDIT : The point here is of course to do this by hand, so no cheating with WolframAlpha etc ! Consult your trigonometry tables if needed ( I definitely had to ), but don't take any other shortcuts, or else don't bother with this in the first place.

So this looked like an interesting question and i took a crack at it, but i think i've come unstuck somewhere. I figure i've either got the wrong answer to part (a) or my algebra sucks, i was hoping someone could give me a clue as to which is the case.

For the distance, measured along the incline, my answer is,



where is the angle, from the incline, at which the ball is launched, and is the angle that the incline makes with the horizontal. Anyone else get this answer?

Originally Posted by wallaby

So this looked like an interesting question and i took a crack at it, but i think i've come unstuck somewhere. I figure i've either got the wrong answer to part (a) or my algebra sucks, i was hoping someone could give me a clue as to which is the case.

For the distance, measured along the incline, my answer is,



where is the angle, from the incline, at which the ball is launched, and is the angle that the incline makes with the horizontal. Anyone else get this answer?

Would you like me to publish the answer ?

Originally Posted by Markus Hanke

Would you like me to publish the answer ?

That would be great. (I should be able to do this question so it's really bothering me that i can't)

Originally Posted by wallaby

Originally Posted by Markus Hanke

Would you like me to publish the answer ?

That would be great. (I should be able to do this question so it's really bothering me that i can't)

Don't beat yourself up - I had to do this exercise twice over also, the first time I arrived at some monster of an expression which was completely wrong. The second time I got it right though. Not getting these basic things right in the first attempt was one of the main motivations for me to go back and re-study the basics.
Anyway, the correct expression for the distance along the incline is



which has a maximum at

Ok so it turns out that i just had the gravitational field oriented in the wrong direction and needed to use some different trig identities in the algebra. Thanks for posting the answers and helping me to clear this up.

Ok then, here is the next chapter's exercise - this one is absolutely wrecking my head, I am missing a term in my own answers, and just can't figure out where it comes from. I'll come back to that in a few days, when my head is a bit clearer. Anyway, here it is :

A small block with mass m is placed inside an inverted cone that is
rotating about a vertical axis such that the time for one revolution
of the cone is T. The walls of the cone make an angle
with the horizontal. The coefficient of static friction between the
block and the cone is u. If the block is to remain at a constant
height h above the apex of the cone, what are (a) the maximum
value of T and (b) the minimum value of T ? (That is, find expressions
for and in terms of the angle and h.)

Good luck

And on to the next chapter - this one is surprisingly easy, I got this within just a few minutes :

"Consider a hanging spring of negligible mass that does not obeyHooke’s law. When the spring is extended by a distance x, the
force exerted by the spring has magnitude ax^2, where a is a positive
constant. The spring is not extended when a block of mass
m is attached to it. The block is then released, stretching the
spring as it falls. (a) How fast is the block moving
when it has fallen a distance x1? (b) At what rate does the spring
do work on the block at this point? (c) Find the maximum distance
x2 that the spring stretches. (d) Will the block remain at the
point found in part (c)?"

Good luck

This one is straightforward as well :

"Sphere A of mass 0.600 kg is initially moving to the right at 4m/s.Sphere B, of mass 1.80 kg, is initially to the right of
sphere A and moving to the right at 2 m/s. After the two
spheres collide, sphere B is moving at 3m/s in the same direction
as before. (a) What is the velocity (magnitude and direction)
of sphere A after this collision? (b) Is this collision elastic or inelastic?
(c) Sphere B then has an off-center collision with sphere C,
which has mass 1.20 kg and is initially at rest. After this collision,
sphere B is moving at 19.0° to its initial direction at 2 m/s.
What is the velocity (magnitude and direction) of sphere C after
this collision? (d) What is the impulse (magnitude and direction)
imparted to sphere B by sphere C when they collide? (e) Is this
second collision elastic or inelastic? (f) What is the velocity (magnitude
and direction) of the center of mass of the system of three
spheres (A, B, and C) after the second collision? No external forces
act on any of the spheres in this problem. "

I'd love to be able to tackle questions like this! But would not have a clue where to start, or even what an incline is supposed to be.

I expected the answers to the OP to be something like:

A, some unit of measurement

B, some kind of angle such as 45`

How you guys can come up with those equations from just those questions without any real quantitative data to work with is a mystery to me and I admire you for knowing what on Earth your talking about. I wish I had a clue. I'm gonna look up trigonometry and see if that is in anyway comprehendable to me.

P.S im embaressed to post on a thread such as this... but I just had to express my awe.

Originally Posted by question for you

I'd love to be able to tackle questions like this! But would not have a clue where to start, or even what an incline is supposed to be.

I expected the answers to the OP to be something like:

A, some unit of measurement

B, some kind of angle such as 45`

How you guys can come up with those equations from just those questions without any real quantitative data to work with is a mystery to me and I admire you for knowing what on Earth your talking about. I wish I had a clue. I'm gonna look up trigonometry and see if that is in anyway comprehendable to me.

P.S im embaressed to post on a thread such as this... but I just had to express my awe.

It's simple practice
Once you do a couple of those it almost becomes second nature.

<img src="http://www.thescienceforum.com/attachment.php?attachmentid=1233&amp;stc=1" attachmentid="1233" alt="" id="vbattach_1233" class="previewthumb">I'll stop there for answer

Originally Posted by bugfrag

<img src="http://www.thescienceforum.com/attachment.php?attachmentid=1233&amp;stc=1" attachmentid="1233" alt="" id="vbattach_1233" class="previewthumb">I'll stop there for answer

The diagram is pretty much correct. You will find the algebraic answer in post 5.

Managed to find R in terms of , then use

I'm assuming the slop of incline is kept constant, and I vary the angle of the cannon.

I guess in that the case the ideal angle would be 45, though we must differentiate to find out.

This one was fun ! Quite easy and straightforward

"A large cylindrical tank with diameter D is open to the air at the
top. The tank contains water to a height H. A small circular hole
with diameter d, where d is very much less than D, is then opened
at the bottom of the tank. Ignore any effects of viscosity. (a) Find y,
the height of water in the tank a time t after the hole is opened, as a
function of t. (b) How long does it take to drain the tank completely?
(c) If you double the initial height of water in the tank, by
what factor does the time to drain the tank increase?"

Interesting one, but also quite easy

"A comet orbits the sun in an elliptical orbit of semimajoraxis a and eccentricity e. (a) Find expressions for the speeds
of the comet at perihelion and aphelion. (b) Evaluate these expressions
for Comet Halley (research the data you need)."

Right down my alleyway...love it

"An astronaut visiting Jupiter’s satellite Europa leaves a canister of1.20 mol of nitrogen gas (28.0 g mol) at 25.0°C on the satellite’s
surface. Europa has no significant atmosphere, and the acceleration
due to gravity at its surface is 1.30 m/s2. The canister springs
a leak, allowing molecules to escape from a small hole. (a) What is
the maximum height (in km) above Europa’s surface that is
reached by a nitrogen molecule whose speed equals the rms speed?
Assume that the molecule is shot straight up out of the hole in the
canister, and ignore the variation in g with altitude. (b) The escape
speed from Europa is 2025 m s. Can any of the nitrogen molecules
escape from Europa and into space?"

What kind of physics exercises would you like?

Originally Posted by Kerling

What kind of physics exercises would you like?

I got plenty, thanks Kerling
Just posting a few key ones from my current textbook here, in case other readers are interested.
For yourself all of these are probably no big deal, but I found some of them challenging.

One of the most challanging exercises I ever had was using the basis of Algebra to prove that A*0=0 :P

Wow I got a lot to learn.

Originally Posted by Kerling

One of the most challanging exercises I ever had was using the basis of Algebra to prove that A*0=0 :P

Well I for one would love to see this proof - please share it with us. More particularly, please state what sort of object is "A"

Originally Posted by PhysicsApple

Wow I got a lot to learn.

Don't worry. You are not alone!

You never appear alone but only your ink will be seen on the paper...my advice for you and for myself is learn how to visualize physics problem as if you are doing the pratical...it then becomes clearer.

Originally Posted by Guitarist

Originally Posted by Kerling

One of the most challanging exercises I ever had was using the basis of Algebra to prove that A*0=0 :P

Well I for one would love to see this proof - please share it with us. More particularly, please state what sort of object is "A"

Well, there is the rules of a certain system of counting. I don't quite remember them all. But I do remember that one of the first few ones was that a number (if it exists) say A has another, or at least one other element in the set (say B) that has the special charactarestic that A + B = 0. Then C*0 = C*A+C*B=0 Assuming that (this step I forgot) C*A has a set element and so does C*B, and assuming that There is one or more set elements that has C*D=1, then I can asssume that C*A+C*B are eachothers opposites and hence zero.
Or something like that. I don't remember exactly, I was a Freshman, and didn't want to do mathematics. :P

Originally Posted by Kerling

Well, there is the rules of a certain system of counting. I don't quite remember them all. But I do remember that one of the first few ones was that a number (if it exists) say A has another, or at least one other element in the set (say B) that has the special charactarestic that A + B = 0. Then C*0 = C*A+C*B=0 Assuming that (this step I forgot) C*A has a set element and so does C*B, and assuming that There is one or more set elements that has C*D=1, then I can asssume that C*A+C*B are eachothers opposites and hence zero.
Or something like that. I don't remember exactly, I was a Freshman, and didn't want to do mathematics. :P

Bzzst!

You are using the axioms of field theory to prove an axiom of field theory. Totally circular argument

Not good, and not what one would expect from a theoretical physicist

Originally Posted by Guitarist

Bzzst!

You are using the axioms of field theory to prove an axiom of field theory. Totally circular argument

Not good, and not what one would expect from a theoretical physicist

So then, how would one correctly prove that statement ? Just out of interest...

Originally Posted by Guitarist

Originally Posted by Kerling

Well, there is the rules of a certain system of counting. I don't quite remember them all. But I do remember that one of the first few ones was that a number (if it exists) say A has another, or at least one other element in the set (say B) that has the special charactarestic that A + B = 0. Then C*0 = C*A+C*B=0 Assuming that (this step I forgot) C*A has a set element and so does C*B, and assuming that There is one or more set elements that has C*D=1, then I can asssume that C*A+C*B are eachothers opposites and hence zero. Or something like that. I don't remember exactly, I was a Freshman, and didn't want to do mathematics. :P

Bzzst!You are using the axioms of field theory to prove an axiom of field theory. Totally circular argumentNot good, and not what one would expect from a theoretical physicist

Actually the proof was for algebra. Not quite field theory.The statement is provable by the first few formal rules of algebra. : http://www.themathpage.com/aprecalc/algebraPre.htmAs I said in my post it was mathematics, not physics, and I already said twice in the same posts of my uncertainty of my answer.Yep, I believe the first 5 formal rules should suffice.

Originally Posted by Markus Hanke

So then, how would one correctly prove that statement ? Just out of interest...

Recall that 0 satisfies so for any it is true that which implies that

But does it in fact imply that? After all, once we have 0x=0x +0x then saying that either x is its own opposite, which would make x zero. Or we subtract 0x on both sides. However then we need to define subtraction. Whereas the more general proof a only require x to have an opposite. Say y. Then we add 0y to both sides. And get 0x + 0y=0x + 0x + 0y => 0(x+y) = 0x + 0(x+y) and since x is y's opposite 0 = 0x.So we can only proof this is true if every element has an opposite!

Originally Posted by Kerling

But does it in fact imply that? After all, once we have 0x=0x +0x then saying that either x is its own opposite, which would make x zero. Or we subtract 0x on both sides. However then we need to define subtraction. Whereas the more general proof a only require x to have an opposite. Say y. Then we add 0y to both sides. And get 0x + 0y=0x + 0x + 0y => 0(x+y) = 0x + 0(x+y) and since x is y's opposite 0 = 0x.So we can only proof this is true if every element has an opposite!

Hi Kerling, you actually only need to know that the additive structure is cancellative.

Originally Posted by river_rat

Hi Kerling, you actually only need to know that the additive structure is cancellative.

But how do I know that?

Umm, it is just a definition.

Look.....

Suppose the commutative ring . An obvious example is the set of integers.

Then is called an integral domain if and only if, for all , that provided only that . This the Cancellation Law that river_rat referred to.

Now it is a fact that every field is an integral domain (but not conversely) and every integral domain is a commutative ring (but not conversely).

This is elementary algebra. I am astonished that a theoretical physicist doesn't know this stuff

Originally Posted by Guitarist

Umm, it is just a definition. Look..... Suppose the commutative ring . An obvious example is the set of integers. Then is called an integral domain if and only if, for all , that provided only that . This the Cancellation Law that river_rat referred to. Now it is a fact that every field is an integral domain (but not conversely) and every integral domain is a commutative ring (but not conversely). This is elementary algebra. I am astonished that a theoretical physicist doesn't know this stuff

But I want the proof to be more general, I don't want to need a commutative ring to proof the above 0*x = 0 question. This proof should be able to be made for non-commutative rings (in multiplication). So addition is associative, has an identity, is commutative and has an inverse element. And multiplication is associative and distributable over addition.
Using just that, it should be possible to proof the above. And I think River_rat did so.

Also, why do you use Ad Hominem arguments? Twice. It doesn't seem to contribute to the fruitfullness of the discussion. Also some of the die-hard maths I do not fully master, as I don't need to. Then I would have become a Mathematical physicist. And I would have loved string theory. But I am not. I am just a theoretical physicist and ironically from a philosophical point of view, these often don't mix very well.

I know you need to assume the annihilation property for semirings - commutativity is not important but you need either additive inverses or more weakly some version of the cancellation property would be my guess.