1) For an isolated system, energy is conserved.

Conside 2 situations:
(i) a ball fallen by 10 m, starting from rest
(ii) a ball attached to a string of length 10 m, and the string is made horizontal (at rest), and then released

Niow the speed at the bottom in both situtations are the same, I don't understand WHY. For the 2nd case, there is centripetal force (tension), external force?? is it still an isolated system in which energy is conserved.

Can someone explain briefly please?

Since the force from the string always acts perpendicular to the direction of the motion, it does no work.

Since the force from the string always acts perpendicular to the direction of the motion, it does no work.

But shouldn't the tension be an external force?

But the force acting on the ball will always be pefectly prependicular to the string so there would be no opposing force. The string is just redirecting the ball and there would be no tensile force on the string, yeah?

this can be easily explained on the basis of potential and kinetic energies, i agree that there is a force action on the ball due to the tension in the string , but this force does no work since the direction of motion of the ball is perpendicular to the force at all times, however the force will call the ball to accelerate, by the continually changing the direction of motion of the ball(the bal thus takes a curved path downwards.) there fore sice no work is done by this force it will in no way affect the intial or final energy states of the body.

Now , velcity of body can be obtained by following formula.

mgh= o.5XmXVXV
hence V^2= 2gh
and v=sqrt(2gh)

the same is observed in ball released from rest

But shouldn't the tension be an external force?
Not if the string is part of the system.

It depends what you define as "the system". If you take the ball alone as the system, then the system is not isolated, since gravity and the tension are both external forces.

If you take the ball plus the string as the system, then it still isn't isolated, since gravity is an external force.

If you take the ball plus the string plus the Earth as the system, then finally you have an isolated enough system, and total energy is conserved. There are no significant external forces on this system.

Defining and sticking to a particular system in a particular problem is very important in physics, as is being clear on which "system" you are using.

While I expect energy conservation, it is not obvious to me that the speed of the ball will be the same in both cases.

Gravity is always acting in the vertical direction. The string constrains the ball to move on a circular path. Only the component of acceleration tangential to the circular path adds to the speed.

There is likely to be some short cut solution, but the obvious way to calculate the speed requires integrating over the circular path.

Since only a component of the aceleration adds to the speed, I would guess that the ball speed is slower for the circular path, except for the first infinitesimal time interval after being released.

You can get an intuitive idea of what's going on by considering a couple of different cases:
1 - A ball falls 10m into a tube like this: http://pete.wildit.net.au/images/sciforums/Tube.GIF
The tube changes the vertical motion to horizontal motion without altering the ball's speed, right?

2 - Even more simply, the ball falls then bounces off a 45° incline so it's now moving horizontally instead of vertically.

In each case, a force acts to change the balls direction without changing its speed. The string works in a similar sort of way to these two examples, but over a longer period.

Like Dinosaur said, integrating acceleration due to gravity over the circular path is a good way to check and be sure.

The speed of the ball on the string will be the same at any particular height as it would be if the ball simply fell and there was no string. As somebody said previously, the string does no work on the ball.

James R: Can you show a proof of the following?The speed of the ball on the string will be the same at any particular height as it would be if the ball simply fell and there was no string. As somebody said previously, the string does no work on the ball.Intuitively, gravity is always directed vertically. Hence only the component tangential to the curved path results in increased speed. This strongly suggests that the speed along the curved path is less than the speed of the falling ball.

Without a cogent argument counter to my intuition, I find it hard to believe that the speed of the ball is the same in both cases. Your above statement does not convince me.

While often correct, my intuition has been shown to be fallible. I trust it until shown a good reason to assume it is wrong.

Dinosaur:

It's a matter of energy conservation. Define ball + string + Earth as the system under consideration. There are no external forces on this system, so total energy is conserved.

Suppose the release height of the ball above the ground is H, and the ball is released from rest. The total energy at height H is:

E = kinetic + potential = 0 + mgH = mgH.

When the ball is at a height h above the ground, later in its motion, the total energy E is still the same (i.e. mgH). Therefore, we can write:

E = mgH = (1/2)mv² + mgh

The speed of the ball at this point is v. That is:

v = sqrt[2g(H-h)]

Next, consider a ball dropped from rest at height H, not attached to a string. Exactly the same energy conservation considerations apply, which leads to an identical result.

As I said before, there is no energy transfer to or from the string (assuming the string's mass is insignificant compared to the mass of the ball), which explains why there are no energy terms relating to the string angle or other string parameter in the above equations.

An afterthought:

The vertical speed of the ball attached to the string at height h will, of course, not be the same as the vertical speed of the ball at height h which falls in a straight line vertically. The speed of the ball attached to the string has two components: horizontal and vertical, whereas the ball which falls vertically only has vertical speed.

Clearly, at the lowest point of its swing, the vertical velocity of the ball attached to the string will be zero, which is obviously not at all true for the falling ball.

The speed in the above equations is the total speed, not just the vertical speed. The vertical speed at any particular height for the ball on the string is:

vertical speed = total speed times sin(theta),

where theta is the angle the string makes with the vertical at that particular position.

Hi Dinosaur,
Try a frictionless block sliding down a ramp compared to a block that falls from the same height. This is intuitively the same sort of scenario, but can be done without calculus:
http://pete.wildit.net.au/images/sciforums/SlidingBlock.JPG


Intuition says that final speed of the falling block will be more than the final speed of the sliding block, right?

But when you work it out, you find that the lower force in the direction of motion for the sliding block is compensated by the greater distance and time of the force's action, and the final speed works out to be the same.

James R & Pete: The conservation of energy equations posted by James R were so convincing that I paid attention to Pete's suggestion. The calculations based on a frictionless ramp checked out.

The above suggested that calculus applied to the circular arc would also show that my intuition was wrong.

However, to be consistent with many posters at SciForums, I refuse to admit that I was wrong. I am sure that if I search the Web, I will find some nut who can show that energy is not conserved. Furthermore, there is no such thing as a frictionless ramp, so why should I accept my own calculations based on a mythical artrifact?

Besides, I was lazy and only did the caclulations for a ramp inclined at 30 degrees. Maybe this is a special case and calculations based on other angles will show that energy is not conserved.

You folks ruthlessly demolished my brilliant intuitive analysis.

However, to be consistent with many posters at SciForums, I refuse to admit that I was wrong. I am sure that if I search the Web, I will find some nut who can show that energy is not conserved. Furthermore, there is no such thing as a frictionless ramp, so why should I accept my own calculations based on a mythical artrifact?
:D
No need to search the web, I'm sure you'll find plenty of support here if you look!