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Quick question regarding work:

I should really derive it but its late and I'm lazy:

Use normal modelling assumptions, no friction, no dissipative forces.

If a ball is rolling down a slope, if its VERTICAL descent is x metres, will it be travelling at the same speed as it would be if it fell vertically x metres, with no slope?

Reply 1

Scipio90

If it's rolling, no, but whether they want that will depend on what level you're studying it at.

Reply 2

QuantumTheory

Won't, by Pythag, the length of the slope always be longer than the direct drop; so by SUVAT, the time will increase?

Reply 3

Moronic Acid

In terms of speed - the same PE is being converted to KE therefore the KE values for both cases is the same and since mass is constant, the velocity should be the same in both cases (i.e. terminal velocity is the same - although acceleration will be greater when it is simply dropped straight down)

Reply 4

teachercol

Some of the grav potential energy is converted into rotational ke so there will be less translational ke.

Reply 5

Moronic Acid

teachercol

Some of the grav potential energy is converted into rotational ke so there will be less translational ke.

Whilst i know not of these translational and rotational KE's, im pretty sure a sixth former wouldnt need to go into this much depth

Reply 6

Westernnimbus

teachercol

Some of the grav potential energy is converted into rotational ke so there will be less translational ke.

Standard Simple Mechanic Modelling... that doesnt include the behaviour of atoms :p:

Anyway I think I've confused people with the question, what I meant was:
(Again, no dissipative forces such as friction, air resistance... and rotational ke)

Imagine a ball at the top of a building, you drop it, and record its final speed before it hits the floor.

Now You make an slope, the exact same height (gradient regardless), when the ball reaches the bottom of the slope, will it be travelling at the same speed as the ball dropped vertically.

I have kinda derived it (for a certain situation with fixed variables, not a full mathematical proof), and found yes they do travel at the same speed.

It sounds kind of counter intuitive, but heres my reasoning:

With no dissipative forces, all the g. potential energy is converted into kinetic energy, just like when a ball is dropped vertically.

Reply 7

QuantumTheory

Then surely you don't mean vertical descent. I was thinking you meant the height of the ramp. :tongue:

I'm pretty sure you're right, if you exclude friction. And the tiny changes in the gravitational constant.
But most mech. calculations treat gravity as constant anyway.

Reply 8

Scipio90

joelio36

Standard Simple Mechanic Modelling... that doesnt include the behaviour of atoms :p:

wtf?

Essentially, is the ball rolling or sliding?

Reply 9

Westernnimbus

Scipio90

wtf?

Essentially, is the ball rolling or sliding?

Sorry I mean a point mass, I used a ball just because it seemed a better analogy... but it wasnt lol

Reply 10

Scipio90

Then yes, the speeds are the same.

Reply 11

Westernnimbus

Scipio90

Then yes, the speeds are the same.

Just to be a bit pedantic, if the ball is rolling or not is not a factor because the modelling assumptions states no friction

Reply 12

teachercol

OP says its rolling though.If you had said sliding then my answer would have been different!

Reply 13

Moronic Acid

joelio36

With no dissipative forces, all the g. potential energy is converted into kinetic energy, just like when a ball is dropped vertically.

Thats what i said :biggrin:

lol

Another thing to consider is that when the particle is dropped straight down it accelerates downwards with the full accelerating force of mg, whereas when it rolls down the slope, it only gets the component of mg down the slope (mgsintheta where theta is the angle between the slope and the horizontal) therefore the accelerating force is smaller so it accelerates at a slower rate down the slope - therefore it takes longer to travel the same vertical distance going down the slope than simply being dropped (which is what you'd expect anyway)

Im pretty sure nobody asked this but i thought id throw it in for good measure :p:

Reply 14

teachercol

If you draw a circle.

Then draw lines from a point at the top of the circle to any other point on the circle , it will take the same time for an object to slide along the line to that point ....

Reply 15

Moronic Acid

teachercol

If you draw a circle.

Then draw lines from a point at the top of the circle to any other point on the circle , it will take the same time for an object to slide along the line to that point ....

. . . because as the points move further down, the angle theta tends to 90 degrees, therefore the force of acceleration (mgsintheta) increases (and tends to its maximum = mg), and at the same time the distance travelled increases - these 2 factors cancel out invariably, meaning that the time taken to travel from the top point to any other point is constant

Assuming what you said was true, this would be my reasoning :smile: