A small lump of ice sliding down a large, smooth sphere.

[MrDaahl]

Homework Statement

A small lump of ice is sliding down a large, smooth sphere with a radius R. The lump is initially at rest. To get it started, it starts from a position slightly right to the sphere's top, but you can count it to start from the top. The lump is fallowing the sphere for a certain amount of time as shown in the figure. The sphere does not move.

http://i.tinyuploads.com/r6tkWK.png

2. Questions
a) Write an expression for the ice lump velocity when it is has slided down the angle.

b) Draw a force diagram on the block when it has slided down to the angle.

c) Determine the normal force size when the ice lump has completed the angle.

When the angle is large enough, the ice block no longer follow the sphere's surface, but fall freely.

d) Decide/Calculate this angle.

The Attempt at a Solution

So I am pretty sure i got the solution for queation a) :
When you guys said "Use conservation energy" i thought about using the radius to calculate the height of the circle. I puzzled abit with some formulas and got this equation:

mgh = 3/4mv^2

In question b) i made a FBD diagram to show where the forces are pointing. It looks like this:
http://i.tinyuploads.com/HSSjC4.png

The c) solution must be like this:
F_n = -F_T

 

[Pythagorean]

I wonder if conservation of energy wouldn't help...

 

[Tanya Sharma]

For part a) use conservation of energy

For part b) , make an FBD of the block .Mark all the forces acting on the block.Resolve the forces tangentially and radially.

 

[MrDaahl]

Ok, so I've to do something with R*2, which will equal the height of the circle

So i got this equation so far:

V₁ = √(2*g*h+v₀²)

Can anyone confirm the equation?

 

[Tanya Sharma]

Do you know how to apply conservation of energy ?

 

[MrDaahl]

Apparently not.. I assume

I thought it should be the block's potentiel + kinetic energy at the initialising state, equal the kinetic energy at the secondary state.

 

[Pythagorean]

What's the initial kinetic energy though?

 

[MrDaahl]

Isnt it zero?

 

[Tanya Sharma]

Apparently not.. I assume

I thought it should be the block's potentiel + kinetic energy at the initialising state, equal the kinetic energy at the secondary state.

KE₁+PE₁=KE₂+PE₂

What is PE₁ ? Assume h=0(reference of potential) to be at the center of the sphere .In other words measure heights from the center of sphere ,not the ground.

 

[Pythagorean]

What's the block's initial potential energy?

 

[Pythagorean]

I think OP was assuming that h=0 was at final position (so that potential energy could be zero).

 

[MrDaahl]

KE₁+PE₁=KE₂+PE₂

What is PE₁ ? Assume h=0(reference of potential) to be at the center of the sphere .In other words measure heights from the center of sphere ,not the ground.

That was wrong.

PE₁ = mgR

 

[Tanya Sharma]

PE₁ = 0.

Do you think the block is at zero height with respect to the center of the sphere ?

 

[Pythagorean]

If h=0 is the center of the circle, the block is never there, so h is never 0. so PE can't be 0, either. You can choose h=0 to be where PE1 is measured or where PE2 measured if you like, just make sure you do the rest of the problem keeping that coordinate system.

 

[MrDaahl]

KE₁+PE₁=KE₂+PE₂

KE₁ = 0

PE₁ = mgR

KE₂ = ½mv²

PE₂ = mgΔR.

→ mgR=½mv²+mgΔR

→ this doesn't make sense

 

[Pythagorean]

if h is initially R, it is not finally 0... The vertical distance traveled by the block (h2-h1) is not equal to the radius of the circle. And you already told us that KE1 = 0 (as it should).

 

[MrDaahl]

I know the R is not the same value in the initialising state as it is in the next.
But i simly cannot see how to calculate PE_2

 

[Pythagorean]

You will have to use some trigonometry to see the vertical distance traveled. Your change in potential energy is going to be mg \Delta h, essentially.

 

[haruspex]

I know the R is not the same value in the initialising state as it is in the next.
But i simly cannot see how to calculate PE_2

R is the radius of the sphere, that does not change.
Call the centre of the sphere O, the initial position of the ice A, and the later position B.
Draw a line horizontally from B to meet the line AO at C. You know the angle COB and the length OB. So what is length OC? Length AC?

 

[MrDaahl]

wait

 

[MrDaahl]

The length OC must be 3/4 R and the length of AC must therefore be 1/4 R

 

[haruspex]

The length OC must be 3/4 R and the length of AC must therefore be 1/4 R

No, it must depend on theta. How did you get 3R/4?