- #1

Zipi Damn

- 11

- 0

## Homework Statement

A particle of mass M is on the top of a vertical circle without initial velocity. It starts to fall clockwise.

Find the angle with respect to the origin, where the particle leaves the circle.

## Homework Equations

v=ωXr

## The Attempt at a Solution

I used two unitary vectors..

**εr**: which direction is the same as the line described by the radio in the separation point.

**εθ**: perpendicular to εr (the direction of the tangential velocity in the separation point)

**r**=R

**er**

**v**=d

**r**/dt = Rd

**er**/dt

**v=ωXr**(counterclockwise, so I changedthe sign to negative)

**v**=-R(

**ωXer**)

**v**= -R(-ω

**eθ**) = ωR

**eθ**

α=dω/dt

**a**= d

**v**/dt = R(α

**eθ**+ ω d

**eθ**/dt)

d

**eθ**/dt = -(ωX

**eθ**) = -ω

**er**

a = α R

**eθ**- ω^2 R

**er**

Forces:

Weight, Normal force

Centripetal is a resultant?

F(eθ)==> mgcosθ = αRm

F(er)==> N-mgsenθ= -ω^2Rm

So:

α=dω/dt= (dω/dθ)(dθ/dt)=ωdω/dθ

θ

∫αdθ =

∏/2

ω

∫ωdω

ω0

From F(er) : α= (gcosθ)/R

θ

∫ (gcosθ)/Rdθ = (gsenθ)/R

∏/2

ω

∫ωdω= (ω^2)/2

ω0ω^2 = (2gsenθ)/R

I come back to the force equations, (with zero normal force) and replace ω^2

And I get nonsenses like 2=1. Did I do something wrong?

Solution: 48,19º (by the teacher)