some help please (1 Viewer)

[polythenepam]

some help pls with ii)
i dont have the answer so an explanation would be good

thanks guys

A conical pendulum consists of a mass of mkg hanging on the end of a light 2
metre string from a hook on a ceiling. The mass is set rotating in a horizontal circle,
and moves with a period of P seconds, with the string making a constant angle θ to
the vertical.


Draw a diagram of the situation, showing the forces acting on the mass.
(ii) By resolving forces vertically and radially, express the period P as a function of
the angle θ between the string and the vertical (leave your answer in terms of g).
(iii) The string can just support a stationary mass of 10mkg hanging vertically on
it, but will break under any further weight. Find the smallest period that the conical
pendulum can have (leave your answer in terms of g).

 

[KFunk]

(iii) The string can just support a stationary mass of 10mkg hanging vertically on
it, but will break under any further weight. Find the smallest period that the conical
pendulum can have (leave your answer in terms of g).

Well, when the pendulum hangs vertically the tension in the string is T=10g (F =m.a) hence 10g is the maximum value of T, i.e. T ≤ 10g .... (1)

When you were resolving forces you should have gotten the result:

∑ horizontal forces = mw²r

Tsinθ = mw²r ... where T is the tension in the string giving us,

T = mw²r/sinθ ...(2)


Subbing (2) into (1) we get

mw²r/sinθ ≤ 10g .... here you want to use the relationship w = 2π/P , P = period

4π²mr/P²sinθ ≤ 10g

P² ≥ 4π²mr/(10gsinθ )

hence the mimimum value of P is 2π√(mr/10gsinθ ) *

also sinθ = r/2 ... so mimimum P = 2π√(m/5g) since P ≥ 2π√(m/5g)

(*P is a positive value so you ignore the negative root)

 

[polythenepam]

ohhhkay i see now... thanks heaps!

 

[KFunk]

Oh, and just something I missed. The question says the maximum weight is 10m (not just 10)... in which case T ≤ 10mg

giving mw²r/sinθ ≤ 10mg ---> w²r/sinθ ≤ 10g

and eventually P ≥ 2π/√(5g)