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M3 elastic springs and strings question

A particle of mass 0.3kg is suspended by two identical elastic springs of natural length 1.5m and modulus lamba N. The other ends of the strings are fixed to two points A and B on a horizontal ceiling where AB=3m. P is released from rest at the mid-point of AB and falsl vertically until it is instantaneously at rest at a point 1m below the leve lof the ceiling.

a) calculate the value of lamba

b) calculate the speed of P when it is 0.5m below the level of the ceiling

Reply 1

didgeridoo12uk

what have you tried so far?

Reply 2

sulexk

Original post by didgeridoo12uk

what have you tried so far?

Well I did the first part and I obtained 49N however in the book the answer is given as 48.1N?

Reply 3

didgeridoo12uk

Original post by sulexk

Well I did the first part and I obtained 49N however in the book the answer is given as 48.1N?

cant be bothered to work though it at the moment, but i imagine you did it the right way. just used a different value of g or rounded early or something.

Reply 4

sulexk

This is in the M3 book exercise 2c question 9

I did write mg=2(lambda)x^2 / 2l

and then worked from there.

Reply 5

sulexk

Original post by didgeridoo12uk

cant be bothered to work though it at the moment, but i imagine you did it the right way. just used a different value of g or rounded early or something.

Thank you very much!

I did round early somewhere.

I have just one more question, can you help me with this please, then I will have completed the exercise.

Here I go:

A light elastic string has natural length L and modulus (lamba). One end is fixed to a point A on a ceiling and a particle P of mass m is attached to the other end. P is held vertically below A so that AP=2L and then released. P has speed v when the extension of the string is x. Show that, while the string remains taut,

1/2mv^2 = L/2 ( (lamba) - 2mg ) + mgx - (lambda)x^2 / 2L

I have completed that part

I am stuck with this part:

By considering the speed of P when x=0 show that the string will never become slack provided 2mg > (lamba)

Thank you :smile:

Reply 6

didgeridoo12uk

Original post by sulexk

1/2mv^2 = L/2 ( (lamba) - 2mg ) + mgx - (lambda)x^2 / 2L

I have completed that part

I am stuck with this part:

By considering the speed of P when x=0 show that the string will never become slack provided 2mg > (lamba)

Thank you :smile:

put x = 0 into that equation and see what happens..

Reply 7

sulexk

Original post by didgeridoo12uk

put x = 0 into that equation and see what happens..

I am still confused.

I have put x=0 and obtained the equation mv^2=L((lambda) - 2mg)

Reply 8

didgeridoo12uk

Original post by sulexk

I am still confused.

I have put x=0 and obtained the equation mv^2=L((lambda) - 2mg)

re-arrange for v (This equation is correct while the string is taught)

look what happens if 2mg < lambda...

i assume this is the root you're meant to be going down. without thinking about it more deeply i'm not sure whether i'm convinced its actually a proof or not

Reply 9

sulexk

Original post by didgeridoo12uk

well if 2mg>lambda then mv^2 would be negative. But v^2 cannot be negative. so then must 2mg be less than lambda?

Reply 10

didgeridoo12uk

Original post by sulexk

well if 2mg>lambda then mv^2 would be negative. But v^2 cannot be negative. so then must 2mg be less than lambda?

exactly

Reply 11

sulexk

Original post by didgeridoo12uk

exactly

However does what I stated before answer the question?

since it says " show that the string will never become slack provided 2mg>(lambda)"

Where is the connection?

Thank you

Reply 12

didgeridoo12uk

Original post by sulexk

However does what I stated before answer the question?

since it says " show that the string will never become slack provided 2mg>(lambda)"

Where is the connection?

Thank you

it shows that at x=0 (where the rope is on the verge of becoming slack) provided that 2mg>lambda, then given that the rope isn't slack then the velocity equation holds...

i guess if lambda = 2mg then the v would be zero and so you could argue that the rope never becomes slack, it just gets to the point then reverses.

basically i'm not really sure if it you can prove it or if the question is correct :/