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A mechanics question for Trinity College Interview Test

consider a mass mm at position x(t)x(t) on a rough horizontal table attached to the origin by a spring with constant kk (restoring force kx-kx) and with a dry friction force ff {f=F,if x˙<0FfF,if x˙=0f=F,if x˙>0\begin{cases}f=F, & \text{if $\dot x \lt 0$}\\-F \le f \le F, & \text{if $\dot x =0$}\\f=-F,& \text{if $\dot x \gt 0$}\end{cases}

a). What is the range of xx where the mass can rest?

b) Show that if the mass moves, the maximum excursion decreases by 2Fk\frac{2F}{k} per half cycle.

c) Discuss the motion

Could someone please give a hint as how to show that the amplitude of oscillation decrease per half cycle?
(edited 8 years ago)
Reply 1
Easiest way is probably energy. Let me know if you need a further prod.

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Reply 2
Original post by Krollo
Easiest way is probably energy. Let me know if you need a further prod.

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Thanks! I've solved it both using energy methods and differential equations. It is too slow to type it here as the differential methods cost a lot of space.

For those interested, see http://physics.stackexchange.com/questions/196537/showing-that-a-mass-moves-a-half-cycle
Reply 3
Original post by 100708244
Thanks! I've solved it both using energy methods and differential equations. It is too slow to type it here as the differential methods cost a lot of space.

For those interested, see http://physics.stackexchange.com/questions/196537/showing-that-a-mass-moves-a-half-cycle


Ah, excellent. Are you planning to try for Trinity? :ahee:

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Reply 4
Original post by Krollo
Ah, excellent. Are you planning to try for Trinity? :ahee:

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No, my ability is not comparable to BMO participants.:frown: But I will try for Cambridge.
Reply 5
Original post by 100708244
No, my ability is not comparable to BMO participants.:frown: But I will try for Cambridge.


Did someone say BMO?

Reply 6
Original post by 100708244
No, my ability is not comparable to BMO participants.:frown: But I will try for Cambridge.


Great to hear it. Will you be applying for 2016?

Don't worry about BMO, it's a very particular style of maths, you won't be disadvantaged one bit.


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