Thank you, that makes sense.
For your 2nd equation, what methods of integration did you use to get the 3rd? I can just about argue the first term because at t = 0, x is variable x to be incremented along until you get to t = t at which x becomes 3a. Not sure how you integrated the rest of the...
It works! And it agrees with the answer I got when I used v = Aw! I can rest now.
But first, time for some plenary learning. Your first force balance equation, what direction are you taking it from? From the ceiling or from point A which is 3a below the ceiling? Because I don't understand...
This is UK A-level Maths (mechanics). We don't deal with spring constants right now as our form of hooks law is actually ## T = \frac{\lambda x}{l_0} ## where ##\lambda## is the modulus of elasticity.
But if that really is the amplitude of the motion between points A and B, then maximum speed...
Why doesn't it make sense? I wrote elastic energy = gravitational potential energy + kinetic energy. I can't see how its not dimensionally correct.
And I know the amplitude of the oscillations. I worked out the equilibrium point and confirmed that the object should perform SHM between point...
## \frac {4/3 m g (2a)^2} {2a} = 5/4mga + 1/2mv^2 ##
Elastic energy in spring at A = gain in potential to equilibrium point (3a to 7/4a) + kinetic energy built up in the mass during that motion.
I get an answer of ## \sqrt{17/6ga} ## as the maximum speed. I don't think that's correct.
From...
a) λ = 4/3 by considering the energy balance as P moves from A (2a) to B (1/2a). The E.E. at A changes into a gain in gravitational potential energy + build up of E.E. at B since the spring compresses.
b) a = 5/3g by considering that the mass P is in dynamic equilibrium immediately after...