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Simple Harmonic Motion
a) Magnitude of force in a spring = kx, where k is spring constant and x is displacement
for case A
net force = kA
for case B
net force = kA + kA = 2kA
keff = 2k
for case C
the two springs are in series so keff = (1/k + 1/k)-1 = k/2
therefore net force = kA/2
for case D
the two springs are parallel so keff = k +k = 2k
Net force = 2kA
more force that means more acceleration which means lesser time to travel.
Which means force and time are inversely related
Work done by each springs in each case
for case A
for case B
for case C
this means potential energy of the two springs in case C is less than potential energy of the spring in Case A
for case D
effective spring constant is already found above for each case in Ans 1 a)
So, rank B=D>A>C
Time period is inversely proportional to spring constant. So, more k means less T
So, rank C>A>D=B
Ranking in terms of potential energy
we see the work done by each springs in the system
so potential energy = work done =1/2 kx2
For case A : potential energy of the spring = 1/2 kA2
For case B : potential energy = 1/2 *2k* A2 =-kA2
For case C : potential energy = 1/2 *k/2 * A2 = -1/4 kA2
For case D : potential energy = 1/2 * 2k* A2 =-kA2
if any doubt feel free to comment
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