Two blocks connected by springs moving in a circle

[ConorDMK]

Homework Statement

Block A and block B are on a frictionless table as shown in Figure 2. Spring 1 connects block A to a frictionless peg at 0 and spring 2 connects block A and block B. Their masses are respectively m_A=0:45 kg and m_B=0:32 kg. When the blocks are in uniform circular motion about 0, the springs have lengths of l_A=0.80 m and l_B=0.50 m. The springs are ideal and massless, and the linear speed of block B is 2.8 m/s. If the distance that spring 2 stretches is 0.070 m, determine the spring constant of spring 2.

Homework Equations

F=ma
a=v²/r
F=kx

The Attempt at a Solution

y-direction is perpendicular to surface and x-direction is parallel to surface
r_A=l_A, r_B=l_A+l_B
T₁=Tension is spring 1, T₂=Tension in spring 2
N_A=Normal acting on block A, N_B=Normal acting on block B
x_B=0.07m (The extension of spring 2)

For block A (From FBD)
Resolving (y-direction): F_Ay=m_Aa_Ay=N_A-m_Ag=0

Resolving (x-direction): F_Ax=m_Aa_Ax=T₁-T₂

⇒(m_Av_A²)/r_A=T₁-T₂For block B (From FBD)
Resolving (y-direction): F_By=m_Ba_By=N_B-m_Bg=0

Resolving (x-direction): F_Bx=m_Ba_Bx=T₂

⇒(m_Bv_B²)/r_B=T₂And this is where I have become stuck, because I know I can't simply say T₂=kx_B as this would give the spring constant for a single spring that has extended to a total length of 1.3m. And continuations from this point do not seem to go anywhere e.g. substituting centripetal force of block B into the equation for block A, does not help as far as I can see.

 

[haruspex]

I know I can't simply say T₂=kx_B as this would give the spring constant for a single spring that has extended to a total length of 1.3m.

why would it do that? T₂ is the tension, x_B is the extension.

 

[ConorDMK]

why would it do that? T₂ is the tension, x_B is the extension.

I thought of it originally, but then I thought it couldn't be because it doesn't require me to use anything from block A, and the block B would be under the same tension at that distance with a single spring, at least I think so. But surely it can't be as simple as to just equate (m_Bv_B²)/r_B with kx_B?

 

[ConorDMK]

I think I managed to convince myself that (m_Bv_B²)/r_B=kx_B was wrong because it seemed too easy and didn't involve values from spring 1 or block A.

 

[haruspex]

I think I managed to convince myself that (m_Bv_B²)/r_B=kx_B was wrong because it seemed too easy and didn't involve values from spring 1 or block A.

I don't see how string 2 could tell the difference if string 1 and block A were replaced by an inextensible massless string of the same (current) length.

 

[ConorDMK]

I don't see how string 2 could tell the difference if string 1 and block A were replaced by an inextensible massless string of the same (current) length.

Thank you for clearing that up for me in my mind. I've been doing this question for the past three days and I kept thinking to myself that the answer couldn't be as simple as I thought it was.