Oscillations of a complicated spring block system

[Abhishekdas]

Homework Statement

A 4kg block M in horizontal plane is attached to a sprig S1 fixed to a light rod OA of length 4m as shown in diagram(refer to attachement). The other end O of the rod is hinged to rotate in the plane about the vertical axis passing through it. A spring S2 is fixed to the mid point B of the rod. If the spring constants of both springs are k=10N/m then calculate the angular frequency of oscillations of the mass.

Homework Equations

The Attempt at a Solution

Firstly i am a little confused about the plane of the rod...If the axis is vertical i am assuming it is in the horizontal plane...Anyway i don't think it has any effect on the sum...

Nw coming to my attempt...I didnt go too far...Correct me wherever i am wrong...

Since in these cases we assume the displacement to be very small i am assuming that the diaplacement of the block and the extensions of the springs to be parallel (ie i am neglecting the curve of the arc)...
Let extensions of :
Spring S1 be x1 and that of spring S2 be x2...and let displacement of block from mean position be x...

Then x= (x1)+2*(x2)...Hope i am correct till this...

What to do after this ? because of the rod i am having difficulty in finding the force on the block...

I feel it should be kx1...But then i don't have any other relation to get x1 or x2...

How to go about furthur? Please help...

 

[anathema]

Thank you for your post and for sharing your thoughts and progress so far. Let me try to help you with the problem.

Firstly, you are correct in assuming that the plane of the rod is in the horizontal plane. This means that the motion of the block will also be in the horizontal plane.

Now, let's look at the forces acting on the block. We have the weight of the block acting downwards, and the normal force from the surface acting upwards. Since the block is attached to the spring S1, there will be a force from the spring acting on the block in the direction of the displacement (x). This force can be calculated using Hooke's Law, F = -kx, where k is the spring constant and x is the displacement.

Now, let's consider the forces acting on the rod. At point B, the spring S2 is attached, which will exert a force on the rod in the direction of the displacement (x2). Similarly, at point A, the rod is hinged and can rotate about the vertical axis. This means that there will be a torque acting on the rod due to the weight of the block and the force from the spring S1. This torque can be calculated using T = F*d, where F is the net force acting on the rod and d is the perpendicular distance from the axis of rotation to the point of application of the force.

Using these forces and torques, we can set up the equations of motion for the system and solve for the angular frequency of oscillations.

I hope this helps you in solving the problem. Let me know if you need any further clarification or assistance.