Please show work and thank you in advance A continuous cord

A continuous cord of total length 4 m is w around the small A, B, C, and D If the stiffness of each rapped frictionless pulleys at of each spring is k is the mass block. Neglect the weight streteched m, determine m. of the pulleys and cords. 0.3 d The springs are unstretched when k a 500 Nr k a 500 Nh

Solution

Let us assume that the tension on the string attached to the masses is T = Mg, as they are at rest.

Also, the tension on the string looping through the pulleys is T1 Newtons. The length of the loop is 4m, hence each of the sides of the four sided quadrilateral it makes would be 1 metre.

Plus, d = 2 metres when the springs are unstretched, that would mean that at the time of equilibrium when the springs are stretched by 0.3 metres, the length d = 2 - 0.6 = 1.4 metres

We will balance the forces for the spring to determine the tension T1 and then balance the forces for the masses to find the mass.

That is, 2 x T1 x d/2 (AB) = Kx [The vertical component for tension T1 would pull the bottom spring up. Similar equation can be written for the top spring]

or T1 = 500 (0.3) / 1.4 x 1 = 107.14286 Newtons.

Now, balancing the force for the masses, we get: 2 T1 Cos(Angle made by CD with horizontal) = Mg

or, 2 x 107.14286 x sqrt(1 - 0.49) = Mg

or M = 15.599 Kgs

Therefore the required mass m is 15.599 Kg