Fig. Q3 shows a rigid light beam which is hinged at O to a wall. The beam carries a point mass of m at the free end. The beam is connected at its mid-point to a spring (stiffness k) which is also connected to a support that moves with a vertical motion y = Y sinwt. A damper (coefficient c) connects the mid-point of the beam to a fixed support.

(i) By taking moments about O of the forces acting on the beam show that an equation of motion can be obtained with the form

4mx + cx + kx = kY sin ωt

Please note that in the equation above the first x should have two dots above it and the second x should have one dot above it only. The reason why they haven't is because I am unsure how to type them in, my apologises.

where x is the displacement of the mid-point of the beam.

(ii) Determine the maximum value for w if the amplitude ratio of the x displacement (i.e. X) to Y is not to exceed 10%.

Take m = 2 kg; k = 6.25 kN/m; c = 45 Ns/m.
(The Answer to part (ii) is given as 87.8 rad/s)

Please could you respond to this question in its simplest form, with any additional notes for me to understand the process of the answer. I look forward to your reply. Many Thanks for your time and effort. Mr P Stones.
PS. I have also included an attachment of the question and sketch.

A 4.0 kg toboggan rests on a frictionless icy surface, and a 2.0 kg block rests
on top of the toboggan. The coefficient of static friction µs between the block
and the surface of the toboggan is 0.58, whereas the kinetic friction
16.T op
coefficient is 0.48. The block is pulled by a horizontal force of 30 N as shown.
a. Ca

In an industrial cooling process, water is circulated through a system. If the water is pumped with a speed of 0.45 m/s under a pressure of 400 torr from the first floor through a 6.0-cm diameter pipe, what will be the pressure on the next floor 4.0 m above in a pipe with a diameter of 2.0 cm?

A man with a mass of 65 kg skis down a friction-less hill that is 4.4m high. At the bottom of the hill the terrain levels out. As the man reaches the horizontal section, he grabs a 19-kgbackpack and skis off a 1.0 m-high ledge. At what horizontal distance from the edge of the ledge does the man land?

An open channel having a rectangular cross section is laid so that the rate of grade is 0.00048. The width of the channel is 10.00 ft, and the value of n in Kutter's formula may be taken as 0.011. When the depth of the water in the channel is 4.00 ft, the velocity computed by the Chezy formula is?

A constant torque of 25.0 N M is applied to a grindstone whose moment of inertia is 0.130kg m^2. Using energy principles and neglecting friction, find the angular speed after the grindstone has made 15.0 revolutions.

A 4.0 kg toy race car can move along an x axis. The figure below gives Fx of the force acting on the car, which begins at rest at time t = 0. (The vertical axis is marked in increments of 2 N.)
See the attached document for the necessary graph.
(a) What is the momentum of the car at t = 6.0 s
(b) What is the moment

The photon is normally assumed to have zero rest mass. If the photon had a small mass, this would alter the potential energy which the electron experiences in the electric field of the proton. Instead of
V(r) = -(e^2)/(4(pi)(epsion_0)(r))
we would have
V(r) = -((e^(2))(e^(-r/r_0)))/(4(pi)(epsilon_0)(r))
where r_0 is