Motion in Compound Constraint

The tangential acceleration is dv/dt where v is the tangential
velocity. The angle q at any time t is 4*t. The velocity of a
fixed point on the straight rod is just r*dq/dt. Only the
component of that velocity projected on the tangent to the circle
at the location of the collar assembly is the tangential
velocity. That reasoning leaves us with tangential
velocity:

v=cos(q)*r*dq/dt=cos(4*t)*1.6*cos(4*t)*4=6.4*cos^{2}(4*t).

Taking the derivative with respect to t we get

dv/dt=6.4*2*cos(4*t)*-sin(4*t)*4

At 45 degrees of the straight rod pi/4 radians=4r/s*t seconds
or t=pi/16seconds. Putting that value in our expression for dv/dt
we get

dv/dt=6.4*2*cos(pi/4)*-sin(pi/4)*4=6.4*4=25.6m/s/s

To accelerate the collar assembly 25.6m/s/s tangent to the circular path requires 25.6N since the total assembly has a mass of 1kg.

We know at 45 degrees the two components of the force of the straight rod on the collar assembly are equal in magnitude so the total force of the straight rod on the collar assembly is the vector sum of 25.6N tangent to the circular path and 25.6N along the outward radial. This is 25.6*1.414=36.2N at an angle of 135 degrees.

The force of the circular rod on the collar has three
components. Since the apparatus is horizontal, the force of
gravity acts downward,into the diagram, on the collar assembly so
the reaction force of the circular rod on the collar assembly is
an upward (out of the diagram) force of
m*g=1*9.8=9.8N. This assumes that the entire collar
assembly is supported by the circular rod and the straight rod
has negligible mass. Another component is a radial force holding
the collar assembly in a circular path. The centripetal
acceleration is v^{2}/r where v is the tangential
velocity of the collar assembly around the circle and r is the
radius of the circular path. The third component is the radially
inward reaction to any radially outward force supplied by the
straight rod.

The magnitude of the instantaneous velocity of the point on the straight rod at the center of the collar assembly is the radius at that instant times the fixed angular velocity of the straight rod. At a straight rod angle of 45 degrees that is 1.6*cos(45)*4=4.52m/s. The direction of motion of this point is 90 degrees from the straight rod or 135 degrees from the reference line. Part of this velocity results in the straight rod slipping through the collar assembly. Part of it causes the collar assembly to slide along the circular rod.

The component of the velocity of the fixed point on the
straight rod which contributes to collar moving around the circle
is the projection of the fixed point velocity on the tangent to
the circle at the collar assembly. The tangent to the circle at
the collar assembly when straight rod angle is 45 degrees is
parallel to the reference line so the angle between the fixed
point velocity and the tangent to the circle is 45 degrees. This
means that 0.707 times the velocity of the fixed point is the
tangential velocity of the collar assembly.

v=4.52*0.707=3.2m/s

The centripetal acceleration then is

ar=3.2^{2}/0.8=12.8m/s/s.

The total force on the collar assembly is the vector sum of 9.8N upward and 12.8N inward along the radial due to centripetal acceleration reaction and 25.6N inward along the radial due to reaction against the outward force from the straight rod.

I will leave the vector arithmetic up to you.

This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can be of more help, please let me know. JDJ