Motion in Compound Constraint

## Question:

Refer to the diagram below. The rod OA makes angle q with the reference line and rotates counterclockwise at a constant angular rate, dq/dt, of 4 radians per second. The collar assembly consists of two collars pinned together so they are free to rotate relative to one another, with one collar sliding over the the rotating rod and the other sliding over the circular rod described by the equation r=1.6*cos(q)m. Each collar in the assembly has mass = 0.5kg. The motion is in the horizontal plane. Determine the force exerted on the lower collar by the circular rod and on the upper collar by the straight rod at the instant that q=45deg.

I assume here that we are neglecting friction. The collar assembly is constrained by the circular rod to travel in a circle. The force of the straight rod on the collar assembly is directed 135 degrees from the reference line when the straight rod is at an angle of 45 degrees. This force may be resolved into a component radial to the circular rod and a component tangent to the circular rod. The radial component is cancelled by the reaction of the circular rod. The tangent component goes into accelerating the collar assembly around the circular path.

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*cos2(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 v2/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=3.22/0.8=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.

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