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Rod and Pin Apparatus: Friction, Acceleration, and Velocity

Rigid rod OA is 2 ft long and rotated around fixed pt O. This rod is pinned at A to rigid connecting rod AB, 6 ft long, which in turn is pinned at B to piston C. This piston moves without friction along the y axis. The pins are also frictionless.

Rod OA is positioned so that it is pi/3 radians above the x-axis and angular velocity = 2 rad/sec counterclockwise and its angular acceleration is 0. What are the velocity and acceleration of the piston in this case?
and...
Weights of the three components are as follows: W(OA)=64.4 lb, W(AB) = 16.1 lb, W(C) = 5 lb. A constant counterclockwise torque is applied to rod OA at O. The system starts from rest with rod OA horizontal to the right. What torque is needed to give rod OA an angular velocity of 5 rad/sec when rod OA is vertical, along the y axis?

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SOLUTION.

Here is a diagram of the rods and the piston.

a) So, lets denote the angle between OA and x-axis as .
Then, for the vertical position of the piston, Y, we can write down:
Y=OA*sin  + AB*cos(OBA)
We know that  = /3 + *t, where  is rod OA's angular
Velosity and t is time.
Now we need to find OBA. For that, we can write down that:
OA*cos  = AB*sin(OBA)
sin(OBA) = OA/AB* cos 
cos(OBA) = (1 - { OA/AB* cos }2)1/2
So we can write down that:
Y = OA*sin  + AB*(1 - { OA/AB* cos }2)1/2

Y = OA*sin  + (AB 2 - {OA* cos }2)1/2
Now, in order to find piston's velocity and acceleration, we need to find first and second derivatives of Y:
V = dY/dt = OA * cos  * d/dt +
+ ((-OA2)*2*cos  * (-sin ) * d/dt) / (AB 2 - {OA* cos }2)1/2 .
Now we remember that d/dt =  and we ...

Solution Summary

A Rod/pin apparatus is discussed. The friction, acceleration and velocity of an apparatus is given.

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