Question:
The leg and cast in the figure below weigh 280N (w1). Determine the weight w2 and the angle 𝛼 needed so that no force is exerted on the hip joint by the leg plus cast.
Solution:
Given:
T3=w1=220NT1=110Nθ1=40°
θ2=?T2=w2=?
The representation of the image is;
Newton’s 3rd Law of Motion has already been used to isolate the joint from its physical connections by way of action-reaction pair.
Newton’s 2nd Law of Motion Use components. The x axis is the horizontal and the y axis the vertical directions.
Because there is no horizontal displacement
ΣFx=0–T1x+T2x=−T1cosθ1+T2cosθ2=0
Because Upward force is equal to downward force
ΣFy=0T1y+T2y–W1=0T1sinθ1+T2sinθ2–W1=0
T2 and θ2 are both unknown. They can be eliminated. These two variables occur together as T2sinθ2 and T2cosθ2. This suggests solving for the angle θ2 by solving for;
tanθ2=T2cosθ2T2sinθ2=T1cosθ1W1−T1sinθ1
Then the angle follows from;
θ2=α=tan−1T1cosθ1W1−T1sinθ1
=tan−1T1cos40°220−110sin40°
=tan−184.3149.3=60.6°
T2=w2=T1cosθ2cosθ1=110cos60.6°cos40°
=172N
Comments
Thank you, the working was simplified well